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1ADVANCESINTHEHOMOTOPYANALYSISMETHOD
2May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank
3ADVANCESINTHEHOMOTOPYANALYSISMETHODEditorShijunLiaoShanghaiJiaoTongUniversity,ChinaWorldScientificNEWJERSEY•LONDON•SINGAPORE•BEIJING•SHANGHAI•HONGKONG•TAIPEI•CHENNAI
4PublishedbyWorldScientificPublishingCo.Pte.Ltd.5TohTuckLink,Singapore596224USAoffice:27WarrenStreet,Suite401-402,Hackensack,NJ07601UKoffice:57SheltonStreet,CoventGarden,LondonWC2H9HELibraryofCongressCataloging-in-PublicationDataAdvancesinthehomotopyanalysismethod/editedbyShijunLiao,professor,deputydirectoroftheStateKeyLabofOceanEngineering,ShanghaiJiaoTongUniversity,China.pagescmIncludesbibliographicalreferences.ISBN978-9814551243(hardcover:alk.paper)1.Homotopytheory.I.Liao,Shijun,1963–editorofcompilation.QA612.7.A3752014514'.24--dc232013028624BritishLibraryCataloguing-in-PublicationDataAcataloguerecordforthisbookisavailablefromtheBritishLibrary.Copyright©2014byWorldScientificPublishingCo.Pte.Ltd.Allrightsreserved.Thisbook,orpartsthereof,maynotbereproducedinanyformorbyanymeans,electronicormechanical,includingphotocopying,recordingoranyinformationstorageandretrievalsystemnowknownortobeinvented,withoutwrittenpermissionfromthePublisher.Forphotocopyingofmaterialinthisvolume,pleasepayacopyingfeethroughtheCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,USA.Inthiscasepermissiontophotocopyisnotrequiredfromthepublisher.In-houseEditor:AngelineFongPrintedinSingapore
5October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.0PrefaceThehomotopyisafundamentalconceptintopology,whichcanbetracedbacktoJulesHenriPoincar´e(1854–1912),aFrenchmathematician.Basedonthehomotopy,twomethodshavebeendeveloped.Oneisthehomotopycontinuationmethoddatingbackto1930s,whichisaglobalconvergentnumericalmethodmainlyfornonlinearalgebraicequations.Theotheristhehomotopyanalysismethod(HAM)proposedin1990sbyShijunLiao,theeditorofthisbook,whichisananalyticapproximationmethodwithguaranteeofconvergence,mainlyfornonlineardifferentialequations.Differentfromperturbationtechniqueswhicharestronglydependentuponsmall/largephysicalparameters(i.e.perturbationquantities),theHAMhasnothingtodowithanysmall/largephysicalparametersatall.Besides,manyanalyticapproximationmethods,suchas“Lyapunovartifi-cialsmallparametermethod”,“Adomiandecompositionmethod”andsoon,areonlyspecialcasesoftheHAM.Unlikeotheranalyticapproximationtechniques,theHAMprovidesusgreatfreedomandflexibilitytochooseequation-typeandsolutionexpressionofhigh-orderapproximationequa-tions.Noticethat“theessenceofmathematicsliesentirelyinitsfreedom”,aspointedoutbyGeorgCantor(1845–1918).Mostimportantly,differentfromallofotheranalyticapproximationmethods,theHAMprovidesusaconvenientwaytoguaranteetheconvergenceofapproximationseriesbymeansofintroducingtheso-called“convergence-controlparameter”.Infact,itistheconvergence-controlparameterthatdifferstheHAMfromallotheranalyticapproximationmethods.Asaresult,theHAMisgenerallyvalidforvarioustypesofequationswithhighnonlinearity,especiallyforthosewithoutsmall/largephysicalparameters.Since1992whentheearlyHAMwasfirstproposedbyLiao,theHAMhasbeendevelopinggreatlyintheoryandappliedsuccessfullytonumer-oustypesofnonlinearequationsinlotsofdifferentfieldsbyscientists,researchers,engineersandgraduatedstudentsindozensofcountries.Alloftheseverifytheoriginality,novelty,validityandgeneralityoftheHAM.v
6October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.0viPrefaceSo,itisnecessarytodescribe,althoughbriefly,thecurrentadvancesoftheHAMinboththeoryandapplications.Thisisthefirstmotivationofthebook,whosechaptersarecontributedbytheleadingresearchersintheHAMcomingfromsevencountries.Anytrulynewmethodshouldgivesomethingnoveland/orbetter.Inthepast20years,hundredsofarticlesrelatedtotheHAMwerepublishedinvariousfields,andsomenewsolutionswereindeedfoundbymeansoftheHAM.Thus,itisnowthetimetosuggestsomevaluablebutchallengingnonlinearproblemstotheHAMcommunity.Thisisthesecondmotivationofthebook.Someoftheseproblemsareveryfamous,withalonghistory.Hopefully,theabove-mentionedfreedomandflexibilityoftheHAMmightcreatesomenovelideasandinspirebrave,enterprising,youngresearcherswithstimulatedimaginationtoattackthemwithsatisfactoryresults.IpersonallybelievethattheapplicationsoftheHAMonthesefamous,chal-lengingproblemsmightnotonlyindicatethegreatpotentialoftheHAM,butalsoleadtogreatmodificationsoftheHAMintheory.AbriefreviewoftheHAMisgiveninChapter1,withsomesuggestedchallengingproblems.Thefascinating“PredictorHAM”and“SpectralHAM”aredescribedinChapters2and3,respectively.Someinterestingtheoreticalworksontheauxiliarylinearoperator,convergence-controlpa-rameterandconvergenceofapproximationseriesaredescribedinChapters4and5.AnattractiveapplicationoftheHAMaboutflowsofnanofluidisgiveninChapter6.AcharmingapplicationoftheHAMfortime-fractionalboundary-valueproblemisillustratedinChapter7.TheHAM-basedMaplepackageNOPH1.0.2(http://numericaltank.sjtu.edu.cn/NOPH.htm)forperiodicoscillationsandlimitcyclesofnonlineardynamicsystemswithvariousapplicationsisdescribedinChapter8.TheHAM-basedMathemat-icapackageBVPh2.0(http://numericaltank.sjtu.edu.cn/BVPh.htm)forcouplednonlinearordinarydifferentialequationsanditsapplicationsaregiveninChapter9.Bothofthemareeasy-to-use,user-friendly,andfreeavailableonlinewithuser’sguide.TheycangreatlysimplifysomeapplicationsoftheHAM.Itisagreatpitythatitisimpossibletodescribe,evenbriefly,thewholeadvancesoftheHAMintheoryandapplicationsinsuchabook.Here,IwouldliketoexpressmysincereandtruthfulacknowledgementstoalloftheHAMcommunityfortheirgreatcontributionstotheHAM.ShijunLiaoJune2013,Shanghai
7October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.0ContentsPrefacev1.ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod1S.-J.Liao2.PredictorHomotopyAnalysisMethod(PHAM)35S.AbbasbandyandE.Shivanian3.SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems85S.MotsaandP.Sibanda4.StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter123R.A.VanGorder5.AConvergenceConditionoftheHomotopyAnalysisMethod181M.Turkyilmazoglu6.HomotopyAnalysisMethodforSomeBoundaryLayerFlowsofNanofluids259T.HayatandM.Mustafavii
8October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.0viiiContents7.HomotopyAnalysisMethodforFractionalSwift–HohenbergEquation291S.DasandK.Vishal8.HAM-BasedPackageNOPHforPeriodicOscillationsofNonlinearDynamicSystems309Y.-P.Liu9.HAM-BasedMathematicaPackageBVPh2.0forNonlinearBoundaryValueProblems361Y.-L.ZhaoandS.-J.Liao
9October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1Chapter1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethodShijunLiaoShanghaiJiaoTongUniversity,Shanghai200240,Chinasjliao@sjtu.edu.cnAbriefreviewofthehomotopyanalysismethod(HAM)andsomeofitscurrentadvancesaredescribed.Weemphasizethattheintroductionofthehomotopy,abasicconceptintopology,isamilestoneofthean-alyticapproximationmethods,sinceitisthehomotopywhichprovidesusgreatfreedomandflexibilitytochooseequationtypeandsolutionex-pressionofhigh-orderapproximationequations.Besides,theso-called“convergence-controlparameter”isamilestoneoftheHAM,too,sinceitistheconvergence-controlparameterthatprovidesusaconvenientwaytoguaranteetheconvergenceofsolutionseriesandthatdifferstheHAMfromallotheranalyticapproximationmethods.RelationsoftheHAMtothehomotopycontinuationmethodandotheranalyticapproximationtechniquesarebrieflydescribed.Someinterestingbutchallengingnon-linearproblemsaresuggestedtotheHAMcommunity.AspointedoutbyGeorgCantor(1845–1918),“theessenceofmathematicsliesentirelyinitsfreedom”.Hopefully,theabove-mentionedfreedomandgreatflex-ibilityoftheHAMmightcreatesomenovelideasandinspirebrave,enterprising,youngresearcherswithstimulatedimaginationtoattackthemwithsatisfactory,betterresults.1
10October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.12S.-J.LiaoContents1.1.Background.....................................21.2.AbriefhistoryoftheHAM............................51.3.SomeadvancesoftheHAM............................101.3.1.Generalizedzeroth-orderdeformationequation..............101.3.2.SpectralHAMandcomplicatedauxiliaryoperator............131.3.3.PredictorHAMandmultiplesolutions..................151.3.4.ConvergenceconditionandHAM-basedsoftware.............161.4.Relationshipstoothermethods..........................181.5.Chanceandchallenge:somesuggestedproblems................201.5.1.Periodicsolutionsofchaoticdynamicsystems..............211.5.2.PeriodicorbitsofNewtonianthree-bodyproblem............221.5.3.Viscousflowpastasphere.........................241.5.4.Viscousflowpastacylinder........................251.5.5.Nonlinearwaterwaves...........................26References.........................................291.1.BackgroundPhysicalexperiment,numericalsimulationandanalytic(approximation)methodarethreemainstreamtoolstoinvestigatenonlinearproblems.Withoutdoubt,physicalexperimentisalwaysthebasicapproach.How-ever,physicalexperimentsareoftenexpensiveandtime-consuming.Be-sides,modelsforphysicalexperimentsareoftenmuchsmallerthantheoriginalones,butmostlyitisveryhardtosatisfyallsimilaritycriteri-ons.Bymeansofnumericalmethods,nonlinearequationsdefinedinrathercomplicateddomaincanbesolved.However,itisdifficulttogainnumeri-calsolutionsofnonlinearproblemswithsingularityandmultiplesolutionsordefinedinaninfinitydomain.Bymeansofanalytic(approximation)methods,onecaninvestigatenonlinearproblemswithsingularityandmul-tiplesolutionsinaninfinityinterval,butequationsshouldbedefinedinasimpleenoughdomain.So,physicalexperiments,numericalsimulationsandanalytic(approximation)methodshavetheirinherentadvantagesanddisadvantages.Therefore,eachofthemisimportantandusefulforustobetterunderstandnonlinearproblemsinscienceandengineering.Ingeneral,exact,closed-formsolutionsofnonlinearequationsarehardlyobtained.Perturbationtechniques[1–4]arewidelyusedtogainanalyticapproximationsofnonlinearequations.Usingperturbationmethods,manynonlinearequationsaresuccessfullysolved,andlotsofnonlinearphenom-enaareunderstoodbetter.Withoutdoubt,perturbationmethodsmakegreatcontributiontothedevelopmentofnonlinearscience.Perturbation
11October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod3methodsaremostlybasedonsmall(orlarge)physicalparameters,calledperturbationquantity.Usingsmall/largephysicalparameters,perturba-tionmethodstransferanonlinearequationintoaninfinitenumberofsub-problemsthataremostlylinear.Unfortunately,manynonlinearequationsdonotcontainsuchkindofperturbationquantitiesatall.Moreimpor-tantly,perturbationapproximationsoftenquicklybecomeinvalidwhentheso-calledperturbationquantitiesenlarge.Inaddition,perturbationtech-niquesaresostronglydependentuponphysicalsmallparametersthatwehavenearlynofreedomtochooseequationtypeandsolutionexpressionofhigh-orderapproximationequations,whichareoftencomplicatedandthusdifficulttosolve.Duetotheserestrictions,perturbationmethodsarevalidmostlyforweaklynonlinearproblemsingeneral.Ontheotherside,somenon-perturbationmethodswereproposedlongago.Theso-called“Lyapunov’sartificialsmall-parametermethod”[5]cantracebacktothefamousRussianmathematicianLyapunov(1857–1918),whofirstrewroteanonlinearequationN[u(r,t)]=L0[u(r,t)]+N0[u(r,t)]=f(r,t),(1.1)whererandtdenotethespatialandtemporalvariables,u(r,t)aunknownfunction,f(r,t)aknownfunction,L0andN0arelinearandnonlinearoperator,respectively,tosuchanewequationL0[u(r,t)]+qN0[u(r,t)]=f(r,t),(1.2)whereqhasnophysicalmeaning.Then,LyapunovregardedqasasmallparametertogainperturbationapproximationsX+∞23mu≈u0+u1q+u2q+u3q+···=u0+umq,(1.3)m=1andfinallygainedapproximationX+∞u≈u0+um,(1.4)m=1bysettingq=1,whereL0[u0(r,t)]=f(r,t),L0[u1(r,t)]=−N0[u0(r,t)],···(1.5)andsoon.ItshouldbeemphasizedthatonehasnofreedomtochoosethelinearoperatorL0inLyapunov’sartificialsmall-parametermethod:itisexactlythelinearpartofthewholeleft-handsideoftheoriginalequationN[u]=f,whereN=L0+N0.Thus,whenL0iscomplicatedor“singular”
12October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.14S.-J.Liao(forexample,itdoesnotcontainthehighestderivative),itisdifficult(orevenimpossible)tosolvethehigh-orderapproximationequation(1.5).Be-sides,theconvergenceoftheapproximationseries(1.4)isnotguaranteedingeneral.Evenso,Lyapunov’sexcellentworkisamilestoneofanalyticapproximationmethods,becauseitisindependentoftheexistenceofphys-icalsmallparameter,eventhoughitfirstregardsqasa“smallparameter”butfinallyenforcesittobe1thatishowevernot“small”strictlyfrommathematicalviewpoints.Theso-called“Adomiandecompositionmethod”(ADM)[6–8]wasde-velopedfromthe1970stothe1990sbyGeorgeAdomian,thechairoftheCenterforAppliedMathematicsattheUniversityofGeorgia,USA.Ado-mianrewrote(1.1)intheformN[u(r,t)]=LA[u(r,t)]+NA[u(r,t)]=f(r,t),(1.6)whereLAoftencorrespondstothehighestderivativeoftheequationunderconsideration,NA[u(r,t)]givestheleftpart,respectively.ApproximationsoftheADMarealsogivenby(1.4),too,whereLA[u0(r,t)]=f(r,t),LA[um(r,t)]=−Am−1(r,t),m≥1,(1.7)withtheso-calledAdomialpolynomial("#)1∂kX+∞nAk(r,t)=kNAun(r,t)q.(1.8)k!∂qn=0q=0SincethelinearoperatorLAissimplythehighestderivativeoftheconsid-eredequation,itisconvenienttosolvethehigh-orderapproximationequa-tions(1.7).ThisisanadvantageoftheADM,comparedto“Lyapunov’sartificialsmall-parametermethod”[5].However,theADMdoesnotpro-videsusfreedomtochoosethelinearoperatorLA,whichisrestrictedtoberelatedonlytothehighestderivative.Besides,like“Lyapunov’sartificialsmall-parametermethod”[5],theconvergenceoftheapproximationseries(1.4)givenbytheADMisstillnotguaranteed.Essentially,bothofthe“Lyapunov’sartificialsmallparametermethod”andthe“Adomiandecompositionmethod”transferanonlinearproblemintoaninfinitenumberoflinearsub-problems,withoutsmallphysicalpa-rameter.However,theyhavetwofundamentalrestrictions.First,onehasnofreedomandflexibilitytochoosethelinearoperatorsL0orLA,sinceL0isexactlythelinearpartofNandLAcorrespondstothehighestderivative,respectively.Second,thereisnowaytoguaranteetheconvergenceoftheapproximationseries(1.4).Thesecondonesismoreserious,sincedivergent
13October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod5approximationsaremostlyuseless.Thus,likeperturbationmethods,thetraditionalnon-perturbationmethods(suchasLyapunov’sartificialsmallparametermethodandtheADM)areoftenvalidforweaklynonlinearprob-lemsinmostcases.Intheory,itisveryvaluabletodevelopanewkindofanalyticapproxi-mationmethodwhichshouldhavethefollowingcharacteristics:(1)itisindependentofsmallphysicalparameter;(2)itprovidesusgreatfreedomandflexibilitytochoosetheequation-typeandsolutionexpressionofhigh-orderapproximationequations;(3)itprovidesusaconvenientwaytoguaranteetheconvergenceofapprox-imationseries.Oneofsuchkindofanalyticapproximationmethods,namelythe“homo-topyanalysismethod”(HAM)[9–17],wasdevelopedbyShijunLiaofrom1990sto2010s,togetherwithcontributionsofmanyotherresearchersintheoryandapplications.ThebasicideasoftheHAMwithitsbriefhistoryaredescribedbelow.1.2.AbriefhistoryoftheHAMThebasicideasof“Lyapunov’sartificialsmall-parametermethod”canbegeneralizedintheframeofthehomotopy,afundamentalconceptoftopol-ogy.ForanonlinearequationN[u(r,t)]=f(r,t),(1.9)Liao[9]proposetheso-called“homotopyanalysismethod”(HAM)byusingthehomotopy,abasicconceptintopology:(1−q)L[ϕ(r,t;q)−u0(r,t)]=c0qH(r,t){N[ϕ(r,t;q)]−f(r,t)},(1.10)whereLisanauxiliarylinearoperatorwiththepropertyL[0]=0,Nisthenonlinearoperatorrelatedtotheoriginalequation(1.9),q∈[0,1]istheembeddingparameterintopology(calledthehomotopyparameter),ϕ(r,t;q)isthesolutionof(1.10)forq∈[0,1],u0(r,t)isaninitialguess,c06=0istheso-called“convergence-controlparameter”,andH(r,t)isanauxiliaryfunctionthatisnon-zeroalmosteverywhere,respectively.Notethat,intheframeofthehomotopy,wehavegreatfreedomtochoosetheauxiliarylinearoperatorL,theinitialguessu0(r,t),theauxiliaryfunctionH(r,t),andthevalueoftheconvergence-controlparameterc0.
14October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.16S.-J.LiaoWhenq=0,duetothepropertyL[0]=0,wehavefrom(1.10)thesolutionϕ(r,t;0)=u0(r,t).(1.11)Whenq=1,sincec06=0andH(r,t)6=0almosteverywhere,Eq.(1.10)isequivalenttotheoriginalnonlinearequation(1.9)sothatwehaveϕ(r,t;1)=u(r,t),(1.12)whereu(r,t)isthesolutionoftheoriginalequation(1.9).Thus,astheho-motopyparameterqincreasesfrom0to1,thesolutionϕ(r,t;q)ofEq.(1.10)varies(ordeforms)continuouslyfromtheinitialguessu0(r,t)tothesolu-tionu(r,t)oftheoriginalequation(1.9).Forthissake,Eq.(1.10)iscalledthezeroth-orderdeformationequation.Here,itmustbeemphasizedonceagainthatwehavegreatfreedomandflexibilitytochoosetheauxiliarylinearoperatorL,theauxiliaryfunctionH(r,t),andespeciallythevalueoftheconvergencecontrolparameterc0inthezeroth-orderdeformationequation(1.10).Inotherwords,thesolutionϕ(r,t;q)ofthezeroth-orderdeformationequation(1.10)isalsodependentuponallaoftheauxiliarylinearoperatorL,theauxiliaryfunctionH(r,t)andtheconvergence-controlparameterc0asawhole,eventhoughtheyhavenophysicalmeanings.ThisisakeypointoftheHAM,whichwewilldiscussindetailslater.AssumethatL,H(r,t)andc0areproperlychosensothatthesolutionϕ(r,t;q)ofthezeroth-orderdeformationequation(1.10)alwaysexistsforq∈(0,1)andbesidesitisanalyticatq=0,andthattheMaclaurinseriesofϕ(r,t;q)withrespecttoq,i.e.X+∞ϕ(r,t;q)=u(r,t)+u(r,t)qm(1.13)0mm=1convergesatq=1.Then,dueto(1.12),wehavetheapproximationseriesX+∞u(r,t)=u0(r,t)+um(r,t).(1.14)m=1Substitutingtheseries(1.13)intothezeroth-orderdeformationequation(1.10)andequatingthelike-powerofq,wehavethehigh-orderapproxima-tionequationsforum(r,t),calledthemth-orderdeformationequationL[um(r,t)−χmum−1(r,t)]=c0H(r,t)Rm−1(r,t),(1.15)aMorestrictly,ϕ(r,t;q)shouldbereplacedbyϕ(r,t;q,L,H(r,t),c0).Onlyforthesakeofsimplicity,weusehereϕ(r,t;q),butshouldalwayskeepthispointinmind.
15October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod7where("#!)1∂kX+∞nRk(r,t)=N(r,t)q−f(r,t),(1.16)k!∂qkn=0q=0withthedefinition0,whenk≤1,χk=(1.17)1,whenk≥2.Forvarioustypesofnonlinearequations,itiseasyandstraightforwardtousethetheoremsprovedinChapter4ofLiao’sbook[11]tocalculatethetermRk(r,t)ofthehigh-orderdeformationequation(1.15).ItshouldbeemphasizedthattheHAMprovidesusgreatfreedomandflexibilitytochoosetheauxiliarylinearoperatorLandtheinitialguessu0.Thus,differentfromallotheranalyticmethods,theHAMprovidesusgreatfreedomandflexibilitytochoosetheequationtypeandsolutionexpressionofthehigh-orderdeformationequation(1.15)sothatitssolutioncanbeof-tengainedwithoutgreatdifficulty.Noticethat“theessenceofmathematicsliesentirelyinitsfreedom”,aspointedoutbyGeorgCantor(1845–1918).Moreimportantly,thehigh-orderdeformationequation(1.15)containstheconvergence-controlparameterc0,andtheHAMprovidesgreatfreedomtochoosethevalueofc0.Mathematically,ithasbeenprovedthattheconvergence-controlparameterc0canadjustandcontroltheconvergenceregionandratiooftheapproximationseries(1.14).Fordetails,pleaserefertoLiao[10,12,13]andespecially§5.2to§5.4ofhisbook[11].So,unlikeallotheranalyticapproximationmethods,theconvergence-controlparameterc0oftheHAMprovidesusaconvenientwaytoguaranteetheconvergenceoftheapproximationseries(1.14).Infact,itistheconvergence-controlparameterc0thatdifferstheHAMfromallotheranalyticmethods.Atthemth-orderofapproximation,theoptimalvalueoftheconvergence-controlparameterc0canbedeterminedbytheminimumofresidualsquareoftheoriginalgoverningequation,i.e.dEm=0,(1.18)dc0whereZ("m#)2XEm=Nun(r,t)−f(r,t)dΩ.(1.19)Ωn=0Besides,ithasbeenprovedbyLiao[16]thatahomotopyseriessolution(1.14)mustbeoneofsolutionsofconsideredequation,aslongasitis
16October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.18S.-J.Liaoconvergent.Inotherwords,foranarbitraryconvergence-controlparameterc0∈Rc,whereRc=c0:limEm(c0)→0(1.20)m→+∞isaninterval,thesolutionseries(1.14)isconvergenttothetruesolutionoftheoriginalequation(1.9).Fordetails,pleaserefertoLiao[16]andChapter3ofhisbook[11].Insummary,theHAMhasthefollowingadvantages:(a)itisindependentofanysmall/largephysicalparameters;(b)itprovidesusgreatfreedomandlargeflexibilitytochooseequationtypeandsolutionexpressionoflinearhigh-orderapproximationequations;(c)itprovidesusaconvenientwaytoguaranteetheconvergenceofapprox-imationseries.Inthisway,nearlyallrestrictionsandlimitationsofthetraditionalnon-perturbationmethods(suchasLyapunov’sartificialsmallparametermethod[5],theAdomiandecompositionmethod[6–8],theδ-expansionmethod[18]andsoon)canbeovercomebymeansoftheHAM.Besides,ithasbeengenerallyproved[10,12,13]thattheLyapunov’sartificialsmallparametermethod[5],theAdomiandecompositionmethod[6–8]andtheδ-expansionmethod[18]areonlyspecialcasesoftheHAMforsomespeciallychosenauxiliarylinearoperatorLandconvergence-controlparameterc0.Especially,theso-called“homotopyperturbationmethod”(HPM)[19]proposedbyJihuanHein1998(sixyearslaterafterLiao[9]proposedtheearlyHAMin1992)wasonlyaspecialcaseoftheHAMwhenc0=−1,andthushas“nothingnewexceptitsname”[20].SomeresultsgivenbytheHPMaredivergenteveninthewholeintervalexceptthegiveninitial/boundaryconditions,andthus“itisveryimportanttoinvestigatetheconvergenceofapproximationseries,otherwiseonemightgetuselessresults”,aspointedoutbyLiangandJeffrey[21].Fordetails,see§6.2ofLiao’sbook[11].Thus,theHAMismoregeneralintheoryandwidelyvalidinpracticeformoreofnonlinearproblemsthanotheranalyticapproximationtechniques.Incalculus,thefamousEulertransformisoftenusedtoacceleratecon-vergenceofaseriesortomakeadivergentseriesconvergent.Itisinter-estingthatonecanderivetheEulertransformintheframeoftheHAM,andgiveasimilarbutmoregeneraltransform(calledthegeneralizedEulertransform),asshowninChapter5ofLiao’sbook[11].Thisprovidesusa
17October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod9theoreticalcornerstoneforthevalidityandgeneralityoftheHAM.Theintroductionoftheso-called“convergence-controlparameter”c0inthezeroth-orderdeformationequation(1.10)isamilestonefortheHAM.Fromphysicalviewpoint,the“convergence-controlparameter”c0hasnophysicalmeaningssothatconvergentseriesofsolutiongivenbytheHAMmustbeindependentofc0.Thisisindeedtrue:thereexistssucharegionRcthat,forarbitraryc0∈Rc,theHAMseriesconvergestothetruesolutionoftheoriginalequation(1.9),asillustratedbyLiao[10,11].However,ifc06∈Rc,thesolutionseriesdiverges!So,fromamathematicalviewpoint,the“convergence-controlparameter”isakeypointoftheHAM,whichprovidesusaconvenientwaytoguaranteetheconvergenceofthesolutionseries.Infact,itistheso-called“convergence-controlparameter”thatdifferstheHAMfromallotheranalyticapproximationmethods.Theintroductionofthebasicconcepthomotopyintopologyisalsoamilestoneoftheanalyticapproximationmethodsfornonlinearproblems.ItisthehomotopythatprovidesusgreatfreedomandlargeflexibilitytochoosetheauxiliarylinearoperatorLandinitialguessu0inthezeroth-orderdeformationequation(1.10),whichdeterminetheequationtypeandsolutionexpressionofthehigh-orderdeformationequations(1.15).Besides,itisthehomotopythatprovidesusthefreedomtointroducetheso-called“convergence-controlparameter”c0in(1.10),whichbecomesnowacor-nerstoneoftheHAM.Notethatitisimpossibletointroducesuchkindof“convergence-controlparameter”intheframeofperturbationtechniquesandthetraditionalnon-perturbationmethods(suchasLyapunov’sartificialsmallparameter,Adomiandecompositionmethodandsoon).ThefreedomonthechoiceoftheauxiliarylinearoperatorLissolargethatthesecond-ordernonlinearGelfandequationcanbesolvedconveniently(withgoodagreementwithnumericalresults)intheframeoftheHAMevenbymeansofaforth-orderauxiliarylinearoperator(fortwodimen-sionalGelfandequation)orasixth-orderauxiliarylinearoperator(forthreedimensionalGelfandequation),respectively,asillustratedbyLiao[14].Al-thoughitistruethattheauxiliarylinearoperator(withthesamehighestorderofderivativeasthatofconsideredproblem)canbechosenstraight-forwardlyinmostcases,suchkindoffreedomoftheHAMshouldbetakenintoaccountsufficientlybytheHAMcommunitywhennecessary,especiallyforsomevaluablebutchallengingproblems(someofthemaresuggestedbelowin§1.5).Inaddition,bymeansoftheabove-mentionedfreedomoftheHAM,theconvergenceofapproximationsolutioncanbegreatlyacceleratedinthe
18October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.110S.-J.LiaoframeoftheHAMbymeansoftheiteration,theso-calledhomotopy-Pad´etechniqueandsoon.Fordetails,pleasereferto§2.3.5to§2.3.7ofLiao’sbook[11].Indeed,“theessenceofmathematicsliesentirelyinitsfreedom”,aspointedoutbyGeorgCantor(1845–1918).SuchkindofgreatfreedomoftheHAMshouldprovideusgreatpossi-bilitytosolvesomeopenquestions.Oneofthemisdescribedbelow.Thesolutionofthehigh-orderdeformationequation(1.15)canbeexpressedintheform−1um(r,t)=−χmum−1(r,t)+L[c0H(r,t)Rm−1(r,t)],(1.21)whereL−1istheinverseoperatorofL.ForafewauxiliarylinearoperatorL,itsinverseoperatorissimple.However,inmostcases,itisnotstraight-forwardtosolvetheabovelineardifferentialequation.Canwedirectlychoose(ordefine)theinverseauxiliarylinearoperatorL−1soastosolve(1.15)conveniently?ThisispossibleintheframeoftheHAM,sinceintheorytheHAMprovidesusgreatfreedomandlargeflexibilitytochoosetheauxiliarylinearoperatorL.Ifsuccessful,itwouldberatherefficientandconvenienttosolvethehigh-orderdeformationequation(1.15).ThisisaninterestingbutopenquestionfortheHAMcommunity,whichdeservestobestudiedindetails.Notethatsomeinterestingproblemsaresuggestedin§1.5.1.3.SomeadvancesoftheHAMSince1992whenLiao[9]proposedtheearlyHAM,theHAMhasbeendevelopinggreatlyintheoryandapplications,duetothecontributionsofmanyresearchersindozensofcountries.Unfortunately,itisimpossibletodescribealloftheseadvancesindetailsinthisbriefreview,andeveninthisbook.Infact,theHAMhasbeensuccessfullyappliedtonumerous,varioustypesofnonlinearproblemsinscience,engineeringandfinance.So,wehadtofocusonarathersmallportoftheseadvanceshere.1.3.1.Generalizedzeroth-orderdeformationequationThestartingpointoftheuseoftheHAMistoconstructtheso-calledzeroth-orderdeformationequation,whichbuildsaconnection(i.e.acontinuousmapping/deformation)betweenagivennonlinearproblemandarelativelymuchsimplerlinearones.So,thezeroth-orderdeformationequationisabaseoftheHAM.
19October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod11Givenanonlinearequation,wehavegreatfreedomandlargeflexibilityintheframeoftheHAMtoconstructtheso-calledzeroth-orderdefor-mationequationusingtheconcepthomotopyintopology.Especially,theconvergence-controlparameterc0playsanimportantroleintheframeoftheHAM.So,itisnaturaltoenhancetheabilityoftheso-called“convergencecontrol”bymeansofintroducingmoresuchkindofauxiliaryparameters.Duetotheabove-mentionedfreedomandflexibilityoftheHAM,therearenumerousapproachestodoso.Forexample,wecanconstructsuchakindofzeroth-orderdeformationequationwithK+1convergence-controlparameters:(1−q)L[ϕ(r,t;q)−u0(r,t)]!XK=cqn+1H(r,t){N[ϕ(r,t;q)]−f(r,t)},(1.22)nn=0whereϕ(r,t;q)isthesolution,NisanonlinearoperatorrelatedtoanoriginalproblemN[u(r,t)]=f(r,t),q∈[0,1]isthehomotopyparameter,u0isaninitialguess,Lisanauxiliarylinearoperator,H(r,t)isanauxiliaryfunctionwhichisnonzeroalmosteverywhere,andc={c0,c1,···,cK}isavectorof(K+1)non-zeroconvergence-controlparameters,respectively.Notethat,whenK=0,itgivesexactlythezeroth-orderdeformationequation(1.10).Thecorrespondinghigh-orderdeformationequationreadsmin{Xm−1,K}L[um(r,t)−χmum−1(r,t)]=H(r,t)cnRm−1−n(r,t),(1.23)n=0whereRn(r,t)andχnaredefinedbythesameformulas(1.16)and(1.17),respectively.WhenK=0,theabovehigh-orderdeformationequation(1.23)isexactlythesameas(1.15).Atthemth-orderofapproximation,theoptimalconvergence-controlparametersaredeterminedbytheminimumoftheresidualsquareoftheoriginalequation,i.e.dEm=0,0≤n≤min{m−1,K},(1.24)dcnwhereEmisdefinedby(1.19).Fordetails,pleaserefertoChapter4ofLiao’sbook[11].WhenK→+∞,itisexactlytheso-called“optimalhomotopyasymptoticmethod”[22].So,the“optimalhomotopyasymptotic
20October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.112S.-J.Liaomethod”[22]isalsoaspecialcaseoftheHAM,asshownin§3.2.2and§6.3ofLiao’sbook[11].Intheory,themoretheconvergence-controlparameters,thelargertheabilitytocontroltheconvergenceoftheHAMseries.However,itisfound[16]thatmuchmoreCPUtimesisneededinpracticewhenmoreconvergence-controlparametersareused.Inmostcases,oneoptimalconvergence-controlparameterisgoodenoughtogainconvergentresultsbymeansoftheHAM.Consideringthecomputationalefficiency,oneuptothreeconvergence-controlparametersaregenerallysuggestedintheframeoftheHAM.Fordetails,pleasereferto§2.3.3,§2.3.4,§4.6.1andChapter3ofLiao’sbook[16].Itshouldbeemphasizedonceagainthat,intheframeofthehomotopyintopology,wehaverathergreatfreedomandlargeflexibilitytoconstructtheso-calledzeroth-orderdeformationequation.Intheory,givenanonlinearequationN[u(r,t)]=f(r,t),wecanalwaysproperlychooseaninitialguessu0(r,t)andanauxiliarylinearoperatorLtoconstructsuchazeroth-orderdeformationequationinarathergeneralformA[u0(r,t),L,ϕ(r,t;q),c;q]=0(1.25)thatitholdsϕ(r,t;0)=u0(r,t),whenq=0,(1.26)andϕ(r,t;1)=u(r,t),whenq=1,(1.27)i.e.,whenq=1thezeroth-orderdeformationequation(1.25)isequivalenttotheoriginalnonlinearequationN[u(r,t)]=f(r,t).UsingthetheoremsgiveninChapter4ofLiao’sbook[11],itiseasytogainthecorrespondinghigh-orderdeformationequations.Here,c={c0,c1,···,cK}isavectorofconvergence-controlparameters,whoseoptimalvaluesarede-terminedbytheminimumofresidualsquareoftheoriginalequation.Notethat(1.25)israthergeneral:thezeroth-orderdeformationequations(1.10)and(1.22),andevenEq.(1.2)forLyapunov’sartificialsmallparametermethod,areonlyspecialcasesof(1.25).Somecommonlyusedzeroth-orderdeformationequationsaredescribedin§4.3ofLiao’sbook[11]asspecialcasesofthegeneralizedzeroth-orderdeformationequation(1.25).Intheory,thereareaninfinitenumberofdifferentwaystoconstructazeroth-orderdeformationequation(1.25).Therefore,intheframeofthe
21October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod13HAM,weindeedhavehugefreedomandflexibility.Suchkindoffreedomandflexibilitycomesfromthehomotopy,abasicconceptintopology.Intheory,thiskindoffreedomandflexibilityprovidesusgreatabilitytosolvesomeinterestingbutchallengingnonlinearproblems(someofthemaresuggestedbelowin§1.5ofthischapter),ifwecanclearlyknowhowtousetheminaproperwaywithstimulatedimagination!Inpractice,itissuggestedtofirstlyusethezeroth-orderdeformationequation(1.10),sinceitworksformostofnonlinearproblems,asillustratedbyLiao[10,11].Ifunsuccessful,onecanfurtherattemptalittlemorecomplicatedzeroth-orderdeformationequations,suchas(1.22).Finally,weemphasizeonceagainthat,intheory,onehashugefreedomtoconstructazeroth-orderdeformationequation(1.25)satisfyingbothof(1.26)and(1.27),aslongasoneclearlyknowshowtousesuchkindoffreedom.1.3.2.SpectralHAMandcomplicatedauxiliaryoperatorAlthoughtheHAMprovidesusgreatfreedomtochoosetheauxiliarylinearoperatorL,itmightbedifficulttosolvethelinearhigh-orderdeformationequation(1.15)or(1.23)exactly,ifLiscomplicated.Thisismainlybecausemostoflineardifferentialequationshavenoclosed-formsolutions,i.e.theirsolutionsaremostlyexpressedbyaninfiniteseries.So,inordertoexactlysolvehigh-orderdeformationequationsintheframeoftheHAM,weoftenshouldchooseareasonablebutsimpleenoughauxiliarylinearoperatorL.This,however,restrictstheapplicationsoftheHAM.Thisisthemainreasonwhyonlyafewsimpleauxiliarylinearoperators,suchasLu=u0,Lu=xu0+u,Lu=u0+u,Lu=u00+uandsoon,havebeenmostlyusedintheframeoftheHAM,wheretheprimedenotesthedifferentiationwithrespecttox.Theseauxiliarylinearoperatorscorrespondtosomefundamentalfunctionssuchaspolynomial,exponential,trigonometricfunctionsandtheircombination.Therearemanyspecialfunctionsgovernedbylineardifferentialequa-tions.Althoughmanysolutionscanexpressedbythesespecialfunctions,theyarehardlyusedintheframeoftheHAMuptonow,becausethecorrespondinghigh-orderdeformationequationsoftenbecomemoreandmoredifficulttosolve.Thisisapity,sinceintheorytheHAMindeedprovidesusfreedomtousespecialfunctionstoexpresssolutionsofmanynonlineardifferentialequations.Currently,VanGorder[23]madeaninter-
22October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.114S.-J.Liaoestingattemptinthisdirection.IntheframeoftheHAM,VanGorder[23]expressedanalyticapproximationsoftheFitzhugh–Nagumoequationbymeansoferrorfunction,Gaussianfunctionandsoon.ThekeyisthatVanGorder[23]chosesuchanauxiliarylinearoperator2z2−1Lu=u00+u0,zwheretheprimedenotesthedifferentiationwithrespecttoz,andespeciallysuchaproperauxiliaryfunctionH(z)=z|z|,thatthecorrespondinghigh-orderdeformationequationscanbesolvedeasily.Fordetails,pleaserefertoVanGorder[23],VajraveluandVanGorder[24]and§4.6ofthisbook.ThisexampleillustratesonceagainthattheHAMindeedprovidesusgreatfreedom,i.e.lotsofpossibilities.Thekeyishowtousesuchkindoffreedom!Generallyspeaking,solutionofacomplicatedlinearODE/PDEshouldbeexpressedinaserieswithaninfinitenumberofterms.Mathematically,suchaseriesleadstothelargerandlargerdifficultytogainhigher-orderanalyticapproximationsofanonlinearproblem.Fortunately,fromphysicalviewpoint,itisoftenaccurateenoughtohaveanalyticapproximationswithmanyenoughterms.Currently,usingtheSchmidt-Gramprocess,Zhao,LinandLiao[25]suggestedaneffectivetruncationtechniqueintheframeoftheHAM,whichcanbeusedtogreatlysimplifytheright-handsideofthehigh-orderdeformationequations,suchas(1.15)and(1.23),priortosolvingthem.Inthisway,muchCPUtimecanbesaved,evenwithoutlossofaccuracy.In2010,Motsaetal.[26,27]suggestedtheso-called“spectralhomotopyanalysismethod”(SHAM)usingtheChebyshevpseudospectralmethodtosolvethelinearhigh-orderdeformationequationsandchoosingtheauxiliarylinearoperatorLintermsoftheChebyshevspectralcollocationdifferentia-tionmatrix[28].Intheory,anyacontinuousfunctioninaboundedintervalcanbebestapproximatedbyChebyshevpolynomial.So,theSHAMpro-videsuslargerfreedomtochoosetheauxiliarylinearoperatorLandinitialguessintheframeoftheHAM.ItisvaluabletoexpandtheSHAMfornonlinearpartialdifferentialequations.Besides,itiseasytoemploytheoptimalconvergence-controlparameterintheframeoftheSHAM.Thus,theSHAMhasgreatpotentialtosolvemorecomplicatednonlinearprob-lems,althoughfurthermodificationsintheoryandmoreapplicationsareneeded.ForthedetailsabouttheSHAM,pleasereferto[26,27]andChap-ter3ofthisbook.
23October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod15Chebyshevpolynomialisjustoneofspecialfunctions.TherearemanyotherspecialfunctionssuchasHermitepolynomial,Legendrepolynomial,Airyfunction,Besselfunction,Riemannzetafunction,hypergeometricfunctions,errorfunction,Gaussianfunctionandsoon.SincetheHAMprovidesusextremelylargefreedomtochooseauxiliarylinearoperatorLandinitialguess,itshouldbepossibletodevelopa“generalizedspectralHAM”whichcanuseproperspecialfunctionsforsomenonlinearproblems.Especially,combinedtheSHAM[26,27]withtheabove-mentionedtrunca-tiontechniquesuggestedbyZhao,LinandLiao[25],itwouldbepossibletouse,whennecessary,morecomplicatedauxiliarylinearoperatorsintheframeoftheHAMsothatsomedifficultnonlinearproblemscanbesolved.1.3.3.PredictorHAMandmultiplesolutionsManynonlinearboundaryvalueproblemshavemultiplesolutions.Ingen-eral,itisdifficulttogainthesedualsolutionsbymeansofnumericaltech-niques,mainlybecausedualsolutionsareoftenstronglydependentuponinitialconditionsbutwedonotknowhowtochoosethemexactly.Compar-ativelyspeaking,itisalittlemoreconvenienttouseanalyticapproximationmethodstosearchformultiplesolutionsofnonlinearproblems,sinceana-lyticmethodsadmitunknownvariablesininitialguess.Forexample,letusconsiderasecond-ordernonlineardifferentialequa-tionofatwo-pointboundaryvalueproblem:N[u(x)]=0,u(0)=a,u(1)=b,(1.28)whereNisa2nd-ordernonlineardifferentialoperator,aandbareknownconstants,respectively.Assumethatthereexistmultiplesolutionsu(x).Thesemultiplesolutionsmusthavesomethingdifferent.Withoutlossofgenerality,assumethattheyhavedifferentfirst-orderderivativeu0(0)=σ,whereσisunknown.Obviously,differentinitialguessu0(x)mightleadtomultiplesolutions.Fortunately,theHAMprovidesusgreatfreedomtochooseinitialguessu0(x).Asmentionedbefore,suchkindoffreedomisonecornerstoneoftheHAM.So,intheframeoftheHAM,itisconvenientforustochoosesuchaninitialguessu0(x)thatitsatisfiesnotonlythetwoboundaryconditionsu(0)=a,u(1)=bbutalsotheadditionalconditionu0(0)=σ.Inthisway,theinitialguessu0(x)containsanunknownparameterσ,calledbyLiao(seeChapter8of[11])themultiple-solution-controlparameter.Then,theanalyticapproximationsgainedbytheHAMcontainatleasttwoun-
24October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.116S.-J.Liaoknownauxiliaryparameters:theconvergence-controlparameterc0andthemultiple-solution-controlparameterσ.AssuggestedbyLiao(seeChapter8of[11]),theoptimalvaluesofc0andσcanbedeterminedbytheminimumoftheresidualsquareofgoverningequations.Inthisway,multiplesolu-tionsofsomenonlineardifferentialequationscanbegained,asillustratedbyLiao(seeChapter8of[11]).IntheframeoftheHAM,AbbasbandyandShivanian[29,30]developedadifferentialbutratherinterestingapproachtogaindualsolutions,namelythePredictorHAM(PHAM).Forsimplicity,letususethesameequation(1.28)asanexampletodescribeitsbasicideas.Firstofall,anadditionalconditionsuchasu0(0)=σisintroducedwiththeunknownparameterσ.Then,intheframeoftheHAM,onesolvesthenonlineardifferentialequationN[u(x)]=0,butwiththetwoboundaryconditionsu0(0)=σandu(1)=b.Then,u(0),theHAMapproximationatx=0,containsatleasttwounknownparameters:oneistheso-calledconvergence-controlparameterc,theotherisσ=u0(0),calledthemultiple-solution-control0parameterbyLiao(seeChapter8of[11])intheabove-mentionedapproach.Substitutingtheexpressionofu(0)intotheboundaryconditionu(0)=agivesanonlinearalgebraicequationaboutc0andσ.Fromthephysicalviewpoint,σ=u0(0)hasphysicalmeanings,buttheconvergence-controlparameterc0doesnot.Iftheorderofapproximationishighenough,onecangainconvergent,accurateenoughmultiplevaluesofσforproperlychosenvaluesofc0inafiniteinterval,asillustratedin[29,30].Inthisway,onecanfindmultiplesolutionsofagivennonlinearproblem.Fordetails,pleaserefertoChapter2ofthisbook.IntheframeoftheHAM,somenewbranchesofsolutionsforviscousboundary-layerflowswerefound[31,32],andthemultipleequilibrium-statesofresonantwavesindeepwater[33]andinfinitewaterdepth[34]werediscoveredforthefirsttime,tothebestofauthor’sknowledge.Alloftheseillustratethepotential,noveltyandvalidityoftheHAMtogivesomethingnewanddifferent.ThisisasuperiorityoftheHAMtonumericalmethodsandsomeotheranalyticapproximationtechniques.Certainly,itisvaluabletoapplytheHAMtodiscoversomenewsolutionsofothernonlinearproblems!1.3.4.ConvergenceconditionandHAM-basedsoftwareTheoreticallyspeaking,theHAMindeedprovidesusgreatfreedomtochooseinitialguess,auxiliarylinearoperator,convergence-controlparame-
25October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod17ter,equation-typeandsolution-expressionofhigh-orderdeformationequa-tion,andsoon.However,itisstillnotveryclearhowtousethesefreedomintheframeoftheHAM,mainlybecauselittlemathematicaltheoremshavebeenprovedinanabstractway.Somestudiesonthestabilityofauxiliarylinearoperatorandconvergence-controlparameteroftheHAMaredescribedinChapter4ofthisbook.SomecurrentworksaboutconvergenceconditionoftheHAMseriesaredescribedinChapter5.ItshouldbeespeciallyemphasizedthatParkandKim[35,36]success-fullyappliedtheHAMtosolveafewclassicproblemsinfinance,andgaveconvergenceconditionsfortheiranalyticapproximations.Itisratherinter-estingthattheyevengaveanerrorestimationfortheiranalyticapproxima-tionsin[36].Currently,ParkandKimusedtheHAMtosolveanabstractlinearproblemwithrespecttoboundedlinearoperatorsfromaBanachspacetoaBanachspace,andrigorouslyprovedthatthehomotopysolutionexistsinthesensethataseriesoftheproblemconvergesinaBanachnormsenseifthelinearoperatorsatisfiessomemildconditions.Theirfantasticworksareveryimportant,andmightpioneeranewresearchdirectionandstyle(i.e.abstractproof)intheframeoftheHAM.SuchkindofabstractmathematicaltheoremsintheframeoftheHAMaremorevaluableanduseful,ifnonlineargoverningequationsandespeciallytheinfluenceoftheconvergence-controlparameterontheconvergencecouldbeconsidered.Ontheotherside,theHAMhasbeensuccessfullyappliedtonumerousnonlinearproblemsinvariousfieldsofscienceandengineering.Theseap-plicationsshowthegeneralvalidityandnoveltyoftheHAM.Unfortunately,itisimpossibletomentionallofthemhereindetails.Asexamplesamongthesenumerousapplications,aHAM-basedapproachaboutboundary-layerflowsofnanofluidisgiveninChapter6ofthisbook.Inaddition,anappli-cationoftheHAMfortime-fractionalboundary-valueproblemisillustratedinChapter7.TosimplifysomeapplicationsoftheHAM,twoHAM-basedsoftwareweredeveloped.TheHAM-basedMaplepackageNOPH(version1.0.2)forperiodicoscillationsandlimitcyclesofnonlineardynamicsystemsisdescribedinChapter8ofthisbookwithvariousapplications.Itisfreeavailableonlineathttp://numericaltank.sjtu.edu.cn/NOPH.htmwithasimpleuser’sguide.Besides,theHAM-basedMathematicapackageBVPh(version2.0)forcouplednonlinearordinarydifferentialequations
26October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.118S.-J.LiaowithboundaryconditionsatmultiplepointsaregiveninChapter9ofthisbook.Itisfreeavailableonlineathttp://numericaltank.sjtu.edu.cn/BVPh.htmwithasimpleuser’sguideandsomeexamplesofapplication.BothofthesetwoHAM-basedsoftwareareeasy-to-useanduser-friendly.TheygreatlysimplifysomeapplicationsoftheHAM,andareespeciallyhelpfulforthebeginnersoftheHAM.1.4.RelationshipstoothermethodsInpuremathematics,thehomotopyisafundamentalconceptintopologyanddifferentialgeometry.TheconceptofhomotopycanbetracedbacktoJulesHenriPoincar´e(1854–1912),aFrenchmathematician.Ahomotopydescribesakindofcontinuousvariation(ordeformation)inmathematics.Forexample,acirclecanbecontinuouslydeformedintoasquareoranellipse,theshapeofacoffeecupcandeformcontinuouslyintotheshapeofadoughnutbutcannotbedistortedcontinuouslyintotheshapeofafootball.Essentially,ahomotopydefinesaconnectionbetweendifferentthingsinmathematics,whichcontainsamecharacteristicsinsomeaspects.Inpuremathematics,thehomotopyiswidelyusedtoinvestigateexistenceanduniquenessofsolutionsofsomeequations,andsoon.Inappliedmathematics,theconceptofhomotopywasusedlongagotodevelopsomenumericaltechniquesfornonlinearalgebraicequations.Theso-called“differentialarclengthhomotopycontinuationmethod”werepro-posedin1970sbyKeller[37,38].However,theglobalhomotopymethodscanbetracedasfarbackastheworkofLahaye[39]in1934.Tosolveanon-linearalgebraicequationf(x)=0bymeansofthehomotopycontinuationmethod,onefirstconstructssuchahomotopyH(x,q)=qf(x)+(1−q)g(x),(1.29)whereq∈[0,1]isthehomotopyparameter,g(x)isafunctionforwhichazeroisknownorreadilyobtained.AsdiscussedbyWayburnandSeader[40],thechoiceofg(x)isarbitrary,butthetwomostwidelyusedfunctionsaretheNewtonhomotopyH(x,q)=qf(x)+(1−q)[f(x)−f(x0)],andthefixed-pointhomotopyH(x,q)=qf(x)+(1−q)(x−x0),
27October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod19wherex0isanarbitrarystartingpoint.Thelocusofsolutionsdefinesthehomotopypaththatistrackedwithsomecontinuationmethod.Conse-quently,homotopycontinuationmethodsconsistnotonlyofthehomotopyequationitself,butalsothehomotopypathtrackingmethod,i.e.ofsomecontinuationstrategy.Homotopycontinuationmethodsareusuallybasedupondifferentiatingthehomotopyequation(1.29)withrespecttothearclengths,whichgivestheequation∂Hdx∂Hdq+=0.(1.30)∂xds∂qdsTakingintoaccountthearc-lengthrelation22dxdq+=1dsdsandtheinitialconditionH(x0,0)=0,weobtainaninitialvalueprob-lem.Then,pathtrackingbasedontheinitialvalueproblemisnumericallycarriedoutwithapredictor-correctoralgorithmtogainasolutionoftheoriginalequationf(x)=0.Someeleganttheoremsofconvergenceareprovedintheframeofthehomotopycontinuationmethod.Fordetailsofthehomotopycontinuationmethod,pleasereferto[41–49].Unlikethehomotopycontinuationmethodthatisaglobalconvergentnumericalmethodmainlyfornonlinearalgebraicequations,theHAMisakindofanalyticapproximationmethodmainlyfornonlineardifferentialequations.So,theHAMisessentiallydifferentfromthehomotopycon-tinuationmethod,althoughbothofthemarebasedonthehomotopy,thebasicconceptofthetopology.NotethattheHAMusesmuchmorecom-plicatedhomotopyequation(1.10)or(1.22)than(1.29)forthehomotopycontinuationmethod.Furthermore,theHAMprovideslargerfreedomtochoosetheauxiliarylinearoperatorL.Mostimportantly,theso-calledconvergence-controlparameterc0isintroducedforthefirsttime,tothebestofourknowledge,inthehomotopyequation(1.10)or(1.22)sothattheHAMprovidesusaconvenientwaytoguaranteetheconvergenceofseriesseries.Notethatthehomotopyequation(1.29)ofthehomotopycontinuationmethoddoesnotcontainsuchkindofconvergence-controlpa-rameteratall.So,theconvergence-controlparameterc0isindeedanovel.Infact,itistheconvergence-controlparameterc0whichdifferstheHAMfromallotheranalyticapproximationmethods.Inaddition,theHAMlogicallycontainsmanyotheranalyticapproxima-tionmethodsandthusisrathergeneral.Forexample,ithasbeengenerally
28October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.120S.-J.Liaoproved[10–13]thattheLyapunov’sartificialsmallparametermethod[5],theAdomiandecompositionmethod[6–8],theδ-expansionmethod[18]areonlyspecialcasesoftheHAMforsomespeciallychosenauxiliarylinearoperatorLandconvergence-controlparameterc0.Furthermore,theso-called“optimalhomotopyasymptoticmethod”[22]developedin2008isalsoaspecialcaseofthehomotopyequation(1.22)whenK→+∞,too,aspointedoutbyLiao(see§6.3ofLiao’sbook[11]).Especially,theso-called“homotopyperturbationmethod”(HPM)[19]proposedbyJihuanHein1998(sixyearslaterafterLiao[9]proposedtheearlyHAMin1992)wasonlyaspecialcaseoftheHAMwhenc0=−1,asprovedin[20],andthusithas“nothingnewexceptitsname”[20].SomeresultsgivenbytheHPMaredivergenteveninthewholeintervalexceptthegiveninitial/boundaryconditions,andthus“itisveryimportanttoinvestigatetheconvergenceofapproximationseries,otherwiseonemightgetuselessresults”,aspointedoutbyLiangandJeffrey[21].Formoredetails,see§6.2ofLiao’sbook[11].Inaddition,eventhefamousEulertransformincalculuscanbederivedintheframeoftheHAM(seeChapter5ofLiao’sbook[11]).ThisprovidesusatheoreticalcornerstoneforthevalidityandgeneralityoftheHAM.Insummary,basedontheconceptofhomotopytopology,theHAMisanovelanalyticapproximationmethodforhighlynonlinearproblems,withgreatfreedomandflexibilitytochooseequation-typeandsolutionex-pressionofhigh-orderapproximationequationsandalsowithaconvenientwaytoguaranteetheconvergence,sothatitmightovercomerestrictionsofperturbationtechniquesandothernon-perturbationmethods.1.5.Chanceandchallenge:somesuggestedproblemsAnytrulynewmethodsshouldgivesomethingnoveland/ordifferent,orsolvesomedifficultproblemsthatcannotbesolvedwithsatisfactionbyothermethods.Unlikeotheranalyticapproximationmethods,theHAMprovidesusgreatfreedomandflexibilitytochooseequation-typeandsolutionexpres-sionofhigh-orderapproximationequations,andespeciallyasimplewaytoguaranteetheconvergenceofsolutionseries.Thus,theHAMprovidesusalargepossibilityandchancetogivesomethingnovelordifferent,andtoattacksomedifficultnonlinearproblems.Forexample,somenewso-lutions[31,32]ofboundary-layerflowshavebeenfoundbymeansoftheHAM,whichhadbeenneglectedevenbynumericaltechniquesandhad
29October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod21beenneverreported.Someanalyticapproximationsfortheoptimalex-erciseboundaryofAmericanputoptionweregiven,whicharevalidfromacoupleofyears(see[50,51])uptoeven20years(seeChapter13ofLiao’sbook[11])priortoexpiry,andthusmuchbetterthantheasymp-totic/perturbationapproximationsthatareoftenvalidonlyforacoupleofdaysorweeks.Besides,theHAMhasbeensuccessfullyemployedtosolvesomecomplicatednonlinearPDEs:themultipleequilibrium-statesofreso-nantwavesindeepwater[33]andinfinitewaterdepth[34]werediscoveredbymeansoftheHAMforthefirsttime,tothebestofourknowledge,whichgreatlydeepenandenrichourunderstandingsaboutresonantwaves.Allofthesesuccessfulapplicationsshowtheoriginality,validityandgeneralityoftheHAMfornonlinearproblems,andencourageustoapplytheHAMtoattacksomefamous,challengingnonlinearproblems.SomeoftheseproblemsaresuggestedbelowfortheHAMcommunity,especiallyforbrave,enterprising,youngresearchers.1.5.1.PeriodicsolutionsofchaoticdynamicsystemsItiswellknownthatchaoticdynamicsystemshavetheso-called“but-terflyeffect”[52,53],say,thecomputer-generatednumericalsimulationshavesensitivedependencetoinitialconditions(SDIC).Forexample,thenonlineardynamicsystemofLorenzequations[52]x˙=σ(y−x),(1.31)y˙=rx−y−xz,(1.32)z˙=xy−bz,(1.33)haschaoticsolutionincaseofr=28,b=8/3andσ=10formostofgiveninitialconditionsx0,y0,z0ofx,y,zatt=0.However,forsomespecialinitialconditionssuchasx0=−13.7636106821,y0=−19.5787519424,z0=27;x0=−9.1667531454,y0=−9.9743951128,z0=27;x0=−13.5683173175,y0=−19.1345751139,z0=27,theabovedynamicsystemofLorenzequationhasunstableperiodicsolu-tions,asreportedbyViswanath[54].Aperiodicsolutionu(t)withtheperiodThasthepropertyu(t)=u(t+nT)
30October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.122S.-J.Liaoforarbitrarytimet≥0andarbitraryintegersn,evenift→+∞andn→∞.Thispropertycannotbecheckedstrictlybymeansofnumericalapproaches,sinceallnumericalintegrationsimulationsaregainedinafi-niteintervaloftime.Naturally,aperiodicsolutionshouldbeexpressedanalyticallybyperiodicbasefunctionssuchastrigonometricfunctions.So,theoreticallyspeaking,itisinherentlybettertouseanalyticapproximationmethodstosearchforperiodicsolutionsofchaoticdynamicsystemsthannumericalones.Infact,asillustratedbyLiaoinChapter13ofhisbook[10],theHAMcanbeemployedtogainperiodicsolutionofnonlineardynamicsystems.CanweemploytheHAMtogaintheabove-mentionedunstableperiodicsolutionsofLorenzequationfoundbyViswanath[54]?Moreimportantly,itwouldbeveryinterestingiftheHAMcouldbeemployedtofindsomenewperiodicsolutionsofLorenzequationwithphysicalparametersleadingtochaos!ThisismainlybecauseLorenzequationisoneofthemostfamousonesinnonlineardynamicsandnonlinearscience.1.5.2.PeriodicorbitsofNewtonianthree-bodyproblemLetusconsideroneofthemostfamousprobleminmechanicsandappliedmathematics:theNewtonianthree-bodyproblem,say,themotionofthreecelestialbodiesundertheirmutualgravitationalattraction.Letx1,x2,x3denotethethreeorthogonalaxes.Thepositionvectoroftheithbodyisexpressedbyri=(x1,i,x2,i,x3,i),wherei=1,2,3.LetTandLdenotethecharacteristictimeandlengthscales,andmithemassoftheithbody,respectively.UsingNewtoniangravitationlaw,themotionofthethreebodiesaregovernedbythecorrespondingnon-dimensionalequationsX3(xk,j−xk,i)x¨k,i=ρj3,k=1,2,3,(1.34)Ri,jj=1,j6=iwhere"#1/2X3R=(x−x)2(1.35)i,jk,jk,ik=1andmiρi=,i=1,2,3(1.36)m1denotestheratioofthemass.
31October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod23AccordingtoH.Poincar´e,orbitsofthree-bodyproblemareunintegrableingeneral.Althoughchaoticorbitsofthree-bodyproblemswidelyexist,threefamiliesofperiodicorbitswerefound:(1)theLagrange–Eulerfamily,datingbacktotheanalyticalsolutionsinthe18thcentury(onerecentorbitwasgivenbyMoore[55]);(2)theBroucke–Hadjidemetriou–H´enonfamily,datingbacktothemid-1970s[56–61];(3)theFigure-8family,discoveredin1993byMoore[55]andextendedtotherotatingcases[62–65].Notethatnearlyallofthesereportedperiodicorbitsareplanar.In2013,SuvakovandDmitraˇsinovi´c[66]foundthatthereexistfourclassesofplanarˇperiodicorbitsofthree-bodyproblem,withtheabovethreefamiliesbelong-ingtooneclass.Besides,theyreportedthreenewclassesofplanarperiodicorbitsandgavethecorrespondinginitialconditionsforeachclass.Forthedetailsoftheir15planarperiodicorbits,pleaserefertothegallery[67].SuvakovandDmitraˇsinovi´c[66]foundthesenewclassesofplanarperi-ˇodicorbitsbymeansofaniterativenumericalintegrationapproachwithoutusingmultipleprecision.So,itisunknownwhetherornotthenumericalsimulationsdepartthecorrespondingperiodicorbitsforratherlargetime,i.e.t→∞.Asmentionedbefore,itisbetterandmorenaturaltoexpressaperiodicsolutionu(t)withtheperiodTinseriesofperiodicbasefunctions(withthesameperiodT)sothatu(t)=u(t+nT)canholdforarbitraryintegernandarbitrarytimetevenift→∞.Thus,itisvaluabletoapplytheHAMtodoublecheckallofthereportedperiodicorbitsin[66],andmoreimportantly,tofindsomecompletelynewperiodicorbits!Notethatnearlyalloftheperiodicorbitsofthree-bodyproblemre-porteduptonowareplanar.Therefore,itisvaluableandinterestingiftheHAMcanbeappliedtofindsomeperiodicorbitsofNewtonianthree-bodyproblems,whicharenotplanar,i.e.threedimensional.Mathematicallyspeaking,weshoulddeterminesuchunknowninitialpositionsr1,r2,r3,unknowninitialvelocitiesr˙1,r˙2,r˙3andunknowncorrespondingmass-ratiosρ1,ρ2,ρ3ofthreebodiesintheframeoftheHAMthatEqs.(1.34)haveperiodicsolutionxk,i(t)=xk,i(t+nT)forarbitrarytimetandintegern,whereTistheunknowncorrespondingperiodtobedetermined,andi,k=1,2,3.Thisisavaluable,interestingbutchallengingproblemfortheHAMcommunity.
32October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.124S.-J.Liao1.5.3.ViscousflowpastasphereOneofthemostfamous,classicalprobleminfluidmechanicsisthesteady-stateviscousflowpastasphere[68–75],governedbytheNavier-Stokesequation,i.e.asystemofnonlinearpartialdifferentialequations.Considerthesteady-stateviscousflowpastasphereinauniformstream.Howlargeisthedragofthesphereduetotheviscosityoffluid?Tostudythesteady-stateviscousflowpastasphere,thesphericalco-ordinates~r=(r,θ,φ)isoftenused.Sincetheproblemhasaxialsymmetry,onecanusetheStokesstreamfunctionψ(r,θ)definedthroughthefollowingrelations:11vr=2ψθ,vθ=−ψr,vφ=0.(1.37)rsin(θ)rsin(θ)Thestreamfunctionψ(r,θ)isgovernedbythedimensionlessequationR∂(ψ,D2ψ)D4ψ=+2D2ψLψ,(1.38)r2∂(r,µ)subjecttotheboundaryconditionsψ(1,µ)=0,(1.38a)∂ψ(r,µ)=0,(1.38b)∂rr=1ψ(r,µ)12lim=(1−µ),(1.38c)r→∞r22whereR=aU∞/νistheReynoldsnumberandµ≡cos(θ),(1.39)∂21−µ2∂2D2≡+,(1.40)∂r2r2∂µ2µ∂1∂L≡+.(1.41)1−µ2∂rr∂µHere,adenotestheradiusofthesphereandU∞theuniformstreamvelocityatinfinity,respectively,accordingtothenotationofProudmanandPearson[71].AsmentionedbyLiao[74],thedragcoefficientreadsZ124∂ψCD=−pµ+dµ,(1.42)R∂r2−1r=1wherethepressurepisgivenbyZ11∂3ψp=−dµ.(1.43)µ(1−µ2)∂r3r=1
33October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod25Unfortunately,neitherthelinearizationmethod[68–70]northepertur-bationtechniques[71,72]canprovideananalyticapproximationofdragcoefficientCDvalidforRd>3,whereRd=dU∞/ν=2Rforthediam-eterdofthesphere.Especially,the3rd-ordermultiple-scaleperturbationapproximationofCDgivenbyChesterandBreach[72]wasvalideveninasmallerintervalofReynoldsnumberthanthe2nd-ordermultiple-scaleperturbationresultofProudmanandPearson[71].Thisimpliestheinva-lidityofperturbationmethodsforthisfamousproblem.So,“theideaofusingcreepingflowtoexpandintothehighReynoldsnumberregionhasnotbeensuccessful”,aspointedoutbyWhiteinhistextbook[76].Besides,themethodofrenormalizationgroupcannotessentiallymodifytheseanalyticresults[75],either.In2002,Liao[74]employedtheHAMtosolvethesteady-stateviscousflowpastasphereandgainedaanalyticapproximationofdragcoefficientCD,whichagreewellwithexperimentaldatainaconsiderablylargerin-tervalRd≤30.However,thecorrespondingexperimentsindicatethatthesteady-stateviscousflowpastasphereexistsuntilRd≈100.So,strictlyspeaking,thisHAMresultgivenin[74]isnotsatisfactory.Theoreticallyspeaking,itisveryinterestingandvaluableifonecangiveanaccurateenoughanalyticresultofthedragcoefficientCDvalidforthesteady-stateviscousflowpastasphereuptoRd≈100,mainlybecauseitisoneofthemostfamous,classicalproblemsinfluidmechanicswithahistoryofmorethan150year!Canwesolvethisfamous,classicalproblembymeansoftheHAM?1.5.4.ViscousflowpastacylinderThesteady-stateviscousflowpastaninfinitecylinderisalsooneofthemostfamous,classicalproblemsinfluidmechanicswithalonghistory.Forthesteady-stateviscousflowpastaninfinitecylinder,itisnaturaltousecylindricalcoordinates~r=(r,θ,z).Sincetheproblemistwodimensional,itisconvenienttousetheLagrangianstreamfunctionψ(r,θ)definedbyProudmanandPearson[71]:1∂ψ∂ψur=,uθ=−,uz=0.(1.44)r∂θ∂rThestreamfunctionψ(r,θ)isgovernedbyR∂(ψ,∇2)∇4ψ(r,θ)=−r,(1.45)rr∂(r,θ)
34October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.126S.-J.Liaosubjecttotheboundaryconditionsψ(r=1,θ)=0,(1.45a)∂ψ(r,θ)=0,(1.45b)∂rr=1ψ(r,θ)lim=sin(θ),(1.45c)r→∞rwhere∂21∂1∂22422∇r≡∂r2+r∂r+r2∂θ2,∇r≡∇r∇r.Here,R=aU∞/νistheReynoldsnumber,aandU∞denotetheradiusofcylinderandtheuniformstreamvelocityatinfinity,respectively.Asreviewedin[75],neitherthelinearizationmethodnorperturbationtechniquecangivegoodanalyticapproximationofthedragcoefficientCDofacylinderforR≥3.Infact,eventhemethodofrenormalizationgroupcannotmodifytheseresultsgreatly[75].So,itisstillanopenquestion.Theoreticallyspeaking,itisvaluabletogainanaccurateanalyticexpres-sionofdragcoefficientCDvalidforlargeReynoldsnumberuptoR≈40,beyondwhichtheperiodicVonKarm´anvortexoccurs.Thisismainlybe-causeitisoneofthemostfamous,historicalprobleminfluidmechanics.Canthisfamous,classicalproblembesolvedbymeansoftheHAM?1.5.5.NonlinearwaterwavesTheHAMhasbeensuccessfullyappliedtosolvesomenonlinearwaveequa-tions.Especially,intheframeoftheHAM,themultipleequilibrium-statesofresonantwavesindeepwater[33]andinfinitewaterdepth[34]werediscoveredforthefirsttime,tothebestoftheauthor’sknowledge.Thus,theHAMprovidesusaconvenienttooltoinvestigatesomecomplicatedwaveproblems.Strictlyspeaking,waterwavesaregovernedbyEulerequationwithtwononlinearboundaryconditionssatisfiedonanunknownfreesurface,whichhoweverareratherdifficulttosolveingeneral.BasedontheexactEulerequation,somesimplifiedwavemodelsforshallowwaterwaves,suchastheKdVequation[77],Boussinesqequation[78],Camassa–Holm(CH)equa-tion[79],andsoon,arederivedbyassumingtheexistenceofsomesmallphysicalparametersinshallowwater.AlthoughtheseshallowwaterwaveequationsaremuchsimplerthantheexactEulerequation,theycanwellex-
35October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.1ChanceandChallenge:ABriefReviewofHomotopyAnalysisMethod27plainmanyphysicalphenomena,suchassolitonwaves,wavepropagationsandinteractionsinshallowwater,wavebreaking,andsoon.Forexample,thecelebratedCamassa–Holm(CH)equation[79]ut+2ωux−uxxt+3uux=2uxuxx+uuxxx,(1.46)subjecttotheboundaryconditionu=0,ux=0,uxx=0,asx→±∞,(1.47)canmodelbothphenomenaofsolitoninteractionandwavebreaking(see[80]),whereu(x,t)denotesthewaveelevation,x,tarethetemporalandspatialvariables,ωisaconstantrelatedtothecriticalshallowwaterwavespeed,thesubscriptdenotesthepartialdifferentiation,respectively.Mathe-matically,theCHequationisintegrableandbi-Hamiltonian,thuspossessesaninfinitenumberofconservationlawsininvolution[79].Inaddition,itisassociatedwiththegeodesicflowontheinfinitedimensionalHilbertmani-foldofdiffeomorphismsofline(see[80]).Thus,theCHequation(1.46)hasmanyintriguingphysicalandmathematicalproperties.AspointedoutbyFushssteiner[81],theCHequation(1.46)even“hasthepotentialtobecomethenewmasterequationforshallowwaterwavetheory”.Especially,whenω=0,theCHequation(1.46)hasthepeakedsolitarywaveu(x,t)=cexp(−|x−ct|),whichwasfoundfirstbyCamassaandHolm[79].Thefirstderivativeofthepeakedsolitarywaveisdiscontinuousatthecrestx=ct.LiketheCHequation,manyshallowwaterequationsadmitpeakedand/orcuspedsolitarywaves.Theseequationswithpeakedand/orcuspedsolitarywaveshavebeenwidelyinvestigatedmathematically,andthousandsofrelatedar-ticleshavebeenpublished.However,tothebestoftheauthor’sknowledge,peakedandcuspedsolitarywaveshaveneverbeengaineddirectlyfromtheexactEulerequation!Thisisverystrange.Logicallyspeaking,sincethesesimplifiedequations(liketheCHequation)aregoodenoughapprox-imationsoftheEulerequationinshallowwater,theexactEulerequationshouldalsoadmitthepeakedand/orcuspedsolitarywavesaswell.Canwegainsuchkindofpeakedand/orcuspedsolitarywavesoftheexactwaveequationbymeansoftheHAM,iftheyindeedexist?Eitherpositiveornegativeanswerstothisquestionhaveimportantscientificmean-ings.Ifsuchkindofpeakedsolutionsoftheexactwaveequationindeedexist,itcangreatlyenrichanddeepenourunderstandingsaboutpeaked
36October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.128S.-J.Liaosolitarywaves.Ifthepeakedsolitarywavesgivenbytheexactwaveequa-tionexistsmathematicallybutisimpossibleinphysics,wehadtocheckthephysicalvalidityofthepeakedsolitarywaves.So,thisisaninterestingandvaluablework,althoughwithgreatchallenge.Forsomeattemptsinthisdirection,pleaserefertoLiao[82],whoproposedageneralizedwavemodelbasedonthesymmetryandthefullynonlinearwaveequations,whichadmitsnotonlythetraditionalwaveswithsmoothcrestbutalsopeakedsolitarywaves.ItisfoundthatthepeakedsolitarywavessatisfyKelvin’stheoremeverywhere.Besides,thesepeakedsolitarywavesincludethefa-mouspeakedsolitarywavesoftheCamassa–Holmequation.So,thegener-alizedwavemodel[82]isconsistentwiththetraditionalwavetheories.Itisfound[82]thatthepeakedsolitarywaveshavesomeunusualcharacter-isticsquitedifferentfromthetraditionalones,althoughitisstillanopenquestionwhetherornottheyarereasonableinphysicsiftheviscosityoffluidandthesurfacetensionareconsidered.Inaddition,theso-called“roguewave”[83,84]isahottopicofnonlinearwaves.Certainly,itisvaluabletoapplytheHAMtodosomeinvestigationsinthisfield.Insummary,itistruethattheproblemssuggestedaboveareindeeddif-ficult,butveryvaluableandinterestingintheory.Infact,therearemanysuchkindofinterestingbutdifficultproblemsinscience,engineeringandfinance.Itshouldbeemphasizedthat,unlikeallotheranalyticapprox-imationmethods,theHAMprovidesusgreatfreedomandflexibilitytochooseequation-typeandsolutionexpressionofhigh-orderapproximationequations,andbesidesaconvenientwaytoguaranteetheconvergenceofsolutionseries.AspointedoutbyGeorgCantor(1845–1918),“theessenceofmathematicsliesentirelyinitsfreedom”.Hopefully,thegreatfreedomandflexibilityoftheHAMmightcreatesomenovelideasandinspiresomebrave,enterprising,youngresearcherswithstimulatedimaginationtoat-tackthemwithsatisfactory,muchbetterresults.Chancealwaysstayswithchallenges!AcknowledgmentThisworkispartlysupportedbyNationalNaturalScienceFoundationofChina(ApprovalNo.11272209),theFoundationforShanghaiLeadingScientists,andStateKeyLaboratoryofOceanEngineering(ApprovalNo.GKZD010056).
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43October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2Chapter2PredictorHomotopyAnalysisMethod(PHAM)SaeidAbbasbandy∗andElyasShivanian†DepartmentofMathematicsImamKhomeiniInternationalUniversityGhazvin,Iran∗abbasbandy@yahoo.com,†shivanian@sci.ikiu.ac.irThischapterintroducesamethodtopredictthemultiplicityoftheso-lutionsofnonlinearboundaryvalueproblems.Thisprocedurecanbeeasilyappliedonnon-linearordinarydifferentialequationswithbound-aryconditionssothatitcalculateseffectivelytheallbranchesofthesolutions(ontheconditionthat,thereexistsuchsolutionsfortheprob-lem)analyticallyatthesametime.Inthismanner,forpracticaluseinscienceandengineering,thismethodmightgivenewunfamiliarclassofsolutionswhichisoffundamentalinterest.Contents2.1.Preliminaries....................................362.2.Descriptionofthemethod.............................372.2.1.Zeroth-orderdeformationequation.....................382.2.2.High-orderdeformationequation......................392.2.3.Predictionofthemultiplesolutions....................402.3.Convergenceanalysis................................482.4.Someillustrativemodels..............................542.4.1.Nonlinearproblemarisinginheattransfer................552.4.2.StronglynonlinearBratu’sequation....................602.4.3.Nonlinearreaction-diffusionmodel.....................662.4.4.Mixedconvectionflowsinaverticalchannel...............712.5.Concludingremarks.................................79References.........................................8135
44October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.236S.AbbasbandyandE.Shivanian2.1.PreliminariesManyofthemathematicalmodelingofthephysicalphenomenainscienceandengineeringoftenleadtononlineardifferentialequations.Therearealotofmethods,fromthepastuptonow,togivenumericallyapproximatesolutionsofnonlineardifferentialequationssuchasEulermethod,Runge-Kuttamethod,multistepmethod,Taylorseriesmethod,Hybridmethods,familyoffinitedifferencemethods[1,2],familyoffiniteelementmeth-ods[3],meshlessmethods,differentialquadrature,spectralmethods[4–6]etc.Therearealsomanyofmethodswhichgiveanalyticallyapproximatesolutionslikeforexampleperturbationmethods[7,8],theartificialsmallparametermethod[9],theδ-expansionmethod[10],andrecentlynotewor-thytechniquesconsistsoftheAdomiandecompositionmethod[11],thevariationaliterationmethod[12]andsoon.Inthisregard,onemayaskthequestion:Dotheapproximatemethodsenabletopredictmultiplicityofsolutionsofthenonlineardifferentialequations?Intheotherwords,canweforecastexistenceofmultiplesolutionsofnonlinearequationsbyap-proximatemethodsandatthesametimeobtainallbranchesofsolutions?Answertothisquestionissomehowdifficultbyconcerningthattheapprox-imatemethodsusuallyconvergetoonesolutionbyoneinitialguessthatisexactlymeaningof“convergence”.Nevertheless,thepresentchapterisgoingtoinfractthisconventionbyintellectualusinghomotopyanalysismethod.Thepurposeofthepresentchapteristointroduceamethod,probablyforthefirsttime,topredictthemultiplicityofthesolutionsofnonlinearboundaryvalueproblems.Thisprocedurecanbeeasilyappliedonnonlin-earordinarydifferentialequationswithboundaryconditions.Thismethod,aswillbeseen,besidesanticipatingofmultiplicityofthesolutionsofthenonlineardifferentialequations,calculateseffectivelyallbranchesoftheso-lutions(ontheconditionthat,thereexistsuchsolutionsfortheproblem)analyticallyatthesametime.Inthismanner,forpracticaluseinscienceandengineering,thismethodmightgivenewunfamiliarclassofsolutionswhichisoffundamentalinterestandfurthermore,theproposedapproachconvincestoapplyitonnonlinearequationsbytoday’spowerfulsymbolicsoftwareprogramssothatitdoesnotneedtediousstagesofevaluationandcanbeusedwithoutstudyingthewholetheory.Infact,thistechniquehasnewpointofviewtohomotopyanalysismethod.AsitiswellknownintheframeofHAM,theconvergence-controllerparameterplaysimportantroletoguaranteetheconvergenceofthesolutionsofnonlineardifferentialequa-
45October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)37tions.Itisshownthattheconvergence-controllerparameterplaysafunda-mentalroleinthepredictionofmultiplicityofsolutionsandallbranchesofsolutionsareobtainedsimultaneouslybyoneinitialapproximationguess,oneauxiliarylinearoperatorandoneauxiliaryfunction.Notwithstandingboththeperturbationandnon-perturbationmethodssuchastheartificialsmallparametermethod,theδ-expansionmethod,theAdomiandecompositionmethodandvariationaliterationmethodcannotprovideuswithasimplewaytoadjustandcontroltheconvergenceregionandrateofgivenapproximateseries,theHAMcanguaranteetheconver-genceoftheseriessolutionsbyconvergence-controllerparameter~.Infact,thistraitmakesHAMtobedifferentfromtheothersanalyticaltechniqueswhichareusedtoapproachtojustonesolutionandsothatpossiblytolosetheothersolutions.Thereforeinthisway,presentproceduremaybegener-atesnewclassofsolutionsforfurtherphysicalinterpretationsinengineeringandsciences.Thelegitimacyandreliabilityofthemethodischeckedbyitsapplica-tiontofourimportantnonlinearequationsnamelynonlinearheattransferequation[13,14],stronglynonlinearBratu’sequation[15,16],nonlinearreaction-diffusionequation[17]andthemodelofmixedconvectionflowsinaverticalchannel[18,19].Allthesenonlinearproblemsadmitmultiplesolutionsforsomevaluesofparametersoftheequations.2.2.DescriptionofthemethodToillustratetheprocedureconsiderthefollowingnonlineardifferentialequation:N[u(r)]=0,r∈Ω,(2.1)withboundaryconditions∂uBu,=0,r∈Γ,(2.2)∂nwhereNisgeneralnonlinearoperator,Bisaboundaryoperator,andΓistheboundaryofthedomainΩ.Thecrucialstepofthetechniqueisthattheboundaryvalueproblem(2.1)and(2.2)shouldbereplacedbyequivalentproblemsothattheconditions(2.2)involveanunknownparameterlikeδ(prescribedparameter)andaresplitto0∂uBu,δ,=0,r∈Γandu(α)=β,(2.3)∂n
46October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.238S.AbbasbandyandE.Shivanianwhereu(α)=βistheforcingconditionthatcomesfromoriginalconditions(2.2).Now,homotopyanalysismethodisappliedontheproblem(2.1)withtheconditions(2.3)exceptforcingconditionasfollows:N[u(r)]=0,r∈Ω,(2.4)0∂uBu,δ,=0,r∈Γ.(2.5)∂n2.2.1.Zeroth-orderdeformationequationWesupposethatallthesolutionsu=u(r)ofproblem(2.4)canbeexpressedbythesetofbasefunctions{ωi(r),i=0,1,2,...}intheformX+∞u=u(r)=anωn(r),(2.6)n=0whereanarecoefficientstobedetermined.Letu0(r,δ)denoteaninitialapproximationguessoftheexactsolutionu(r)whichsatisfiesboundaryconditions(2.5)automatically.Also,asthatiswellknownintheframeofHAM,assume~6=0denoteconvergence-controllerparameter,H(r)6=0anauxiliaryfunction,andLanauxiliarylinearoperator.Nowusingp∈[0,1]asanembeddingparam-eter,weconstructthegeneralzeroth-orderdeformationequationandthecorrespondingboundaryconditionsasfollow:(1−p)L[ϕ(r,δ;p)−u0(r,δ)]=p~H(r)N[ϕ(r,δ;p)],(2.7)0ϕ(r,δ;p)Bϕ(r,δ;p),δ,=0,r∈Γ,(2.8)∂nwhereϕ(r,δ;p)isanunknownfunctiontobedetermined.Whenp=0,thezeroth-orderdeformationequation(2.7)becomesL[ϕ(r,δ;0)−u0(r,δ)]=0,(2.9)whichgivesϕ(r,δ;0)=u0(r,δ).Whenp=1,Eq.(2.7)leadstoN[ϕ(r,δ;1)]=0,(2.10)whichisexactlythesameastheoriginalEq.(2.1)providedthatϕ(r,δ;1)=u(r,δ).
47October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)39Wenowexpandthefunctionϕ(r,δ;p)inaTaylorseriestotheembeddingparameterp.ThisTaylorexpansioncanbewrittenintheformX+∞ϕ(r,δ;p)=u(r,δ)+u(r,δ)pm,(2.11)0mm=1where1∂mϕ(r,δ;p)um(r,δ)=,m=0,1,2,...,+∞.(2.12)m!∂pmAsitiswellknowninduringtheframeofHAM[20,21],whenthelinearoperatorL,theinitialapproximationu0(r,δ),theauxiliaryparameter~6=0,andtheauxiliaryfunctionH(r)6=0arechosenproperly,theseries(2.11)convergesforp=1,andthusX+∞X+∞u(r,δ)=u0(r,δ)+um(r,δ)=anωn(r),(2.13)m=1n=0willbethesolutionofthenonlinearproblem(2.4)and(2.5)aswillbeprovedlater.2.2.2.High-orderdeformationequationAssumethatthelinearoperatorL,theinitialapproximationu0(r,δ),andtheauxiliaryfunctionH(r)6=0arechosenproperly(itisworthmen-tioningherethat~6=0so-calledconvergence-controllerparameterwillbedeterminedlater),theunknownfunctionsum(r,δ)inEq.(2.13)canbedeterminedwiththeaidofthehigh-orderdeformationequationsasfol-−→lows.Atfirstwedefinethevectorun={u0(r),u1(r),...,un(r)}then,differentiatingthezeroth-orderdeformationequation(2.7)mtimeswithrespecttotheembeddingparameterp,dividingitbym!,settingsubse-quentlyp=0andtakingintoaccounttheboundaryconditions(2.8),oneobtainsthemth-orderdeformationequation−→L[um(r,δ)−χmum−1(r,δ)]=~H(r)Rm(um−1,r,δ),(2.14)subjecttotheboundaryconditions∂mϕ(r,δ;p)B0ϕ(r,δ;p),δ,=0,r∈Γ,(2.15)∂pm∂np=0where0,m≤1,χm=(2.16)1,m>1,
48October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.240S.AbbasbandyandE.Shivanianand1∂m−1N[ϕ(r,δ;p)]Rm(−→um−1,r,δ)=(m−1)!∂pm−1p=0hPi∂m−1Nn=+∞u(r,δ)pn1n=0n=.(2.17)(m−1)!∂pm−1p=0Thehigh-orderdeformationequation(2.14)obviously,isjusttheordinarydifferentialequationwithboundarycondition(2.15)and,canbeeasilysolvedbyusingsomesymbolicsoftwareprogramssuchasMathematicaorMaple.Inthisway,startingbyu0(r,δ),weobtainthefunctionsum(r,δ)form=1,2,3,...fromEqs.(2.14)and(2.15)successively.Accordingly,theMth-orderapproximatesolutionoftheproblems(2.4)and(2.5)isgivenby0XMXMu(r,δ)≈UM(r,δ,~)=u0(r,δ)+um(r,δ)=anωn(r).(2.18)m=1n=02.2.3.PredictionofthemultiplesolutionsItisnoteworthytoindicatethatuptothisstage,thelinearoperatorL,theinitialapproximationu0(r,δ),andtheauxiliaryfunctionH(r)6=0havebeenchosenproperlysothattheseriessolutions(2.18)wouldbeconver-gence.However,therearestilltwounknownparametersinseries(2.18)namelyδ(prescribedparameter)and~(convergence-controllerparameter)whichshouldbedetermined.Itisessentialthatexistenceofuniqueormul-tiplesolutionsintermsofthebasicfunctions(2.6)fortheoriginalboundaryvalueproblem(2.1)dependsonthefactwhethertheforcingcondition(2.3)(u(α)=β),admitsuniqueormultiplevaluesfortheformallyintroducedparameterδintheboundaryconditions(2.3).Thisstageiscalledruleofmultiplicityofsolutionsthatisacriterioninordertoknowhowmanysolutionstheboundaryvalueproblem(2.1)admits.Theso-calledruleofmultiplicityofsolutionsisappliedasfollows:ConsidertheMth-orderapproximatesolution(2.18)andsetintheforcingcondition(2.3)(u(α)=β),sothefollowingequationisderivedu(α)≈Um(α,δ,~)=β.(2.19)Theaboveequationhastwounknownparametersnamelyδ,and~whichcontrolstheconvergenceoftheHAMseries(2.18).ItisabasicfeatureofHAMthattheseriessolution(2.18)convergesatr=αonlyinthatrange
49October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)41of~,wheretheparameterδdoesnotchangewiththevariationof~.Thismeansthatintheplotofδasfunctionof~accordingtoEq.(2.19)inanimplicitlyway,intheconvergencerangeoftheseriesu(α)aplateauoccurs.Thenumberofsuchhorizontalplateauswhereδ(~)becomesconstant,givesthemultiplicityofthesolutionsofproblems(2.1)and(2.2),(wecallthispointasruleofmultiplicityofsolutions).Inthefollowing,wegiveaprooftotheaboveassertion.Wenoticethatthereisadirectconnectionbetweentheexistencesofmultiplesolutionsandthenumberofhorizontalplateausintheplotofδ(~),soatfirst,letusdiscussaboutthefundamentalof~-curveanditsrelationtotheTaylor’sseries.Thefollowingtheoremisbasicandsimpletounderstand.Theorem2.1.Iff(x)iscontinuouson[a,b]andisdifferentiablein(a,b)andf(a)=f(b)=0,thenthereexistsatleastonepointξ∈(a,b)suchthat0f(ξ)=0.Proof.SeeRef.[22].Now,wediscussaboutoccurrenceof~-curveforthearbitrarybutsmoothfunctionf(x).Theorem2.2.Supposethatg(~)beacontinuousfunctionontointerval[a,b]andallderivativesoff:[a,b]→RexistandhaveacommonMsothatmax|f(k)(x)|≤M,forallk.(2.20)x∈[a,b]Furthermore,assumethatGn(x,α)betheTaylorpolynomialofdegreenforf(x)aboutsomeα∈(a,b),sayα=g(~),then∀ε>0andγ∈(a,b)thereexistsN∈Nandinterval(c,d)sothat∀~∈(c,d)andn≥N:|f(γ)−Gn(γ,g(~))|<ε.(2.21)Proof.Letβ∈[a,b]bethepointatwhichwewanttodeterminetheerror.Wesuppose(withoutlossofgenerality)thatβ>α.LetXn(β−x)k(k)s(x)=f(β)−f(x)−f(x),(2.22)k!k=10thens(x)existsforx∈(a,b)andn0(β−x)(n+1)s(x)=−f(x).(2.23)n!
50October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.242S.AbbasbandyandE.ShivanianNow,considerthefunctionn+1β−xU(x)=s(x)−s(α),(2.24)β−αthenU(α)=U(β)=0.(2.25)n+1β−xFromthedifferentiabilityofs(x)and,itfollowsthatU(x)isβ−αdifferentiableonanysubintervalof(a,b).Now,weapplyTheorem2.1toU(x)ontheinterval[α,β].Thereforethereexistsξβ∈(α,β)sothat0U(ξβ)=0,(2.26)whichimpliesnn(β−ξβ)(n+1)(β−ξβ)−f(ξβ)+(n+1)n+1s(α)=0,(2.27)n!(β−α)then,sinceβ6=ξβ,n+1(β−α)(n+1)s(α)=f(ξβ).(2.28)(n+1)!Now,fromEq.(2.22),wehaven+1(β−α)(n+1)s(α)=f(β)−Gn(β,α)=f(ξβ).(2.29)(n+1)!Sincetheparameterβhasbeenchosenarbitrarythenn+1(x−α)(n+1)∀x∈[α,b],α∈[a,b]:f(x)−Gn(x,α)=f(ξx),(2.30)(n+1)!whereξx∈(α,x).Supposethatγ∈(a,b)andε>0,letα∈(a,γ)thenitisclearthatthereexistsNsuchthatn+1(γ−α)ε∀n≥N:<,(2.31)(n+1)!MthenfromEqs.(2.20)and(2.30),n+1(γ−α)(n+1)ε|f(γ)−Gn(γ,α)|=|f(ξγ)|<·M=ε.(2.32)(n+1)!MTherefore,wehaveproved∀γ∈(a,b),α∈(a,γ),ε>0,∃N⇒∀n≥N:|f(γ)−Gn(γ,α)|<ε.
51October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)43Sinceg(~)iscontinuousfunctionontointerval[a,b]thenthereexistsin-terval(c,d)suchthatg{(c,d)}=(a,γ).Hencethestatement(2.32)isreadequivalentlyas∀γ∈(a,b),~∈(c,d),ε>0,∃N⇒∀n≥N:|f(γ)−Gn(γ,g(~))|<ε,andtheproofiscompleted.Corollary2.1.Supposethatf(x)issufficientlysmoothoninterval[a,b],g(x)beacontinuousfunctionontointerval[a,b]andGn(x,g(~))betheTaylorpolynomialofdegreenaboutg(~).Then,asngoesinfinity,intheplotofGn(γ,g(~)),a<γ 52October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.244S.AbbasbandyandE.Shivanian6Γ=-2,-1,0,0.25,0.5,0.75,1,1.25,1.5543210-2-1012Fig.2.1.PlotofG10(γ,g1(~))via~inExample2.1.6Γ=-2,-1,0,0.25,0.5,0.75,1,1.25,1.5543210-4-3-2-10123Fig.2.2.PlotofG10(γ,g2(~))via~inExample2.1.Thecondition(2.20)isverysevereinTheorem2.2,forexample,theTaylorseriesofthefunctionf(x)=1aboutzerohasradiusofconver-1+xgenceR=1andmoreoverthederivativesoff(x)areunboundedbut,aswewillseeinthissection,Corollary2.1stillholdsforγ6=−1.ThefollowingtheoremdescribesthatTheorem2.2stillholdsbyaweaklyconditions.Theorem2.3.Supposethatg(~)beacontinuousfunctionontointerval00Pnk[a,b]andGn(x,α)=k=0ak(α)(x−α)betheTaylorpolynomialofde-00greenforf(x)aboutsomeα∈(a,b),sayα=g(~).Moreover,assume 53October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)451.00.50.0-0.5-1.0Γ=0,6,7,1,5,8,2,4,3-1.5-4-2024Fig.2.3.PlotofG25(γ,g1(~))via~inExample2.2.1.00.50.0-0.5-1.0Γ=0,6,7,1,5,8,2,4,3-1.5-15-10-50510Fig.2.4.PlotofG25(γ,g2(~))via~inExample2.2.that00ak(α)∀k∈N,α∈[a,b],x∈[a,b]:|x−α|≤||.(2.37)ak+1(α)Thenforε>0andγ∈[a,b],thereexistsN∈Nandinterval(c,d)sothat∀~∈(c,d)andn≥N:|f(γ)−Gn(γ,g(~))|<ε.(2.38)Proof.Fromthestatement(2.37),itfollowsthatthereexists0<θ<1sothat00∀k∈N,α∈[a,b],x∈[a,b]:|ak+1(α)||x−α|≤θ|ak(α)|.(2.39) 54October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.246S.AbbasbandyandE.ShivanianToprovethesentence(2.38),weshowthat,forafixedγ∈[a,b],Gn(γ,α),00∞α∈(a,b)isconvergencesequence.Weshowthat{Gn(γ,α)}n=0isCauchysequenceintheHilbertspaceR.From(2.39),itisobtainedn+1nkGn+1(γ,α)−Gn(γ,α)k=|an+1(α)|k(γ−α)k≤θ|an(α)|k(γ−α)k≤θ2|a(α)|k(γ−α)kn−1≤···≤θn−p+1|a(α)|k(γ−α)kp,(2.40)n−1pnow,foreveryn,m∈N,n≥m>p,wehavekGn(γ,α)−Gm(γ,α)k=k(Gn(γ,α)−Gn−1(γ,α))+(Gn−1(γ,α)−Gn−2(γ,α))+···+(Gm+1(γ,α)−Gm(γ,α))k≤k(Gn(γ,α)−Gn−1(γ,α))k+k(Gn−1(γ,α)−Gn−2(γ,α))k+···+kG(γ,α)−G(γ,α)k≤θn−p|a(α)|k(γ−α)kpm+1mp+θn−p−1|a(α)|k(γ−α)kp+···+θm−p+1|a(α)|k(γ−α)kppp1−θn−m=θm−p+1|a(α)|k(γ−α)kp.(2.41)p1−θTherefore,wearrivetolimkGn(γ,α)−Gm(γ,α)k=0,(2.42)m,n→∞∞thenitfollowsthat{Gn(γ,α)}n=0converges.OntheotherhandGn(γ,α)istheTaylorpolynomialoff(x)atx=γ.Henceforeachε>0,thereexistsN∈Nsothat∀n≥N:|f(γ)−Gn(γ,α)|<ε.(2.43)00Nowbythisfactthatαischosenarbitrarilyfrom(a,b),wecanequiv-alentlysaythatforeachε>0thereexistsN∈Nandinterval(c,d)sothat∀~∈(c,d)andn≥N:|f(γ)−Gn(γ,g(~))|<ε,(2.44)andtheproofiscompleted.Example2.3.Considerf(x)=1andg(~)=−1−1(g(~)can1+x~bechosenotherpiecewise-continuousfunctionsontoRexceptsomepoints,wehavechosenthisfunctionduetoconformingaforesaidrefer-00ences.Let[a,b]=[−6,−2]and[a,b]=[−8,−7]inTheorem2.3,sincek(−1)ak(α)=k+1forallk∈Nthen(1+α)ak(α)=1+α.(2.45)ak+1(α) 55October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)47Itiseasytoseethatak(α)∀α∈[−8,−7]andx∈[−6,−2]:|x−α|≤|1+α|=,(2.46)ak+1(α)thenconditionsofTheorem2.3hold.TheTaylorpolynomialofdegreethreeforf(x)aboutg(~)isgivenasfollows3!24131G3(γ,g(~))=~−+γ+1−~+γ+1~~21−~+γ+1−~.(2.47)~Wehaveshown~-curveofG20(γ,g(~))atγ=−6,−5,−4,−3,and−2inFig.2.5.Asitisseen,itcanbefoundhorizontallineforeachcase.Infact,0000ifweleta 56October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.248S.AbbasbandyandE.Shivanian0.0Γ=–2,–3,–4,–5,–6-0.2-0.4-0.6-0.8-1.0-1.20.00.51.01.52.0Fig.2.5.PlotofG20(γ,g(~))via~inExample2.3.1.00.8Γ=0,1/3,1/20.60.40.20.0-0.2-0.4-2.0-1.5-1.0-0.50.0Fig.2.6.PlotofG10(γ,g(~))via~inExample2.4.2.3.ConvergenceanalysisTheorem2.4.Let0<γ<1andthesolutioncomponentsu0(r,δ),u1(r,δ),u2(r,δ),...obtainedby(2.14)satisfythefollowingcondition∃k0∈N,∀k≥k0:kuk+1(r,δ)k≤γkuk(r,δ)k,(2.50)P+∞thentheseriessolutionk=0uk(r,δ)isconvergent. 57October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)49+∞Proof.Definethesequence{Sn}n=0as,S0=u0(r,δ)S1=u0(r,δ)+u1(r,δ)S2=u0(r,δ)+u1(r,δ)+u2(r,δ)(2.51)...Sn=u0(r,δ)+u1(r,δ)+···+un(r,δ)+∞andweshowthat{Sn}n=0isaCauchysequenceintheHilbertspaceR.Forthispurpose,consider,kSn+1−Snk=kun+1(r,δ)k≤γkun(r,δ)k≤γ2ku(r,δ)k≤...≤γn−k0+1ku(r,δ)k.(2.52)n−1k0Foreveryn,m≥N,n≥m>k0,wehavekSn−Smk≤k(Sn−Sn−1)+(Sn−1−Sn−2)+···+(Sm+1−Sm)k≤k(Sn−Sn−1)k+k(Sn−1−Sn−2)k+···+k(Sm+1−Sm)kn−k0n−k0−1≤γkuk0(r,δ)k+γkuk0(r,δ)k+...+γm−k0+1ku(r,δ)kk01−γn−mm−k0+1=γkuk0(r,δ)k,(2.53)1−γandsince0<γ<1,weget,limkSn−Smk=0.(2.54)n,m→∞+∞Therefore,{Sn}n=0isaCauchysequenceintheHilbertspaceRanditimpliesthattheseriessolutiondefinedin(2.18),converges.ThiscompletestheproofofTheorem2.4.P+∞Theorem2.5.Assumethattheseriessolutionk=0uk(r,δ)definedin(2.18),isconvergenttothesolutionu(r).IfthetruncatedseriesPMUM(r,δ,~)=m=0um(r,δ)isusedasanapproximationtothesolutionu(r)oftheproblem(2.14),thenthemaximumabsolutetruncatederrorisestimatedas,1M−k0+1ku(r)−UM(r,δ,~)k≤γkuk0(r,δ)k.(2.55)1−γ 58October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.250S.AbbasbandyandE.ShivanianProof.FromTheorem2.4,followinginequality(2.53),wehave1−γn−MkS−Sk≤γM−k0+1ku(r,δ)k,(2.56)nMk01−γforn≥M.Now,asn→∞thenS→u(r)andγn−M→0.So,n1M−k0+1ku(r)−UM(r,δ,~)k≤γkuk0(r,δ)k.(2.57)1−γTheorems2.3and2.4togetherconfirmthattheconvergenceofseriesso-lution(2.18)leadstooccurrenceofhorizontalplateausin~-curve,inwhichtheygivevalidregionfortheconvergencecontrollerparameter~,whereUM(r,δ,~)converges.Now,wediscussaboutuniquenessofthesolutionofinitialvalueproblem(2.4)–(2.5)whichplaysfundamentalruleintheexis-tenceofmultiplesolutionsoftheboundaryvalueproblem(2.1)–(2.2),forthatwebringthebelowtheorem.Theorem2.6.Letfbeacomplex-valuedcontinuousfunctiondefinedonR:|x−x0|≤a,ky−y0k≤b,(a,b>0),(2.58)suchthat|f(x,y)|≤N,forall(x,y)inR.SupposethereexistsaconstantL>0suchthat|f(x,y)−f(x,z)|≤Lky−zk,(2.59)forall(x,y)and(x,z)inR.Thenthereexistsone,andonlyone,solutionϕofy(n)=fx,y,y0,...,y(n−1),(2.60)ontheintervalbI:|xx0|≤mina,,(M=N+b+ky0k),(2.61)Mwhichsatisfiesϕ(x)=α,ϕ0(x)=α,...,ϕ(n−1)(x)=α,(2.62)01020nwithy0=(α1,...,αn).Proof.PleaseseeRef.[23]. 59October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)51Now,assumingtheinitialvalueproblems(2.4)–(2.5)isintheform(2.60)and(2.61),weconcludethatthereexistsoneandonlyonesolutionto(2.4)–(2.5)foreachvalueoftheprescribedparameterδ.Theorem2.7.Considertheboundaryvalueproblems(2.1)–(2.2)andsup-posethattheconditionsofTheorem2.6holdfortheinitialvalueproblem(2.4)–(2.5)andmore,theseriesX+∞X+∞um(r,δ),L[um(r,δ)],(2.63)m=0m=0converge.IfthenumberofKhorizontalplateausoccurintheplaneof(~,δ)whereEq.(2.19)isplottedimplicitly,thentheproblems(2.1)–(2.2)admitthenumberofKmultiplesolutionsintermsofthebasisfunctions(2.6).Proof.Supposethatthenumberofhorizontalplateausoccurringintheplane(~,δ)isKnamelyδ1(~1),δ2(~2),...,δK(~K),(2.64)where(~j,δj(~j)),j=1,2,...,Kareproperorderedpairwhichcho-senfromtheplane(~,δ).SincetheconditionsofTheorem2.4holdandbyuniquenessoftheTaylor’sseriesweconcludealltheseriesP+∞m=0um(r,δj(~j)),j=1,2,...,Kconverge.SupposeX+∞s1(r)=um(r,δ1(~1)),m=0X+∞s2(r)=um(r,δ2(~2)),m=0...(2.65)X+∞sK(r)=um(r,δK(~K)).m=0Now,weshowalltheaboveseriesarethesolutionsoftheproblem(2.1)–(2.2).Toshowthis,itissufficienttoprovetheyarethesolutionsofprob-lems(2.4)–(2.5)becausetheconditionu(α)=βisautomaticallysatisfied.LetusconsiderX+∞sj(r)=um(r,δj(~j)),(2.66)m=0 60October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.252S.AbbasbandyandE.Shivanianbyhigh-orderdeformationequations(2.14)–(2.16),wegetX+∞−→~jH(r)Rm(um−1,r,δj(~j))m=1X+∞=L[um(r,δj(~j))−χmum−1(r,δj(~j))]m=1()X+∞=L[um(r,δj(~j))−χmum−1(r,δj(~j))]m=1=L{u1(r,δj(~j))}+L{u2(r,δj(~j))−u1(r,δj(~j))}+L{u3(r,δj(~j))−u2(r,δj(~j))}+...+L{un(r,δj(~j))−un−1(r,δj(~j))}+...no=Llimun(r,δj(~j))=L{0}=0,(2.67)n→∞NoticethatthelinearoperatorLisLipschitzintheaboveequationsfromtheconditionsofTheorem2.6.Now,since~j6=0andH(r)6=0thenX+∞−→Rm(um−1,r,δj(~j))=0,(2.68)m=1thus,X+∞m1∂N[Φ(r,δj(~j);q)]m!∂qmm=0q=0X+∞m−11∂N[Φ(r,δj(~j);q)]==0.(m−1)!∂qm−1m=1q=0Considertheresidualoftheoriginalequation(2.1)asR(r,δj(~j);q)=N[Φ(r,δj(~j);q)].(2.69)Now,whatitremainsistoshowR(r,δj(~j);1)=0,becausethenN[Φ(r,δj(~j);1)]=0,(2.70)and,ontheothersidewehaveΦ(r,δj(~j);1)=sj(r).TheTaylor’sseriesofR(r,δj(~j);q)respecttoqisX+∞m1∂R(r,δj(~j);q)mR(r,δj(~j);q)=mq,(2.71)m!∂qm=0q=0 61October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)53andfinallytheaboveequationgivesX+∞m1∂R(r,δj(~j);q)R(r,δj(~j);1)=mm!∂qm=0q=0X+∞m1∂N[Φ(r,δj(~j);q)]=m=0,(2.72)m!∂qm=0q=0thentheproofiscompleted.Example2.5.Consideraone-dimensionalsemi-linearproblem00u+u3=0,x∈[0,π],u(0)=u(π)=0.(2.73)Supposingu0(0)=δ,whereδisaprescribedparameter,wecanhavethefollowinginitialvalueproblem0030u+u=0,u(0)=0,u(0)=δ,(2.74)withu(π)=0,asforcingcondition,attachedtotheaboveproblem.Letusgetthesetofbasefunctionsas{xn|n=0,1,2,...}.(2.75)Undertheruleofsolutionexpressionandaccordingtotheinitialconditions,itiseasytochooseu0(x)=δxasinitialguessofsolutionu(x),H(x)=1asauxiliaryfunction,andtochooseauxiliarylinearoperator∂2ϕ(x,δ;p)L[ϕ(x,δ;p)]=,(2.76)2∂xwiththepropertyL[c1+c2x]=0.(2.77)Thus,aftertwosubsequentintegrations,theMth-orderdeformationequa-tionyieldsform≥1ZxZs−→um(x,δ)=χmum−1(x,δ)+~Rm(um−1,τ,δ)dτds00+c1+c2x,(2.78)wherefrom(2.74)−→00Rm(um−1,τ,δ)=um−1(x,δ)mX−1Xj+um−1−j(x,δ)ui(x,δ)uj−i(x,δ),(2.79)j=0i=0 62October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.254S.AbbasbandyandE.Shivanianandintegrationconstantsc1andc2areobtainedfromtheboundarycondi-tions0um(0)=0,um(0)=0.(2.80)Inthiswayweobtainthefunctionsum(x,δ)form=1,2,3,...fromEq.(2.78)successively.Then,wecanobtainMth-orderapproximatesolutionXMUM(x,δ,~)=um(x,δ).(2.81)m=0SoEq.(2.19),withthehelpofadditionalforcingconditionu(π)=0,becomesu(π)≈UM(π,δ,~)=0.(2.82)AccordingtotheaboveequationinFig.2.7,δasafunctionofconvergencecontrollerparameter~,forM=30,hasbeenplotted.Thenumberofthreeδ-plateauscanbeidentifiedinthisfigure,namelyδ=−0.9851intherange[−0.35,−0.05],δ=0intherange[−0.5,0]andδ=0.9851intherange[−0.35,−0.05]of~.Accordingly,weconcludethattheHAMfurnishestriplesolutionsintermsofbasisfunctions(2.75).21∆0-1-2-0.5-0.4-0.3-0.2-0.10.0ÑFig.2.7.Theplotofδasfunctionof~throughEq.(2.82).2.4.SomeillustrativemodelsInthispart,weapplyaforesaidpredictorhomotopyanalysismethodtodiscovermultiplesolutionsofsomeimportantnonlinearmodelandobtain 63October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)55approximationsofthesesolutionssimultaneouslyaswell.2.4.1.Nonlinearproblemarisinginheattransfer2.4.1.1.ModelandexactsolutionsConsiderastraightfinoflengthLwithauniformcross-sectionareaA.ThefinsurfaceisexposedtoaconvectiveenvironmentattemperatureTaandthelocalheattransfercoefficienthalongthefinsurfaceisassumedtoexhibitapower-law-typedependenceonthelocaltemperaturedifferencebetweenthefinandtheambientfluidasnh=(T−Ta),(2.83)whereaisadimensionalconstantdefinedbyphysicalpropertiesofthesurroundingmedium,Tisthelocaltemperatureonthefinsurface,andtheexponentndependsontheheattransfermode.Thevalueofncanvaryinawiderangebetween−4and5[14].Forexample,theexponentnmaytakethevalues−4,−0.25,0,2and3,indicatingthefinsubjecttotransitionboiling,laminarfilmboilingorcondensation,convection,nucleateboiling,andradiationintofreespaceatzeroabsolutetemperature,respectively.Forone-dimensionalsteadystateheatconduction,theequationintermsofdimensionlessvariablesXT−Tax=,h=,(2.84)LTb−Tacanbewrittenasd2θ−ψ2θn+1=0,(2.85)dx2wheretheaxialdistancexismeasuredfromthefintip,Tbisthefinbasetemperature,andψistheconvective-conductiveparameterofthefindefinedas11hPL22aPL22bnh==(Tb−Ta).(2.86)kAkAIntheaboveequationhb,Pandkrepresenttheheattransfercoefficientatfinbase,theperipheryoffincross-section,andtheconductivityofthefin,respectively.Forsimplicity,assumethefintipisinsulatedandtheboundaryconditionstoEq.(2.85)canbeexpressedasdθ(0)=0,θ(1)=1.(2.87)dx 64October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.256S.AbbasbandyandE.ShivanianEquation(2.85)withboundaryconditions(2.87)hasbeenconsideredlatelybyresearchers.In[24,25],theauthorshavetakenψ,convective-conductiveparameter,asasmallparameterandgaveapproximatesolutionsbypertur-bationmethodandvariationaliterationmethod.M.S.H.ChowdhuryandI.Hashim[14]transformedtheboundaryvalueproblems(2.85)and(2.87)toaninitialvalueproblemwithunknownparameterandthenappliedtheperturbationmethodsothattheycouldgiveapproximatesolutionsforpos-itiveexponentofθinEq.(2.85).Abbasbandy[13]consideredEq.(2.85)withboundaryconditions(2.87)insomespecialcasesand,usedhomotopyanalysismethodinordertoguaranteeconvergenceofapproximateseriessolutionsbyconvergence-controllerparameter.Itcanbeshownthattheproblems(2.85)and(2.87)admitunitsolutionfor−1≤n≤5,bothunitanddualsolutionsfor−2 65October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)57Usingδ=θ(0)andbyintegration(2.92),thefinalsolutionofEq.(2.88)inimplicitformisgivenbyZ√θdτδlogδx=p=−qδ2ψ2(δ−1−τ−1)22ψ2δ2rr√q(θ−δ)ψ222δθlog2δ(θ−δ)ψ3ψδθ+√θ−δ2log2+p2ψ2δθδ2(θ−δ)√q√p(θ−δ)ψ2δθlogθ+(θ−δ)δθ+p.(2.93)(θ−δ)Theparameterδcanbeeasilyobtainedwiththehelpoftheboundaryconditionθ(1)=1from(2.93)asfollows√rr22δlogδδ(1−δ)ψ3ψ−q+√−δ2log22ψ22ψ2δδ22δ2qqp(1−δ)ψ2(1−δ)ψ2δδlog2δδlog1+(1−δ)+p+p=1.(2.94)(1−δ)(1−δ)Theaboveequationinimplicitway,δasfunctionofψ,hasbeenplottedinFig.2.8.1.00.80.6B:H0.5,0.830017L∆A:H0.5,0.348961L0.4Ψmax=0.5916110.20.00.00.10.20.30.40.50.60.7ΨFig.2.8.TheplotofδasfunctionofψthroughEq.(2.94). 66October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.258S.AbbasbandyandE.ShivanianFirstinsighttoFig.2.8,revealsthatcorrespondingto0≤ψ≤ψmax=0.591611thereexisttwoδsodualsolutionsoccur,forexamplecor-respondingtoψ=0.5,wehaveδ=0.348961andδ=0.830017asshowninFig.2.8bypointsAandB,respectively.2.4.1.2.PredictionofdualsolutionsbytheruleofmultiplicityofsolutionsConsideringψ=0.5,thetwo-pointboundaryvalueproblem(2.85)and(2.87)canbereplacedformallybytheinitialvalueproblemd2θ−2−0.25θ=0,(2.95)dx2dθθ(0)=δ,(0)=0,(2.96)dxsubjecttotheadditionalforcingconditionθ(1)=1,whereδdenotestem-peratureofthefintipanditisanunknownparameteroftheproblemandwillbedeterminedlaterbytheruleofmultiplicityofsolutions.Now,HAMisappliedontheproblems(2.95)and(2.96)asfollows:Itisstraightforwardtousethesetofbasefunctions2nx|n=0,1,2,....(2.97)Undertheruleofsolutionexpressionandaccordingtotheinitialconditions,itiseasytochooseθ(x)=x2+δasinitialguessofsolutionθ(x),H(x)=10asauxiliaryfunction,andtochooseauxiliarylinearoperator∂2ϕ(x,δ;p)L[ϕ(x,δ;p)]=,(2.98)2∂xwiththepropertyL[c1+c2x]=0.(2.99)Thus,aftertwosubsequentintegrations,theMth-orderdeformationequa-tion(2.14)yieldsforM≥1ZxZs−→θm(x,δ)=χmθm−1(x,δ)+~Rm(θm−1,τ,δ)dτds00+c1+c2x,(2.100)wheremX−1Xj−→00Rmθm−1,τ,δ=θm−1−j(x)θi(x)θj−i(x)j=0i=0−0.25(1−χm),(2.101) 67October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)59andintegrationconstantsc1andc2areobtainedfromtheboundarycon-ditions0θm(0)=0,θm(0)=0.(2.102)Theabovehomogeneousboundaryconditionsimplythatboththeintegra-tionconstantsc1andc2occurringinEq.(2.100)arezero.Inthiswayweobtainthefunctionsθm(x)form=1,2,3,...fromEq.(2.100)successively.Inthisway,wecanobtainMth-orderapproximatesolutionXMΘM(x,δ,~)=θm(x,δ).(2.103)m=0SoEq.(2.19),withthehelpofadditionalforcingconditionθ(1)=1,be-comesθ(1)≈ΘM(1,δ,~)=1.(2.104)AccordingtotheaboveequationinFig.2.9,δasafunctionofconver-gencecontrollerparameter~,hasbeenplottedinthe~-range[−1.1,−0.1],forM=35.Twoδ-plateauscanbeidentifiedinthisfigure,namelyδ=0.3489intherange[−0.95,−0.3]andδ=0.8300intherange[−0.45,−0.25]of~.Accordingly,weconcludethattheHAMfurnishesdualsolutions,inafullagreementwiththeexactresultshowninFig.2.8.1.00.8∆0.60.40.2-1.0-0.8-0.6-0.4-0.2ÑFig.2.9.Theplotofδasfunctionof~throughEq.(2.104). 68October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.260S.AbbasbandyandE.Shivanian2.4.1.3.EffectivecalculationofthetwobranchesofsolutionAssoonasthemultiplicityofsolutions(heredualsolutions)inthepa-rameterplane(~,δ),havebeenidentified,wemayturntocalculatethemexplicitlytoanydesiredorderMofHAM-approximationaccordingtoEq.(2.103).Inthepresentsection,wedothisforbothofthedualsolutionscor-respondingtoδ=0.3489andδ=0.8300asbeingidentifiedinFig.2.9andcomparetheHAMapproximateseriessolutionsgivenbyEq.(2.103)withtheexactsolutions(2.93)identifiedintherespectivepointpairs(A,B)ofFig.2.8.Weremarkhere,asmentionedinintroduction,boththelowerbranchandupperbranchofsolutionsarecalculatedatthesametimeonlybyEq.(2.103)withdifferentδand~whicharespecifiedfromFig.2.9.Fur-thermore,weemphasisagainthatthereisnoneedtousemorethanoneinitialapproximationguess,oneauxiliarylinearoperator,andoneaux-iliaryfunctionthatisinasharpcontrasttoallapproximationmethodswhichareusedtoconvergetoonesolution.IntheplotshowninFig.2.10(a),correspondtoδ=0.3489and~=−0.6,theapproximateHAMsolutionsΘ3(x,0.3489,−0.6),Θ5(x,0.3489,−0.6)andΘ10(x,0.3489,−0.6)givenbyEq.(2.103)arecomparedtotheexactlowerbranchsolutionθ(x)givenbyEq.(2.93)forδ=0.348961andψ=0.5(PointAofFig.2.8).Inasameaction,inFig.2.10(b),correspondtoδ=0.8300and~=−0.4,theapproximateHAMsolutionsΘ3(x,0.8300,−0.4),Θ5(x,0.8300,−0.4)andΘ10(x,0.8300,−0.4)givenbyEq.(2.103)arecomparedtotheexactup-perbranchsolutionθ(x)givenbyEq.(2.93)forδ=0.830017andψ=0.5(PointBofFig.2.8).OneseesthatwiththeincreasingorderM,theapproximatesolutionsΘM(x)approachtheexactsolutionsmoothly.InFig.2.11thedualHAMsolutionsΘM(x)oforderM=35arecomparedtotheexactdualsolutions.TothisorderofapproximationtheHAMresults(markedbyboldredcircle)andtheexactresults(solidbluelines),atthescaleofFig.2.11becomeundistinguishable.2.4.2.StronglynonlinearBratu’sequation2.4.2.1.ProblemandexactsolutionsThenonlinearBratu’sproblemhasbeenlatelyinvestigatedbyresearcherswithvarioustechniques[15,16],[26,27].ShuicaiLiandShijunLiao[15]appliedsuccessfullytheHAMineruditewaytoobtainmultiplebranchesofsolutionsofthisnonlinearproblem.Theproblemisgivenby00u+λeu=0,x∈(0,1),(2.105) 69October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)61(a)(b)1.11.21.00.91.10.8QHxLQHxL0.71.00.60.50.90.40.00.20.40.60.81.00.00.20.40.60.81.0xxFig.2.10.(a)Thecomparisonofapproximatelowersolutionswiththeexactone:Θ3(x)-Brown,Θ5(x)-BlueandΘ10(x)-Red;theexactlowersolution-Black.(b)Thecomparisonofapproximatelowersolutionswiththeexactone:Θ3(x)-Brown,Θ5(x)-BlueandΘ10(x)-Red;theexactlowersolution-Black.1.00.80.6LxHQ0.40.20.00.00.20.40.60.81.0xFig.2.11.Thecomparisonofapproximatedualsolutions(Θ35(x,0.3489,−0.6)andΘ35(x,0.8300,−0.4))withexactdualsolutions:Boldredcircle-approximateso-lutions;Solidblueline-exactsolutions.withboundaryconditionsu(0)=u(1)=0.(2.106) 70October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.262S.AbbasbandyandE.ShivanianByreductionordertheexactsolutionofEq.(2.105)isgivenby[50]"#a2u(x)=log,(2.107)2λcosh2−a(x+b)2wherea,baretwoarbitraryconstants.Introducingtheboundarycon-ditionstodeterminetheseconstants(Eq.(108)),thenb=−1andais2determinedsuchthathi22aa=2λcosh.(2.108)4Supposethatα=a,thusthesolutionofBratuproblemisgivenby4coshαu(x)=2log,(2.109)cosh[α(1−2x)]whereαsatisfies4coshα=√α.(2.110)2λThedependenceofαonλhasbeenplottedinFig.2.12.Itiseasytofindoutthat,accordingtothevalueofλrelativetoamaximumvalueλmax,theproblems(2.105)and(2.106)hasnosolutionforλ>λmax,onesolutionforλ=λmaxandtwosolutionsforλ<λmax,forexample,asindicatedinFig.2.12,correspondingtoλ=3therearetwoα(α=0.84338andα=1.64414(pointsCandD))thereforethereexisttwosolutions.Fur-thermore,differentiating(2.109)respecttoxoncetimeandsettingx=0gives0u(0)=4αtanhα.(2.111)0InFig.2.13also,forourfuturepurpose,wehaveplottedu(0)asfunc-tionofαaccordingtoEq.(2.111).Aswesee,forthosepointsCand000DinFig.2.12,correspondingu(0)(u(0)=2.3196andu(0)=6.1034(PointsEandF))havebeenmarked.Intheotherwords,forλ=3,dual0solutionsoccursothat,wehaveu(0)=2.3196forthefirstsolutionand0u(0)=6.1034forthesecondsolution.2.4.2.2.PredictionofmultiplesolutionsbytheruleofmultiplicityofsolutionsThepurposeofthissubsectionistoforecasttheexistenceofmultipleso-lutionsfortheproblem(2.105)and(2.106)incaseλ=3bytheruleofmultiplicityofsolutions.Assumingλ=3,thetwo-pointboundaryvalue 71October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)636Λ5max=3.51383071914Α32D:H3,1.64414L1C:H3,0.84338L001234ΛFig.2.12.αasfunctionofλaccordingtoEq.(2.110).65F:H1.64414,6.1034L4E:H0.84338,2.3196Lu'H0L32100.00.51.01.5Α0Fig.2.13.u(0)asfunctionofαaccordingtoEq.(2.111).problem(2.105)and(2.106)canbechangedformallytotheinitialvalueproblemasfollows00u+3eu=0,x∈(0,1),(2.112)u(0)=0,u0(0)=δ,(2.113)subjecttotheadditionalforcingconditionu(1)=0.(2.114)InordertopreventsufferingfromthestronglynonlineartermeuintheframeofHAM,wecantransformtheproblem(2.112),(2.113)andaddi- 72October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.264S.AbbasbandyandE.Shivaniantionalcondition(2.114)toanequivalentone.Supposethaty(x)=e−u(x)oru(x)=−log[y(x)].(2.115)UndertheabovetransformationEq.(2.112)isconvertedtohi2000y(x)y(x)−y(x)−3y(x)=0,x∈(0,1).(2.116)0FromEq.(2.115)y(x)=−u0(x)e−u(x)so,theboundaryconditions(2.113)become0y(0)=1,y(0)=γ=−δ,(2.117)andadditionalconditionbyEq.(2.115)thatcomesfrom(2.114),isy(1)=1.(2.118)Now,insteadoftheproblems(2.112)–(2.114),thetechniqueisappliedontheproblems(2.116)–(2.118)andtheparameterγ,whichplayedanimpor-tantroletorealizeaboutmultiplicityofsolutions,willbeobtainedwiththehelpofruleofmultiplicityofsolutions.Itisstraightforwardtousethesetofbasefunctionsn{x|n=0,1,2,...}.(2.119)Undertheruleofsolutionexpressionandaccordingtotheinitialcon-ditions(2.117),itiseasytochoosey0(x)=γx+1asinitialguessofsolutiony(x),H(x)=1asauxiliaryfunction,andtochooseauxiliarylin-earoperator∂2ϕ(x,γ;p)L[ϕ(x,γ;p)]=,(2.120)2∂xwiththepropertyL[c1+c2x]=0.(2.121)Therefore,aftertwosubsequentintegrations,theMth-orderdeformationequation(2.14)yieldsforM≥1ZxZs−→ym(x,γ)=χmym−1(x,γ)+~Rm(ym−1,τ,γ)dτds00+c1+c2x,(2.122)where mX−1mX−1−→0000Rmym−1,τ,γ=ym−1−j(x)yj(x)−ym−1−j(x)yj(x)j=0j=0−3ym−1(x),(2.123) 73October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)65andintegrationconstantsc1andc2areobtainedfromtheboundarycon-ditions0ym(0)=0,ym(0)=0.(2.124)Theabovehomogeneousboundaryconditionsimplythatboththeintegra-tionconstantsc1andc2occurringinEq.(2.122)arezero.Inthiswayweobtainthefunctionsym(x)form=1,2,3,...fromEq.(2.122)successively.Finally,wecanobtainMth-orderapproximatesolutionXMYM(x,γ,~)=ym(x,γ).(2.125)m=0SotheEq.(2.19),withthehelpofadditionalforcingconditiony(1)=1,becomesy(1)≈YM(1,γ,~)=1.(2.126)AccordingtotheaboveequationinFig.2.14,γasafunctionofconvergencecontrollerparameter~,hasbeenplottedinthe~-range[−2,0],forM=40.Twoγ-plateauscanbeidentifiedinthisfigure,namelyγ=−6.1034(δ=6.1034)intherange[−0.6,−0.4]andγ=−2.3196(δ=2.3196)intherange[−0.8,−0.3]of~.Accordingly,weconcludethattheHAMfur-nishesdualsolutions,inafullagreementwiththeexactresultshowninFigs.2.12and2.13(ItisworthtoremarkthatFigs.2.12and2.13indicate0existenceoftwosolutionsforλ=3sothat,u(0)=2.3196forthefirst0solutionandu(0)=6.1034forthesecondsolution.Ontheotherhand,0Fig.2.14alsoindicatestwosolutionssothatu(0)=δ=−γ=2.31960forthefirstoneandu(0)=δ=−γ=6.1034fortheotherone).2.4.2.3.EffectivecalculationofthetwobranchesofsolutionAfterthatthemultiplicityofsolutionshavebeenidentified,wemayturntocalculatethemexplicitlytoanydesiredorderMofHAM-approximationaccordingtoEq.(2.125).Inthepresentsection,wedothisforthetwobranchesofsolutionscorrespondingtoδ=2.3196andδ=6.1034asbeingidentifiedinFig.2.14andcomparetheHAMapproximateseriessolutionsgivenbyEqs.(2.115)and(2.125)withtheexactsolutions(2.107)iden-tifiedintherespectivepointpairs(C,D)ofFig.2.12.TheMth-orderapproximatesolutionisgivenbyequations(2.115)and(2.125)asfollows:UM(x,δ,~)=−log[YM(x,γ,~)].(2.127) 74October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.266S.AbbasbandyandE.Shivanian-2-3-4Γ-5-6-7-1.0-0.8-0.6-0.4-0.20.0ÑFig.2.14.Theplotofγasfunctionof~throughEq.(2.126).IntheplotshowninFig.2.15,correspondtoδ=2.3196and~=−0.5,theapproximateHAMsolutionsU5(x,2.3196,−0.5),U8(x,2.3196,−0.5)andU10(x,2.3196,−0.5)givenbyEq.(2.127)arecomparedtotheex-actlowerbranchsolutionu(x)givenbyEq.(2.107)forα=0.84338(PointCofFig.2.12).Inasimilarmanner,inFig.2.16,correspondtoδ=6.1034and~=−0.5,theapproximateHAMsolutionsU17(x,6.1034,−0.5),U20(x,6.1034,−0.5)andU22(x,6.1034,−0.5)givenbyEq.(2.127)arecomparedtotheexactupperbranchsolutionu(x)givenbyEq.(2.107)forα=1.64414(PointDofFig.2.12).Obviously,withtheincreasingorderM,theapproximatesolutionsUM(x)approachtheexactsolutionsmoothly.InFig.2.17thebothtwobranchesofHAMsolutionsUM(x)(withM=20forthelowerbranchandM=30fortheupperbranch)arecomparedtotheexactdualsolutions.Weobserveagain,asmentionedbe-fore,boththelowerbranchandupperbranchofsolutionsarecalculatedatthesametimeonlybyEq.(2.127)withdifferentδand~whicharespecifiedfromFig.2.8.Moreover,weemphasizeagainthatthereisnoneedtousemorethanoneinitialapproximationguess,oneauxiliarylinearoperator,andoneauxiliaryfunction.2.4.3.Nonlinearreaction-diffusionmodel2.4.3.1.EquationandexactsolutionsAnonlinearmodelofdiffusionandreactioninporouscatalystshasbeeninvestigatedbyapproximateanalyticalmethods[17,28–30].E.Magyari 75October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)671.00.80.6Fig.2.15.Thecompari-sonofapproximatelowerUHxLsolutionswiththeexact0.4one:U5(x)-Brown,U8(x)-BlueandU10(x)-Red;the0.2exactlowersolution-Black.0.00.00.20.40.60.81.0x2.01.5Fig.2.16.Thecompari-sonofapproximateupperUHxL1.0solutionswiththeexactone:U17(x)-Brown,U20(x)-BlueandU22(x)-Red;the0.5exactuppersolution-Black.0.00.00.20.40.60.81.0x2.01.5Fig.2.17.Thecomparisonofapproximatedualsolu-Ltions(U20(x,2.3196,−0.5)xH1.0UandU30(x,6.1034,−0.5))withtheexactdualso-0.5lutions:Boldredcircle-approximatesolutions;Solidblueline-exactsolutions.0.00.00.20.40.60.81.0xhaveconsideredthismodel[17]andgivensuccessfullyexactanalyticalso-lutionsinimplicitformforallvaluesofparametersoftheproblem.Wejustconsiderhereaspecialcaseofthatproblemwhenthemodeltakes−0.75 76October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.268S.AbbasbandyandE.Shivanianforreaction-orderand0.8forThielemodulus,asfollows00uu0.75−0.64=0,(2.128)withtheboundaryconditionsu0(0)=0,u(1)=1,(2.129)whereuisthedimensionlessconcentrationofthereactant,theprimesde-notedifferentiationwithrespecttothedimensionlesstransversecoordinatex,0≤x≤1.In[17],ithasbeenshownthattheaboveproblems(2.128)and(2.129)admittwofollowingsolutionss10.1836422.6539x=22−2u1r3!10.183640.18360.183640.183685+6+8+16u,(2.130)uuuus10.5330425.8821x=22−2u1r3!10.533040.53300.533040.533085+6+8+16u.(2.131)uuuuAsimpleinspectionoftheabovetwosolutionsgivesu(0)=0.1836,(2.132)forthefirstsolution,andu(0)=0.5330,(2.133)forthesecondone.2.4.3.2.PredictionofmultiplesolutionsbytheruleofmultiplicityofsolutionsLetusconsiderthefollowinginitialvalueproblem00uu0.75−0.64=0,(2.134)u(0)=δ,u0(0)=0,(2.135) 77October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)69withadditionalforcingconditionu(1)=1,(2.136)whereδisanunknownparameteroftheprobleminthistimeandwillbedeterminedlaterbytheruleofmultiplicityofsolutions.Now,HAMisappliedontheproblem(2.134)and(2.135)asfollows:Itisstraightforwardtousethesetofbasefunctionsx2n|n=0,1,2,....(2.137)Undertheruleofsolutionexpressionandaccordingtotheinitialconditions,itiseasytochooseu0(x)=δasinitialguessofsolutionu(x),H(x)=1asauxiliaryfunction,andtochooseauxiliarylinearoperator∂2ϕ(x,δ;p)L[ϕ(x,δ;p)]=,(2.138)2∂xwiththepropertyL[c1+c2x]=0.(2.139)Thus,theMth-orderdeformationequationforM≥1becomes−→L[um(x,δ)−χmum−1(x,δ)]=~Rm(um−1,x,δ),(2.140)wheremX−1−→00Rm(um−1,x,δ)=um−1−j(x)zj(x)−0.64(1−χm),(2.141)j=0with1∂n[ϕ(x,δ;p)]0.75zn(x)=n(n)!∂pp=0hPi0.75∂nk=+∞u(x,δ)pk1k=0k=n.(2.142)(n)!∂pp=0Forinstance0.75z0(x)=[u0(x)],(2.143)0.75u1(x)z1(x)=0.25,(2.144)[u0(x)] 78October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.270S.AbbasbandyandE.Shivanian20.09375[u1(x)]0.75u2(x)z2(x)=−1.25+0.25.(2.145)[u0(x)][u0(x)]Thehigh-orderdeformationequation(2.140)withinitialconditionsu(0)=0,u0(0)=0,(2.146)mmwhichcomesfrom(2.135),canbeeasilysolvedbystartingu0(x,δ)=δ,inthiswayweobtainthefunctionsum(x,δ)form=1,2,3,...successivelyandMth-orderapproximatesolutionXMUM(x,δ,~)=um(x,δ).(2.147)m=0SotheEq.(2.19),withthehelpofadditionalforcingconditionu(1)=1,becomesu(1)≈UM(1,δ,~)=1.(2.148)AccordingtotheaboveequationinFig.2.18,δasafunctionofconvergencecontrollerparameter~,hasbeenplottedinthe~-range[−2,0],forM=25.Twoδ-plateauscanbeidentifiedinthisfigure,namelyδ=0.1836intherange[−1.3,−0.4]andδ=0.5330intherange[−1.7,−0.3]of~.Accordingly,weconcludethattheHAMfurnishesdualsolutions,inafullagreementwiththeexactresult(2.130)and(2.131).0.60.50.4∆0.30.20.1-1.5-1.0-0.50.0ÑFig.2.18.Theplotofδasfunctionof~throughEq.(2.148). 79October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)712.4.3.3.CalculationofthetwobranchesofsolutionNow,wejumptocalculateexplicitlytoanydesiredorderMofHAM-approximatesolutionsaccordingtoEq.(2.147)forbothofdualsolutionscorrespondingtoδ=0.1836andδ=0.5330asbeingidentifiedinFig.2.18andtocomparethemwiththeexactsolutions(2.130)–(2.131).IntheplotshowninFig.2.19,correspondtoδ=0.1836and~=−0.8,theapproximateHAMsolutionsU3(x,0.1836,−0.8),U5(x,0.1836,−0.8)andU7(x,0.1836,−0.8)givenbyEq.(2.147)arecomparedtotheex-actlowerbranchsolutionu(x)givenbyEq.(2.130).Inasameac-tion,inFig.2.20,correspondtoδ=0.5330and~=−1,theapprox-imateHAMsolutionsU0(x,0.5330,−1)=0.5330,U1(x,0.5330,−1)andU2(x,0.5330,−1)givenbyEq.(2.147)arecomparedtotheexactupperbranchsolutionu(x)givenbyEq.(2.131).Oneseesthatwiththeincreas-ingorderM,theapproximatesolutionsUM(x),rapidlyapproachtotheexactsolutionsmoothly.InFig.2.21thedualHAMsolutionsUM(x)oforderM=15arecomparedtotheexactdualsolutions.TothisorderofapproximationtheHAMresults(markedbyboldredcircle)andtheexactresults(solidbluelines),atthescaleofFig.2.21becomeundistinguish-able.Wenoticehere,boththelowerbranchandupperbranchofsolutionsarecalculatedatthesametimeonlybyEq.(2.147)withdifferentδand~whicharespecifiedfromtheruleofmultiplicityofsolutionsand,thereisnoneedtousemorethanoneinitialapproximationguess,oneauxiliarylinearoperator,andoneauxiliaryfunction.2.4.4.MixedconvectionflowsinaverticalchannelTheaimofthissectionistoapplyPredictorhomotopyanalysismethodtoanalyzeakindofmodelinmixedconvectionflowsnamelycombinedforcedandfreeflowinthefullydevelopedregionofaverticalchannelwithisothermalwallskeptatthesametemperature[31,32].Inthismodel,thefluidpropertiesareassumedtobeconstantandtheviscousdissipationeffectistakenintoaccount.Thesetofgoverningbalanceequationsforthevelocityfieldisreducedto42duΞdu=,(2.149)dy416dywithconditionsZ10000u(0)=u(0)=u(1)=0,u(y)dy=1,(2.150)0 80October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.272S.AbbasbandyandE.Shivanian1.00.8Fig.2.19.Thecomparison0.6ofapproximatelowersolu-UHxLtionswiththeexactone:U3(x)-Brown,U5(x)-Blue0.4andU7(x)-Red;theexactlowersolution-Black.0.20.00.20.40.60.81.0x1.00.90.8Fig.2.20.Thecomparisonofapproximateuppersolu-UHxL0.7tionswiththeexactone:U0(x)-Brown,U1(x)-Blue0.6andU2(x)-Red;theexactuppersolution-Black.0.50.40.00.20.40.60.81.0x1.00.8Fig.2.21.Thecompari-0.6sonofapproximatedualso-LxHlutions(U15(x,0.1836,−0.8)UandU15(x,0.5330,−1))with0.4theexactdualsolutions:Boldredcircle-approximate0.2solutions;Solidblueline-exactsolutions.0.00.00.20.40.60.81.0xwhereuandyaredimensionlessvelocityandtransversalcoordinate,re-spectivelyandalsoUY4Lgβµcp4LUmu=,y=,Ge=,Pr=,Re=,Ξ=GePrRe,UmLcpkν 81October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)73inwhichUm,L,cp,µ,k,ν,Ge,PrandRearemeanfluidvelocity,channelhalf-width,specificheatatconstantpressure,dynamicviscosity,thermalconductivity,kinematicviscosity,Gebhartnumber,PrandtlnumberandReynoldsnumber,respectively.InthecaseΞ=0,correspondeithertoaverysmallviscousdissipationheatingortonegligiblebuoyancyeffects,theEqs.(2.149)–(2.150)iseasilysolvedandadmittheuniquesolution3 u(y)=1−y2.(2.151)2Ithasbeenshownin[30,31]byperturbationandnumericalmethodsthatEqs.(2.1)–(2.2)admitdualsolutionsforanygivenΞintheinter-val(−∞,0)∪(0,Ξmax)inwhichΞmax∼=228.128.2.4.4.1.PredictionofdualsolutionsbytheruleofmultiplicityofsolutionsThepurposeofthissubsectionistoshowhowonecanfindouttheexistenceofdualsolutionsforEqs.(2.149)–(2.150)inaforesaidrangeforΞ.Consider00Eqs.(2.149)–(2.150)andsupposethatu(0)=δ,sotheproblembecomes42duΞdu=,(2.152)dy416dysubjecttoboundaryconditions000000u(0)=u(0)=u(1)=0,u(0)=δ,(2.153)withadditionalforcingconditionZ1u(y)dy=1.(2.154)0Now,weapplyPredictorhomotopyanalysismethodonEqs.(2.152)-(2.153)whereprescribedparameterδ,whichisplayedimportantroletorealizeaboutmultiplicityofsolutions,willbeobtainedwiththehelpofruleofmultiplicityofsolutions.Itisstraightforwardtousethesetofbasefunctionsn{y,n=0,1,2,...}.(2.155)Undertheruleofsolutionexpressionandaccordingtotheinitialconditions(2.153),itiseasytochooseδ u(y,δ)=y2−1,02 82October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.274S.AbbasbandyandE.Shivanianasinitialguessofsolutionu(y),H(y)=1asauxiliaryfunction,andtochooseauxiliarylinearoperator∂4φ(y,δ;p)L[φ(y,δ;p)]=,(2.156)∂y4withthepropertyLc+cy+cy2+cy3=0.(2.157)1234Therefore,afterfoursubsequentintegrations,theMth-orderdeformationequation(2.14)yieldsforM≥1um(y,δ)=χmum−1(y,δ)ZyZη1Zη2Zη3−→+~Rm(um−1,τ,δ)dτdη3dη2dη10000+c+cy+cy2+cy3,(2.158)1234wheremX−1−→0000Ξ00Rm(um−1,τ,δ)=um−1(τ,δ)−uj(τ,δ)um−1−j(τ,δ),(2.159)16j=0andintegrationconstantsc1,c2,c3,c4areobtainedbytheconditions000000um(1,δ)=um(0,δ)=um(0,δ)=um(0,δ)=0.(2.160)Inthiswayweobtainthefunctionsum(y,δ)form=1,2,3,...fromEq.(2.158)successively.Finally,wecanobtainMth-orderapproximatesolu-tionXMUM(y,δ,~)=um(y,δ),(2.161)m=0wegivebelowtheseriessolution(2.161)fromtheorderM=1untilthe 83October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)75orderM=3initsformvalidforanyΞ:y6δ2Ξ~1 δ2Ξ~2U1(y,δ,~)=−+y−1δ+,576025760y10δ3Ξ2~2y6δ2Ξ~2y6δ2Ξ~1 δ3Ξ2~22U2(y,δ,~)=−−+y−1δ−3870720057602880238707200δ2Ξ~2δ2Ξ~++,5760288031y14δ4Ξ3~3y10δ3Ξ2~3y10δ3Ξ2~2y6δ2Ξ~3U3(y,δ,~)=−++−743921418240019353600129024005760y6δ2Ξ~2y6δ2Ξ~1 31δ4Ξ3~3−−+y2−1δ+1920192027439214182400δ3Ξ2~3δ3Ξ2~2δ2Ξ~3δ2Ξ~2δ2Ξ~−−+++.1935360012902400576019201920SotheEq.(2.161),withthehelpofadditionalforcingcondition(2.154),becomesZ1Z1ΓM(δ,~,Ξ)=UM(y,Ξ,δ,~)dy≈u(y,Ξ)dy=1.(2.162)00Now,tobespecific,weconsidertwocaseconsistofΞ=20andΞ=−20.AccordingtotheaboveequationinFig.2.22,δ(prescribedparameter)asafunctionofconvergencecontrollerparameter~,hasbeenplottedinthe~-range[–2.6,0.6]implicitly,forM=25andΞ=20.Twoδ-plateauscanbeidentifiedinthisfigure,namelyδ=−3.08411intherange[–1.6,–0.4]of~andδ=−161.726intherange[–0.95,–0.55]of~.ItisnoticeablethatwehavetomagnifytheFig.2.22toobtainvaluesofδwithhighaccuracy(Fig.2.23).Consequently,weconcludethatthePHAMfurnishesdualsolutions,inafullagreementwiththoseobtainedin[31](ItisworthmentioningherethatFig.2.22indicatesexistenceoftwosolutionsforΞ=20sothat,0000u(0)=−3.08411forthefirstbranchsolutionandu(0)=−161.726forthesecondbranchsolution).Anothertechniquetofindouthowmanysolutionsthenonlinearproblems(2.5)–(2.6)admitistousethisfactthatcrosspointofresidualof(2.15)(i.e.ΓM(δ,~,Ξ)−1)byhorizontalaxisdosenotvarywiththevariationof~.Figure2.24showsthattherearetwocrosseswithhorizontalaxiswhichdonotvarywithchangeof~soweturnoutthatthereexistdualsolutions.ThesameprocedurehasbeendoneforthecaseΞ=−20.AsweseeinFigs.2.25–2.26orequivalentlyinFig.2.27,thereexistdualsolutions0000namelyu(0)=−2.92300forthefirstbranchsolutionandu(0)=170.039forthesecondbranchsolutionfortheproblems(2.152)–(2.153). 84October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.276S.AbbasbandyandE.Shivanian500-50Fig.2.22.Prescribedpa-∆rameterδviaconvergencecontrollerparameter~in-100accordingto(2.162)withM=25forΞ=20.-150-200-2.5-2.0-1.5-1.0-0.50.00.5Ñ-3.08409-161.725-3.08410-161.726∆-3.08411∆-161.726-3.08412-161.727-3.08413-161.727-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.2-1.2-1.0-0.8-0.6-0.4ÑÑFig.2.23.Prescribedparameterδviaconvergencecontrollerparameter~inaccordingto(2.162)withM=25forΞ=20.(MagnificationofFig.2.22).100∆:2ndbranch50Fig.2.24.Theresidualof(2.162)i.e.ΓM(δ,~,Ξ)0withdifferentvaluesof~whenM=20forΞ=20.Dashedline:~=0.8;-50boldline:~=1;dot-∆:1stbranchdashedline:~=1.2.-100-400-300-200-1000100 85October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)77200150100Fig.2.25.Prescribedpa-∆rameterδviaconvergencecontrollerparameter~in50accordingto(2.162)withM=25forΞ=−20.0-50-2.5-2.0-1.5-1.0-0.50.00.5Ñ170.041-2.92296170.040-2.92298∆-2.92300∆170.039-2.92302170.038-2.92304170.037-1.5-1.0-0.5-1.1-1.0-0.9-0.8-0.7-0.6-0.5-0.4ÑÑFig.2.26.Prescribedparameterδviaconvergencecontrollerparameter~inaccordingto(2.162)withM=25forΞ=−20.(MagnificationofFig.2.25).150∆:1stbranch∆:2ndbranch100Fig.2.27.Theresidualof(2.162)i.e.ΓM(δ,~,Ξ)50withdifferentvaluesof~whenM=20forΞ=−20.Dashedline:~=–00.8;boldline:~=–1;dot-dashedline:~=–1.2.-50-1000100200300400 86October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.278S.AbbasbandyandE.Shivanian2.4.4.2.EffectivecalculationofthetwobranchesofsolutionAssoonasthemultiplicityofsolutions(heredualsolutions)oftheproblem(2.152)–(2.154)orequivalentlyproblems(2.149)–(2.150),forΞ=−20andΞ=20intheparameterplane(~,δ),havebeenidentified,wemayturntocalculatethemexplicitlytoanydesiredorderMofPHAM-approximationaccordingtoEq.(2.161).Inthepresentsection,wedothisforthebothofdualsolutionscorrespondingtoδ=−3.08411andδ=−161.726asbeingidentifiedinFig.2.23.Weremarkhere,asmentionedinintroduction,boththefirstbranchandsecondbranchofsolutionsarecalculatedatthesametimeonlybyEq.(2.161)withdifferentδand~whicharespecifiedfromFig.2.23.Furthermore,weemphasizeagainthatthereisnoneedtousemorethanoneinitialapproximationguess,oneauxiliarylinearoperator,andoneauxiliaryfunctionthatisinasharpcontrasttoallapproximationmethodswhichareusedtoconvergetoonesolution.IntheplotshowninFig.2.28,correspondtoδ=−3.08411andδ=−161.726,theapproximatePHAMsolutionsU25(y,−3.08411,−1)andU25(y,−161.726,−0.75)givenbyEq.(2.161)havebeennormalizedbydividingtoU25(0,−3.08411,−1)andU25(0,−161.726,−0.75),respec-tivelyinordertohavebetterview.Inasameaction,correspondtoδ=−2.92300andδ=170.039,theapproximatenormalizedPHAMso-lutionsU25(y,−2.92300,−1)andU25(y,170.039,−0.75)areshowninFig.2.29.2.4.4.3.FurtherresultsIntwoprevioussubsections,weappliedPredictorhomotopyanalysismethodontheproblem(2.152)–(2.154)orequivalentlyproblems(2.149)–(2.150),forΞ=−20andΞ=20.Ithasbeenshownthatdualsolutions,whichareidentifiedintheparameterplane(~,δ),arecalculatedeffectivelyonlybyputtingorderedcouples(~,δ)inPHAM-series(2.161)toanydesiredofM,simultaneously.WehavedoneasameprocedureforotherspecificvaluesofΞinthissubsection.TogetafirstinsightintotheFigs.2.30and2.31revealsthat,forsomeothervaluesofΞ(−200,−150,−100,−50,50,100,150and200),dualsolu-tionsoccurforthevelocityprofileaswell.Furthermorewehaveprovided00Table2.1,bythesameactioninprevioussubsections,containingδ=U25(0)andU25(0)fordifferentvaluesofΞinthevalidregionof~. 87October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)791.0∆=-3.08411u(0)=1.509330.5U25HyL0.0U25H0L∆=-161.726-0.5u(0)=17.8165-1.00.00.20.40.60.81.0yFig.2.28.DualprofileU25(y)viadimensionlesstransversalcoordinateyforΞ=20.U25(0)1.0∆=-2.92300u(0)=1.491430.5U25HyL0.0U25H0L∆=170.039u(0)=-16.2435-0.5-1.00.00.20.40.60.81.0yFig.2.29.DualprofileU25(y)viadimensionlesstransversalcoordinateyforΞ=−20.U25(0)2.5.ConcludingremarksThepurposeofthischapteristointroduceamethodtopredictthemul-tiplicityofthesolutionsofthenonlinearboundaryvalueproblemssothatitcouldbeeasilyappliedonnonlinearordinarydifferentialequationswithboundaryconditions.Ourgoalistointroduceamethodnotonlytoantici-patemultiplicityofthesolutionsofthenonlineardifferentialequationsbutalsotocalculateeffectivelyallbranchesofthesolutions(ontheconditionthat,thereexistsuchsolutionsfortheproblem)analyticallyatthesame 88October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.280S.AbbasbandyandE.Shivanian0-10-20∆-30-40-50-60-2.5-2.0-1.5-1.0-0.50.00.5ÑFig.2.30.Prescribedparameterδviaconvergencecontrollerparameter~.Boldline:Ξ=50;dottedline:Ξ=100;dashedline:Ξ=150;dot-dashedline:Ξ=200.6040∆200-2.5-2.0-1.5-1.0-0.50.00.5ÑFig.2.31.Prescribedparameterδviaconvergencecontrollerparameter~.Boldline:Ξ=–50;dottedline:Ξ=–100;dashedline:Ξ=–150;dot-dashedline:Ξ=–200.time.Inthismanner,forpracticaluseinscienceandengineering,suchmethodmightgivenewunfamiliarclassofsolutionswhichisoffundamen-talinterest.ThePHAMhasanewviewpointtothehomotopyanalysismethodbyanewapplicationoftheconvergence-controllerparameter.Sincethisparam-eterplaysimportantroletoguaranteetheconvergenceofthesolutionsofnonlineardifferentialequations.IntheframeofthePHAMthisparameterplaysafundamentalroleinthepredictionofmultiplicityofsolutions.Onecanobtainedallbranchesofsolutionssimultaneouslybyoneinitialapprox- 89October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.2PredictorHomotopyAnalysisMethod(PHAM)8100Table2.1.Calculationofδ=U(0)andU25(0)fordifferentvaluesofΞaccord-25ingtoEq.(2.161)1stbranchsolution2ndbranchsolutionΞU00(0)U25(0)U00(0)U25(0)2525–300–2.244671.414214.4022–0.265373–280–2.278291.4181215.2299–0.350282–260–2.313431.422216.1808–0.447839–240–2.350191.4264617.2853–0.561177–220–2.388741.4309118.5851–0.694569–200–2.429231.4355720.1383–0.853988–180–2.471841.4404622.0289–1.04806–160–2.516781.445624.383–1.28971–140–2.564291.4510227.3982–1.59926–120–2.614651.4567531.4042–2.01054–100–2.668181.4628136.994–2.58444–80–2.725281.4692645.3536–3.4427–60–2.786391.4761459.2494–4.86935–40–2.852081.483586.9805–7.71641–20–2.9231.49143170.039–16.243520–3.084111.50933–161.72617.816540–3.176661.51955–78.64549.2869760–3.27941.53086–50.87716.4359680–3.394691.54349–36.92685.00354100–3.525811.55779–28.4934.1374120–3.677531.57427–22.80463.55307140–3.857241.5937–18.66883.12807160–4.077351.61737–15.48052.80024180–4.361391.64775–12.88692.53331200–4.76481.69064–10.63492.3012220–5.507781.76902–8.37852.06803225–5.904061.81059–7.645781.99206228–6.503211.8732–6.851831.90953imationguess,oneauxiliarylinearoperatorandoneauxiliaryfunction.References[1]J.C.Butcher,NumericalMethodsforOrdinaryDifferentialEquations(2nded.),JohnWiley&Sons,England,2008.[2]G.D.Smith,NumericalSolutionofPartialDifferentialEquations,OxfordUniv.Press,1985.[3]C.Johnson,Numericalsolutionsofpartialdifferentialequationsbyfiniteelementsmethods,CambridgeUniv.Press,1987.[4]Y.Nath,M.Prithviraju,andA.A.Mufti,Nonlinearstaticanddynamicsofantisymmetriccompositelaminatedsquareplatessupportedonnonlinear 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92May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 93October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3Chapter3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblemsSandileMotsa∗andPreciousSibandaSchoolofMathematics,StatisticsandComputerScienceUniversityofKwaZulu-Natal,PrivateBagX01Scottsville,Pietermaritzburg3209,SouthAfrica∗sandilemotsa@gmail.comInthischapterweprovideageneralreviewofthespectralhomotopyanalysismethod(SHAM)forthesolutionofnonlinearboundaryvalueproblems.WedemonstratehowtheSHAMmaybeusedtofindmulti-plesolutionsofnonlinearboundaryvalueproblems(BVPs)andtosolvenonlineareigenvalueproblems.Twoapproachesaresuggestedtodeter-minetheoptimalconvergence-controlparameter~,usingrespectivelytheso-called~-curveandresidualerroranalysis.WealsointroducetheiterativeversionoftheSHAM,whichleadstoenhancedaccuracyandefficiency,andacceleratesconvergencethroughsystematicallyupdatingtheinitialapproximation.Contents3.1.Introduction.....................................863.2.Basicideasofthespectralhomotopyanalysismethod..............863.3.Someapplicationsofthespectralhomotopyanalysismethod..........893.3.1.Falkner–Skanboundarylayerflow.....................893.3.2.Eigenvalueproblems.............................953.3.3.Boundaryvalueproblemswithmultiplesolutions............1003.3.4.Couplednonlinearboundaryvalueequations...............1023.4.Convergenceacceleration..............................1123.4.1.Convergenceaccelerationthroughchoiceoflinearoperator.......1123.4.2.Convergenceaccelerationbyiteration...................1163.5.Conclusion......................................119References.........................................12085 94October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.386S.MotsaandP.Sibanda3.1.IntroductionThespectral-homotopyanalysismethod(SHAM)wasintroducedin2010byMotsaetal.[20,21],whousedChebyshevspectralcollocationmethodstosolvethehigh-orderdeformationequationsintheframeoftheHAM.Theinitialapproximationwasalsofoundsystematicallyasthesolutionofthenon-homogeneouslinearpartofthedifferentialequationtobesolved.Theimmediatebenefitoftheseinnovationswasbetteraccuracyandfasterconvergenceofthesolutionseries,requiringfeweriterationsandlesscom-putationaleffort.OtherbenefitsoftheSHAMarethat(i)therangeofad-missible~valuesismuchwiderinthespectralhomotopyanalysismethodthanintheoriginalhomotopyanalysismethod,(ii)themethodallowsforamuchwiderrangeoflinearandnonlinearoperators.Theuseofthespectralhomotopyanalysismethodhaslargelybeenrestrictedtothesolutionofnonlinearboundaryvalueproblems[2,11,20,21,23,26,27,29]However,Atabakanetal.[3]recentlyusedthemethodtosolveVolterraandFredholmintegro-differentialequations.AslightlydifferentversionoftheSHAMthatusesChebyshev-TaumethodtoconvertaBVPtoalgebraicequationsisproposedinKazemandShaban[13].InthischapterweshowhowtheSHAMcanusedtosolveBVPswithmultiplesolutionsandeigenvalueproblems.Weproposetwomethodsforidentifyingtheoptimal~fortheSHAMusingresidualerroranalysis.Lastly,wepresentaniteratedversionoftheSHAMwhichseekstoacceleratecon-vergenceoftheSHAMthroughsystematicupdatingoftheinitialapproxi-mationusedatthestartofthealgorithm.3.2.BasicideasofthespectralhomotopyanalysismethodInthissectionwepresentthebasicideabehindthedevelopmentofthespectralhomotopyanalysismethod(SHAM).ForillustrationpurposeswedescribetheSHAMapproachforthesolutionofgeneralone-dimensionalnonlineardifferentialequations.Extensiontohigherordersystemsofnon-linearBVPscanbedoneinastraight-forwardmanner.Consideranon-linearordinarydifferentialequationoftheformL[y(x)]+F[y(x)]=Φ(x)(3.1)whereΦ(x)isaknownfunctionoftheindependentvariablexandy(x)isanunknownfunction.ThefunctionsLandFrepresentthelinearandnon-linearcomponentsofthegoverningequationrespectively.Forillustrative 95October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems87purposes,weassumethatequation(3.1)istobesolvedinthedomainx∈[a,b]subjecttotheseparatedboundaryconditionsBa(y(a))=0,Bb(y(b))=0,(3.2)whereBaandBbarelinearoperators.Intheframeworkofthehomotopyanalysismethod(HAM)[15,16],wedefinethefollowingzeroth-orderdeformationequations(1−q)L[Y(x;q)−y0(x)]=q~{N[Y(x;q)]−Φ(x)},(3.3)whereq∈[0,1]denotesanembeddingparameter,Y(x;q)isakindofcontinuousmappingfunctionofy(x),~istheconvergence-controllingpa-rameter.ThenonlinearoperatorNisdefinedfromthegoverningequation(3.1)asN[Y(x;q)]=L[Y(x;q)]+F[Y(x;q)].(3.4)Bydifferentiatingthezeroth-orderequations(3.3)mtimeswithrespecttoq,settingq=0andfinallydividingtheresultingequationsbym!,weobtainthefollowingmth-orderdeformationequations,L[ym(x)−(χm+~)ym−1(x)]=~Rm−1[y0,y1,...,ym−1],(3.5)where1∂m−1{F[Y(x;q)]−Φ(x)}Rm−1[y0,y1,...,ym−1]=m−1,(3.6)(m−1)!∂qq=0and0,m61,χm=(3.7)1,m>1.Afterobtainingsolutionsforequation(3.5),theapproximatehomotopy-seriessolutionfory(x)isdeterminedastheseriessolutionX+∞y(x)=yk(x).(3.8)k=0AHAMsolutionissaidtobeoforderMiftheaboveseriesistruncatedatk=M,thatis,ifXMy(x)=ym(x).(3.9)m=0 96October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.388S.MotsaandP.SibandaInusingtheSHAM,theinitialguessisobtainedsimplyasasolutionofthelinearpartofthegoverningequation(3.1)subjecttotheunderlyingboundaryconditions(3.2).Thatis,wesolveL[y0]=Φ(x).(3.10)Inmostcases,equation(3.10)togetherwiththesubsequenthigherorderdeformationequations(3.5)cannotbesolvedexactlybymeansofanalyticaltechniques.Numericalmethodssuchasfinitedifferences,finiteelementmethod,spectralmethodandmanyotherscanbeusedtosolveequationsoftheform(3.5)and(3.10).Spectralmethods,suchastheChebyshevpseudo-spectralmethod,havebeenfoundtobeveryconvenienttoolsforsolvingtheresultingHAMdecomposedhigherorderdeformationequations.Itisforthisreasonthatthemethodisreferredtoasthespectralhomotopyanalysismethod.Spectralmethodsarenowbecomingthepreferredtoolsforsolvingordinaryandpartialdifferentialequationsbecauseoftheireleganceandhighaccuracyinresolvingproblemswithsmoothfunctions.Forbrevity,weomitthedetailsofthespectralmethods,andreferin-terestedreaderstoRefs.[7,30].Beforeapplyingthespectralmethod,itisconvenienttotransformthedomainonwhichthegoverningequationisde-finedtotheinterval[−1,1]wherethespectralmethodcanbeimplemented.Weusethetransformationx=(b−a)(τ+1)/2tomaptheinterval[a,b]to[−1,1].ThebasicideabehindthespectralcollocationmethodistheintroductionofadifferentiationmatrixDwhichisusedtoapproximatethederivativesoftheunknownvariablesy(x)atthecollocationpoints(gridpoints)asthematrixvectorproductXNdy=Dlky(xk)=DY,l=0,1,...,N(3.11)dxk=0whereN+1isthenumberofcollocationpoints,D=2D/(b−a),andTY=[y(τ0),y(τ1),...,y(τN)]isthevectorfunctionatthecollocationpoints.HigherorderderivativesareobtainedaspowersofD,thatisy(p)=DpY.(3.12)wherepistheorderofthederivative.WechoosetheGauss-Lobattocollo-cationpointstodefinethenodesin[−1,1]asπjτj=cos,j=0,1,...,N.(3.13)N 97October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems89ThematrixDisofsize(N+1)×(N+1)anditsentriesaredefined[7,30]asc(−1)j+kjDjk=,j6=k;j,k=0,1,...,N,ckτj−τkτkDkk=−2,k=1,2,...,N−1,(3.14)2(1−τ)k2N2+1D00==−DNN,6with2k=0,Nck=.(3.15)1−1≤k≤N−1Inthenextsection,weconsiderspecificexampleswhichhighlightthemainfeaturesoftheSHAManddemonstratetheimplementationoftheSHAMalgorithm.3.3.Someapplicationsofthespectralhomotopyanalysismethod3.3.1.Falkner–SkanboundarylayerflowInthissectionwedemonstratetheapplicationoftheSHAMalgorithminsolvingtheFalkner–Skanboundarylayerflowequationsinfluidmechanics,governedbythenonlineardifferentialequationf000(η)+βf(η)f00(η)+β(1−f0(η)2)=0,η∈[0,∞),(3.16)01subjecttotheboundaryconditionsf(0)=f0(0)=0,limf(η)=1.(3.17)η→∞whereβ0andβ1areparameterswhosevaluesforcertainspecialclassesofflowsaregivenas1(1)BlasiusFlow:β0=,β1=0.2(2)PohlhausenFlow:β0=0,β1=1.(3)HomannFlow:β0=2,β1=1.InapplyingtheSHAM,thelinearoperatorischosentobethelinearpartofthegoverningequation.However,inthecaseofequation(3.16),weobserve 98October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.390S.MotsaandP.Sibandathatthelinearpartisjustf000whichsuggeststhattheinitialguessshouldbedeterminedfromsolving0000f=β0,f(0)=f(0)=0,f(∞)=1.(3.18)Thesolutionof(3.18)isnotexponentialandwouldnotbeagoodinitialapproximationforthesolutionof(3.16)whichiswellknowntohaveex-ponentialsolutionprofiles.ToobtaintheappropriatelinearoperatorandinitialguesstobeusedintheSHAMsolutionof(3.16),wesetf(η)=f(η)+g(η),f(η)=η−1+e−η(3.19)00wheref0hasbeenchosenasanexponentialfunctionthatsatisfiestheboundaryconditions.Substitutingequation(3.19)inequation(3.16)gives,g000+βfg00−2βf0g0+βf00g+βgg00−β(g0)2+φ(η)=0,(3.20)00100001subjecttotheboundaryconditionsg(0)=g0(0)=g0(∞)=0,whereφ(η)=f000(η)+βf(η)f00(η)+β(1−f0(η)2).000010Byconsideringthelinearpart,theinitialguessg0forsolvingthenon-linearequation(3.20)usingtheSHAMisobtainedasasolutionofg000+βfg00−2βf0g0+βf00g+φ(η)=0,0000100000(3.21)g(0)=g0(0)=g0(∞)=0.000Usingthelinearpartof(3.20),wechoosethelinearoperatorL(g)=g000+βfg00−2βf0g0+βf00g.(3.22)001000WeremarkthatthelinearoperatorischoseninsuchawaythatL(g0)+φ(η)=0.Basedonthegoverningequation(3.20),thenonlinearoperatorisdefinedasN(g)=L(g)+F(g),F(g)=βgg00−β(g0)2.(3.23)01Thus,intheframeworkoftheHAM,thezeroth-orderdeformationequa-tionbecomes(1−q)L[G(η;q)−g0(η)]=q~{N[G(η;q)]+φ(η)},(3.24) 99October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems91whereq∈[0,1]istheembeddingparameter,G(η;q)isacontinuousmap-pingofg(η)and~istheconvergence-controlparameter.Thecorrespondingmth-orderdeformationequationreadsL[gm(η)−(χm+~)gm−1(η)]=~Rm−1[g0,g1,...,gm−1],(3.25)where1∂m−1{N[G(η;q)]+φ(η)}Rm−1[g0,g1,...,gm−1]=m−1.(3.26)(m−1)!∂qq=0Thus,thehomotopyseriessolutionthatapproximatesf(η)readsX+∞f(η)=f0(η)+g0(η)+gk(η).(3.27)k=1Usingthedefinitions(3.23)and(3.26),thehigh-orderdeformationequa-tionsareL(gm)=~(1−χm)φ(η)+(χm+~)L(gm−1)mX−1mX−1+~βgg00−~βg0g0,(3.28)0mm−1−n1mm−1−nn=0n=0subjecttotheboundaryconditiong(0)=g0(0)=g0(∞)=0,(3.29)mmmwhereLisdefinedby(3.22).Whenm=1,wehavethe1st-orderdeformationequationg000+βfg00−2βf0g0+βf00g=~βgg00−~β(g0)2,100110100100010(3.30)g(0)=g0(0)=g0(∞)=0.111Whenm>1,wehavemX−1mX−10000L(gm)=(1+~)L(gm−1)+~β0gmgm−1−n−~β1gmgm−1−n,(3.31)n=0n=0subjecttotheboundaryconditionsg(0)=g0(0)=g0(∞)=0.mmmApplyingthespectralmethodonequations(3.21),(3.30)and(3.31)gives, 100October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.392S.MotsaandP.Sibanda−Φ,m=00002~β0g0g0−~β1(g0),m=1Agm=mX−1mX−1Ag+~βgg00−~βg0g0,m>1m−10mm−1−n1mm−1−nn=0n=0(3.32)subjecttotheboundaryconditionsXNXNgm(ξN)=0,DNkgm(ξk)=0,D0kgm(ξk)=0,(3.33)k=0k=0whereξ=2η/η∞−1isavariableusedtomapthedomain[0,η∞]to[−1,1],η∞isafinitevalueusedtonumericallyapproximatetheconditionsatinfinity,and32000A=D+β0diag(f0)D−2β1diag(f0)D+β0diag(f0)Tgm=[gm(ξ0),gm(ξ1),...,gm(ξN)],Φ=[φ(η),φ(η),...,φ(η)]T,01Nf=[f(η),f(η),...,f(η)]T.000010NHere,diag()isafunctionthatputsthevector()onthemaindiagonal.AteachlevelmoftheSHAMalgorithm,thetermsontheright-handsideof(3.32)areknownfromthepreviousm−1level.Thus,startingfromtheinitialguessg0,thesolutionsgmcanbeobtainedbyrecursivelysolvingequation(3.32).ToimprovecomputationalefficiencyandaccuracyoftheSHAMtheoptimalvalueoftheconvergencecontrollingparameter~mustbecarefullyselected.InpreviousstudiesusingtheSHAM(seeforexample[11,20–22]),admissiblevaluesof~wereselectedfromarangeofvaluesthatlieonahorizontalsegmentofthe~-curve,whichisaplotofthederivativeofanunknownfunctionagainst~.Thedisadvantageofthisapproachisthatdifferentvaluesof~chosenfromdifferentlocationsoftheflatsegmentofthe~-curvegivedifferentspeedsofconvergenceandaccuracyoftheSHAM.Inthiswork,weproposetwonewapproachesforidentifyingtheoptimalvalueof~.Thefirstapproachisbasedontheobservationthatdifferent~-curvesplottedatdifferentSHAMordersseemtointersectatornearonepoint.Wedeterminedthroughnumericalexperimentationthatthispointofintersectionofthedifferent~-curvesgivestheoptimal~.Thesecondapproachofchoosingtheoptimal~isbasedontheresidualofthegoverningequation.Wedefinethemaximumresidualvalueon(0,∞)as 101October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems93!XmXmXmXm2E~=maxf000+βff00+β1−f0,(3.34)maxm0mm1mk=0k=0k=0k=0wherefistheapproximatevalueoffatthecollocationpoints.Theoptimalvalueof~isselectedtobethevaluethatcorrespondstotheminimumofthemaximumresidualcurve.The~-curveandmaximumresidualcurvesareshowninFigs.3.1,3.2and3.3fortheBlasius,HomannandPohlhausenflows,respectively.ItcanbeseenfromFigs.3.1,3.2and3.3thattheoptimal~obtainedusingthe~-curvesandthemaximumresidualcurvesineachcaseareequal.0.350.334−20.345100.3320.340.33−1.4−1.3−1.2−4100.335(0)00¯hmaxf0.33E−610order=2order=20.325order=4order=40.32order=8−8order=810order=12order=120.315−2−1.5−1−0.5−2−1.5−1−0.5¯h¯h(a)~-curveforBlasiusflow.(b)MaximumresidualforBlasiusflow.Fig.3.1.Optimal~fortheBlasiusflow.01.34101.3141.331.31210−21.31−0.85−0.8−0.751.32−410(0)00¯hmaxfE1.31order=2−6order=210order=4order=4order=81.3order=8order=10−8order=10101.29−1.5−1−0.50−1.2−1−0.8−0.6−0.4−0.20¯h¯h(a)~-curveforHomannflow.(b)MaximumresidualforHomannflow.Fig.3.2.Optimal~fortheHomannflow. 102October24,201311:27WorldScientificReviewVolume-9inx6inAdvances/Chap.394S.MotsaandP.Sibanda−21.164101.158order=21.1621.156order=3−4order=4101.161.154order=6−1.2−1−0.81.158(0)−600¯hmax10order=2f1.156Eorder=3order=41.154−8order=6101.152−101.1510−2−1.5−1−0.50−2−1.5−1−0.50¯h¯h(a)~-curveforPohlhausenflow.(b)MaximumresidualforPohlhausenflow.Fig.3.3.Optimal~forthePohlhausenflow.Table3.1.Skinfrictionf00(0).OrderBlasiusflowHomannflowPohlhausenflow20.33121811981.31259285751.154712416240.33200817571.31195293321.154700559360.33205397851.31193812081.154700538480.33205709081.31193770691.1547005384100.33205731771.31193769431.1547005384120.33205733481.31193769391.1547005384140.33205733611.31193769391.1547005384160.33205733621.31193769391.1547005384Ref.[12]0.33205733621.31193769391.1547005384Optimal~–1.32–0.79–0.94Table3.1presentstheresultsforthewallskinfrictionrate(definedbyf00(0))atdifferentordersoftheSHAMfortheBlasius,HomannandPohlhausenflow.TheSHAMresultsarecomparedagainsttherecentlyreportedresultsofGanapol[12]whoreportedhighlyaccurateresultsofbe-tween10and30decimalplacesusingarobustalgorithmbasedonMaclaurinserieswithconvergenceaccelerationandanalyticalcontinuationtechniques.WeobservethatalltheiterationschemesrapidlyconvergetotheresultsofciteGANAPOLtoall11displayeddecimalplaces.Fullconvergenceisachievedafter16iterationsfortheBlasiusflow,12iterationsfortheHomannandafteronly6iterationsinthecaseofthePohlhausenflow.TheresultsforBlasiusandHomannflowsweregeneratedusingη∞=30andN=100collocationpointsandthePohlhausenflowresultsweregeneratedusingη∞=15andN=50collocationpoints. 103October24,201311:27WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems953.3.2.EigenvalueproblemsInthissectionweillustratehowthespectralHAMcanbeusedtosolvenonlineareigenvalueproblems.WeconsiderasourfirstexampletheLane–Emdenequationdescribedbythenonlinearsingularinitialvaluedifferentialequation0020py(x)+y(x)+y=0,(3.35)xwithinitialconditionsy(0)=1,y0(0)=0,(3.36)wherep∈[0,5]isaconstantparameter.Thisequationhasapplicationsinastrophysicsinthestudyofpolytropicmodelsandstellarstructures[9,10].Forthespecialcaseswhenp=0,1,5exactanalyticalsolutionswereobtainedbyChandrasekhar[9].ForallothervaluesofpapproximateanalyticalmethodsandnumericalmethodsareusedtoapproximatethesolutionoftheLane–Emdenequation.Here,weconsiderthecasewhenp=2.Tosolveequation(3.35)usingtheSHAM,itisconvenienttorecasttheproblemfromaninitialvalueproblemtoaboundaryvalueproblembyconsideringonlythedomainx∈[0,α]whereαisthefirstzeroofy(x).InmostpracticalapplicationsoftheLane–Emdenequation(3.35),thegoalistointegratethegoverningequationfrom0toα.Sinceαisanunknownparameterwerescaletheproblembysettingx=αz.(3.37)Substituting(3.37)inequation(3.35)andsimplifyinggives22yzz+yz+λy=0,y(0)=1,y(1)=0,yz(0)=0,(3.38)zwhichisanonlineareigenvalueproblemwithλ=α2astheeigenvalue.ToobtainthelinearoperatorandinitialguesstobeusedintheSHAMsolutionof(3.38),wesety(z)=y(z)+g(z),λ=λ+γ,y(z)=1−z2(3.39)000wherey0hasbeenchosentosatisfytheboundaryconditionsandλ0isaninitialapproximationtoλ.SubstitutingEq.(3.39)inEq.(3.38)givesg00+2g0+2λyg+y2γ+2yγg+λg2+γg2+φ(z)=0,z00000(3.40)g(0)=g0(0)=g(1)=0, 104October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.396S.MotsaandP.Sibandawheretheprimesdenotedifferentiationwithrespecttoz,and00202φ(z)=y0+y0+λ0y0.zByconsideringthelinearpart,theinitialguessesg0andγ0areobtainedassolutionsof002020g0+g0+2λ0y0g0+y0γ0=−φ,g0(0)=g0(0)=g0(1)=0.(3.41)zApplyingtheChebyshevspectralmethodto(3.41)andimposingthebound-aryconditionsgivesXNAG+y2γ=−Φ,g(τ)=g(τ)=0,Dg(τ)=0,(3.42)0000N00Nk0kk=0whereτ=2z−1isusedtotransformtheintervalz∈[0,1]toτ∈[−1,1].Thus,D=2DwhereDistheChebyshevderivativematrixdefinedby(3.14)andG=[g(τ),g(τ),...,g(τ),g(τ)]T,000010N−10NΦ=[φ(z),φ(z),...,φ(z),φ(z)]T,01N−1NTy0=[y0(z0),y0(z1),...,y0(zN−1),y0(zN)],22A=D+diagD+2λ0diag[y0],zz=[z,z,z,...,z,z]T.012N−1NTheequationsystem(3.42)canbewrittenasthefollowingmatrixequa-tion10···000g0(τ0)02y0(τ1)g0(τ1)φ(z1)......A..=..2y0(τN−1)g0(τN−1)φ(zN−1)00···010g0(τN)0DN0DN1···DNN−1DNN0γ00ThelinearoperatorfortheSHAMalgorithmisdefinedas00202L(g,γ)=g+g+2λ0y0g+y0γ.(3.43)z 105October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems97WeremarkthatthelinearoperatorischoseninsuchawaythatL(g0,γ0)+φ(z)=0.Fromthegoverningequation(3.40),thenonlinearoperatorisdefinedasN(g,γ)=L(g,γ)+2yγg+λg2+γg2.(3.44)00IntheframeworkoftheHAM,thezeroth-orderdeformationequationreads(1−q)L[{G(z;q),Γ}−{g0(z),γ0}]=q~{N[G(z;q),Γ]+φ(z)},(3.45)whereq∈[0,1]istheembeddingparameter,G(z;q),Γarecontinuousmappingsofg(z)andγ,respectively,and~istheconvergencecontrollingparameter.ThemthorderdeformationequationsreadsL(gm,γm)=(χm+~)L(gm−1,γm−1)+~(1−χm)φ(z)mX−1mX−1+2~y0γngm−1−n+~λ0gngm−1−nn=0n=0mX−1Xn+~γm−1−ngign−i,(3.46)n=0i=0subjecttotheboundaryconditiong(0)=g(1)=g0(0)=0,(3.47)mmmwhereL(g,γ)isdefinedby(3.43).Applyingthespectralmethodtoequation(3.46)andimposingtheboundaryconditionsgivesthefollowingmatrixequation,10···000gm(τ0)2y0(τ1)gm(τ1)....A..2y0(τN−1)gm(τN−1)00···010gm(τN)DN0DN1···DNN−1DNN0γm 106October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.398S.MotsaandP.Sibanda00···000gm−1(τ0)2y0(τ1)gm−1(τ1)....=(χm+~)A..y2(τ)g(τ)0N−1m−1N−100···000gm−1(τN)00···000γm−100φ(z1)Q(z1)....+~(1−χm).+~.,(3.48)φ(zN−1)Q(zN−1)0000wheremX−1mX−1mX−1XnQ(z)=2~y0γngm−1−n+~λ0gngm−1−n+~γm−1−ngign−1.n=0n=0n=0i=0Thus,thehomotopyseriessolutionofy(z)andλreadX+∞X+∞y(z)=y0(z)+g0(z)+gk(z),λ=λ0+γ0+γm.(3.49)k=1k=1Forafixedλ0,theminimumofthegraphofthemaximumresidualagainst~givestheoptimal~.Optimalλcanbefoundbyfixing~andlocatingtheminimumofthegraphofthemaximumresidualagainstλ0.Themaximumresidualisdefinedas!XmXmXm22E~,λ0=maxy00+y0+y,(3.50)maxmxmmk=0k=0k=0whereyistheapproximatevalueofyatthecollocationpoints.Themax-imumresidualcurvesfor~andλ0areshowninFig.3.4.Itcanbeseen 107October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems99fromFig.3.4thattheminimaofthemaximumresidualsarewelldefined.Thevalueofthe~(andλ0)atwhichtheminimumoftheresidualcurveislocatedistheoptimalvaluethatgivesthebestconvergenceresultsfortheSHAMalgorithm.501010order=5order=10order=200order=30−510100¯hmaxλmaxEEorder=5−5−10order=101010order=20order=30order=40−10−151010−1.5−1−0.501012141618¯hλ0(a)~-Residualwhenλ0=15.(b)λ0-Residualwhen~=−1.Fig.3.4.MaximumresidualcurvesfortheSHAMsolutionofLane–Emdenequation.InTable3.2,wepresenttheSHAMapproximatesolutionoftheLane–Emdenequationforthefirstzeroαwhichisobtainedusingλ0=15andanoptimal~=−1.03.Theresultsarecomparedwiththerecentlyreportedaccurateresultsof[6,25].WeseethattheSHAMresultsconvergetotheresultsof[6,25].Table3.2.Firstzeroαusingλ0=15,N=30and~=−1.03.OrderFirstzeroα04.393212092624.355030047144.352897496184.3528452341164.3528742136204.3528745567244.3528745919304.3528745958324.3528745959344.3528745959Ref.[6,25]4.3528745959 108October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3100S.MotsaandP.Sibanda3.3.3.BoundaryvalueproblemswithmultiplesolutionsInthissection,weillustratetheapplicationoftheSHAMinsolvingnon-linearboundaryvalueproblemswithmultiplesolutions.Weconsiderthefollowingmodelofmixedconvectioninaporousmediumwithboundaryconditionsonthesemi-infiniteinterval[0,∞)whichadmitsmultiple(dual)solutions[1,18],000002002f+f−(f)=0,f(0)=0,f(0)=1+b,f(∞)=1,(3.51)wheretheprimesdenotedifferentiationwithrespecttoasimilarityvariableη,fisadimensionlessstreamfunction,andbisaconstant.Thenonlinearequation(3.51)wasreported[18]tohavemultiplesolutions"√√!#0132η13+3+2bf(η)=−+tanh√±ln√√(3.52)222223−3+2bforanygivenvalueofb∈[−3/2,0).TosolveEq.(3.51)usingtheSHAM,itisconvenienttoreducetheorderofthedifferentialequationbyintroducingthetransformationf0=u.Thisresultsinthereducedequation,0022u+u−u=0,u(0)=1+b,u(∞)=1.(3.53)InthecontextoftheSHAM,webeginbychoosingtheinitialguessu(η)=1+(b+ση)e−η,(3.54)0whereσisanunknownconstant.Weremarkthatchoosingdifferentval-uesofσleadstodifferentinitialguesses.Inparticular,weobservethatwhenvaryingσbetweennegativeandpositivevalues,theconcavityoftheprofileofu0changesintheregionnearη=0.Byfixingthevalueoftheconvergence-controlparameter~andκ,andvaryingthevaluesofσ,intheSHAMimplementation,optimalvaluesofσcanbeidentifiedfromtheresidualof(3.53).ToobtainthelinearoperatorandinitialguesstobeusedintheSHAMsolutionof(3.53),wesetu(η)=u0(η)+w(η).(3.55)SubstitutingEq.(3.55)inEq.(3.53)gives00112w+w−u0w−w+φ(η)=0,w(0)=w(∞)=0,(3.56)22 109October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems101where00112φ(η)=u0+u0−u0.22Byconsideringthelinearpart,theinitialguessw0forsolvingthenon-linearequation(3.56)usingtheSHAMisobtainedasasolutionof001w0+−u0w0+φ(η)=0,w0(0)=w0(∞)=0.(3.57)2Thelinearoperator,fordevelopingtheSHAMalgorithmisdefinedas001L(w)=w+−u0w.(3.58)2ThelinearoperatorischoseninsuchawaythatL(w0)+φ(η)=0.Thus,thecorrespondingnonlinearoperatorbecomes12N(w)=L(w)−w.(3.59)2IntheframeworkoftheHAM,thezeroth-orderdeformationequationbecomeshino(1−q)LW˜(η;q)−w0(η)=q~N[W˜(η;q)]+φ(η),(3.60)whereq∈[0,1]istheembeddingparameter,W˜(η;q)isthecontinuousmappingofw(η)and~istheconvergencecontrollingparameter.Themth-orderdeformationequationsreads001wm+−u0wm2001=(χm+~)wm−1+−u0wm−12mX−11+~(1−χm)φ(η)−~wnwm−1−n,(3.61)2n=0subjecttotheboundaryconditionswm(0)=wm(∞).(3.62)ApplyingtheSHAMin(3.61)itcaneasilybeshownthattheSHAMre-cursiveschemeisobtainedas"#mX−1−11Wm=(χ+~)Wm−1+Ec~(1−χm)φ(η)−~WmWm−1−n,2n=0 110October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3102S.MotsaandP.Sibandawhere21Ec=D−I−diag[u0],2Disthedifferentiationmatrix,Iistheidentitymatrixandu0istheinitialguessevaluatedatthecollocationpoints.Figure3.5showsthemaximumresidualcurveforafixed~=−1plottedagainstvaryingvaluesofσ.ThegraphhastwolocalminimaatwhichthemaximumresidualisverysmallanddecreaseswithanincreaseintheorderoftheSHAMapproximation.Figure3.6givesthetwosolutionsofthevelocityprofilegeneratedusingσ=0andσ=−1.3.Theresultsarecomparedwiththeexactsolution(3.52)andgoodagreementisobservedbetweenthetworesults.1001051080100100601010−510−10−1.5−11040σmax10−20E10−0.500.52010010−2010−2.5−2−1.5−1−0.500.511.52σFig.3.5.Maximumresidualcurvewhenb=−1,~=−1order4(solidline),order8(dottedline),order12(dashedline).3.3.4.CouplednonlinearboundaryvalueequationsInthissectionwediscusstheextensionoftheSHAMalgorithmtocoupledsystemsofnonlinearboundaryvalueproblems.Forillustrationpurposesweconsidersystemsoftwoandthreecouplednonlineardifferentialequations.WebeginwiththeoriginalvonK´arm´anequationsforthesteady,laminar,axially-symmetricviscousflowinducedbyaninfinitelyrotatingdisk.Thegoverningequationsfortheproblemaregiven[15,17,19]insimilarity 111October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems10310.5)η(0f0−0.5051015ηFig.3.6.ComparisonbetweenexactandSHAMsolutionwhenb=−1,~=−1,κ=0.5usingσ=0(upperbranch)andσ=−1.3(lowerbranch).variableform,by000001002H(η)−H(η)H(η)+H(η)H(η)−2G(η)=0,(3.63)2G00(η)−H(η)G0(η)+H0(η)G(η)=0,(3.64)subjecttotheboundaryconditionsH(0)=H0(0)=H0(∞)=0,G(0)=1,G(∞)=0,(3.65)whereGistheazimuthalvelocityandHistheaxialvelocity.ForeffectiveapplicationoftheSHAMon(3.63)–(3.65),wehomogenizethesystembyintroducingthefollowingtransformationsH(η)=h(η)+H0(η),G(η)=g(η)+G0(η),(3.66)whereH0andG0areinitialapproximationsthatarechosentosatisfytheboundaryconditions(3.65).Following[17],wechoose,asinitialguesses,thefollowingfunctionsH(η)=−1+e−η+ηe−η,G(η)=e−η.(3.67)00Substitutingequation(3.66)in(3.63)–(3.65)gives,000000000001022h−H0h+H0h−H0h−4G0g−hh+h−2g=φ1(η),(3.68)200000000g−H0g+H0g−G0h+G0h−hg+hg=φ2(η),(3.69) 112October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3104S.MotsaandP.Sibandasubjecttotheboundaryconditions00h(0)=h(0)=h(∞)=0,g(0)=0,g(∞)=0,(3.70)where000001002φ1(η)=−H0+H0H0−H0H0+2G0,20000φ2(η)=−G0+H0G0−H0G0.TheinitialapproximationtobeusedintheSHAMalgorithmisobtainedbysolvingthelinearpartofequations(3.68)–(3.70),namelyh000−Hh00+H0h0−H00h−4Gg=φ(η),(3.71)0000000001g00−Hg0+H0g−G0h+Gh0=φ(η),(3.72)0000000002subjecttotheboundaryconditionsh(0)=h0(0)=h0(∞)=0,g(0)=0,g(∞)=0.(3.73)00000TheChebyshevspectralcollocationmethodisthenappliedtosolveh0andg0ofEqs.(3.71)–(3.73).Thisgives,AF0=Φ,(3.74)subjecttotheboundaryconditionsXNXND0kh0(ξk)=0,DNkh0(ξk)=0,h0(ξN)=0,(3.75)k=0k=0g0(ξ0)=0,g0(ξN)=0,(3.76)whereξ=2η/η∞−1isavariableusedtomapthedomain[0,η∞]to[−1,1],η∞isafinitevalueusedtonumericallyapproximatetheconditionsatinfinity,Nisthenumberofcollocationpoints,and32000D−H0D+H0D−H0−4G0A=020,(3.77)G0D−G0D−H0D+H0F=[h(ξ),h(ξ),...,h(ξ),g(ξ),g(ξ),...,g(ξ)]T,000010N00010NΦ=[φ(η),φ(η),...,φ(η),φ(η),φ(η),...,φ(η)]T,(3.78)10111N20212NH0=diag[H0(η0),H0(η1),...,H0(ηN)],G0=diag[G0(η0),G0(η1),...,G0(ηN)]. 113October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems105ToobtaintheSHAMsolutionof(3.68)and(3.69)webeginbydefiningthelinearoperatorsL[h,˜g˜]=h˜000−Hh˜00+H0h˜0−H00˜h−4Gg,˜(3.79)h0000L[h,˜g˜]=˜g00−Hg˜0+H0g˜−G0˜h+Gh˜0,(3.80)g0000whereq∈[0,1]istheembeddingparameter,andh˜(ξ;q)and˜g(ξ;q)areunknownfunctions.Weobservethatthelinearoperators(3.79–3.80)arecoupled.ThisisoneofthemainfeaturesoftheSHAMwhenappliedtononlinearsystemsofBVPgoverningbytwoormorecoupledequations.Thezeroth-orderdeformationequationsreadno(1−q)Lh[{h˜(η;q),g˜(η;q)}−h0]=q~Nh[h˜(η;q),g˜(η;q)]−φ1,(3.81)no(1−q)Lg[{h˜(η;q),g˜(η;q)}−g0]=q~Ng[h˜(η;q),g˜(η;q)]−φ2,(3.82)where001022Nh[h,g]=Lh[h,g]−hh+h−2g,(3.83)200Ng[h,g]=Lg[h,g]−hg+hg.(3.84)From(3.81)–(3.84),itcanbeshownthatthehigh-orderdeformationequationsaregivenbyLh[hm,gm]=(~+χm)Lh[hm−1,gm−1]−φ1(η)~(1−χm)mX−110000+~hnhm−1−n−hnhm−1−n−2gngm−1−n,(3.85)2n=0Lg[hm,gm]=(~+χm)Lg[hm−1,gm−1]−φ2(η)~(1−χm)mX−100+~(hngm−1−n−gnhm−1−n),(3.86)n=0subjecttotheboundaryconditionsh(0)=h0(0)=h0(∞)=0,g(0)=g(∞)=0.(3.87)mmmmmNotethatthehigh-orderdeformationequations(3.85)–(3.87)arecoupled.ApplyingtheChebyshevpseudo-spectralmethodtoEqs.(3.85)–(3.87)givesAFm=(χm+~)AFm−1−~(1−χm)Φ+~Qm−1,(3.88) 114October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3106S.MotsaandP.SibandasubjecttotheboundaryconditionsXNXND0khm(ξk)=0,DNkhm(ξk)=0,hm(ξN)=0,(3.89)k=0k=0gm(ξ0)=0,gm(ξN)=0,(3.90)whereAandΦarerespectivelydefinedby(3.77)and(3.78),andF=[h(ξ),h(ξ),...,h(ξ),g(ξ),g(ξ),...,g(ξ)]T,mm0m1mNm0m1mNmX−112(Dhn)(Dhm−1−n)−hn(Dhm−1−n)−2gngm−1−n2Qm−1=n=0.mX−1(Dhn)gm−1−n−(Dgn)hm−1−nn=0Thus,startingfromtheinitialapproximation,whichisobtainedfrom(3.74),higherorderapproximationsFm(ξ)form≥1,canbeobtainedthroughtherecursiveformula(3.88).InTable3.3wegivetheSHAMcomputedvaluesofH(∞),H00(0)andG0(0)atdifferentordersofapproximation.Theresultsarecom-paredagainstresultsgeneratedusingMATLAB’sbvp4croutineforsolvingboundaryvalueproblems.Itcanbeseenthatfullconvergencetothebvp4cresultsisachievedaftertenortwelveiterations.Table3.3.SHAMvaluesofH(∞),H00(0),G0(0)atdifferentordersofapproximationwhen~=−1,N=100andη∞=22.OrderH(∞)H00(0)G0(0)2−0.88424618−1.02167240−0.614995614−0.88447172−1.02040332−0.615946916−0.88447466−1.02046906−0.615921068−0.88447401−1.02046498−0.6159220510−0.88447410−1.02046526−0.6159220112−0.88447409−1.02046524−0.6159220114−0.88447409−1.02046524−0.61592201bvp4c−0.88447409−1.02046524−0.61592201InFig.3.7wegiveacomparisonbetweentheSHAMandbvp4cgener-atedresultsforthevelocityprofilesH(η)andG(η).Weobservethatthereisgoodagreementbetweenthetworesults.Next,weconsidertheSHAMapplicationonathree-equationsystemthatmodelstheproblemofunsteadyfreeconvectiveheatandmasstransfer 115October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems107110.80.80.60.6)η)(η(HG−0.40.40.20.20002468100246810ηη(a)Profilefor−H(η).(b)ProfileforG(η).Fig.3.7.ComparisonbetweenSHAM(solidline)andbvp4cresultsforthevelocityprofiles.onastretchingsurfaceinaporousmediuminthepresenceofachemicalreaction.Thegoverningequations[8,24]forthisproblemaregivenasthefollowingdimensionlesssystemofequations000000200η00f+ff−(f)−Kf−Af+f+Grθ+Gcφ=0,(3.91)21000010θ−fθ+fθ−Aθ+ηθ=0,(3.92)Pr21000010φ−fφ+fφ−Aφ+ηφ−γφ=0,(3.93)Sc2subjecttotheboundaryconditionsf(0)=f,f0(0)=1,θ(0)=1,φ(0)=1,(3.94)wf0(∞)=0,θ(∞)=0,φ(∞)=0,(3.95)wheref(η),θ(η)andφ(η)are,respectively,thedimensionlessvelocity,tem-peratureandconcentration,fwisthesuction/injectionparameter,γisthechemicalreactionconstant,PristhePrandtlnumber,ScistheSchmidtnumber,Kisthepermeabilityparameter,GrandGcarethetemperatureandconcentrationdependentGrashofnumbersrespectively.ToapplytheSHAMon(3.91)–(3.95),webeginbyhomogenizingthesystembyintroducingthefollowingtransformationsf(η)=F(η)+f0(η),θ(η)=G(η)+θ0(η),φ(η)=H(η)+φ0(η)(3.96)wheref0,θ0andφ0areinitialapproximationsthatarechosentosatisfytheboundaryconditions(3.94)–(3.95).Theappropriateinitialguessesare 116October24,201311:45WorldScientificReviewVolume-9inx6inAdvances/Chap.3108S.MotsaandP.Sibandachosenas−η−η−ηf0(η)=fw+1−e,θ0(η)=e,φ0(η)=e.(3.97)Substitutingequation(3.96)in(3.91)–(3.95)gives,00000η00000000F−KF−AF+F+f0F−2f0F+f0F2+FF00−(F0)2+GrG+GcH=φ,(3.98)1100100000G−AG+ηG−f0G+f0G−θ0F+θ0FPr2−F0G+FG0=φ,(3.99)2100100000H−AH+ηH−γH−f0H+f0H−φ0F+φ0FSc2−F0H+FH0=φ,(3.100)3subjecttotheboundaryconditions0F(0)=0,F(0)=0,G(0)=0,H(0)=0,(3.101)F0(∞)=0,G(∞)=0,H(∞)=0,(3.102)where000000200η00φ1(η)=−f+ff−(f)−Kf−Af+f+Grθ+Gcφ,21000010φ2(η)=−θ−fθ+fθ−Aθ+ηθ,Pr21000010φ3(η)=−φ−fφ+fφ−Aφ+ηφ−γφ.Sc2TheinitialapproximationtobeusedintheSHAMalgorithmisob-tainedbysolvingthelinearpartofequations(3.98)–(3.100),subjecttotheboundaryconditions(3.101)–(3.102),thatis,wesolve00000η00000000F0−KF0−AF0+F0+f0F0−2f0F0+f0F0+GrG0+GcH0=φ1,2100100000G0−AG0+ηG0−f0G0+f0G0−θ0F0+θ0F0=φ2,Pr2100100000H0−AH0+ηH0−γH0−f0H0+f0H0−φ0F0+φ0F0=φ3,Sc2subjecttotheboundaryconditions0F0(0)=0,F0(0)=0,G0(0)=0,H0(0)=0,(3.103)0F0(∞)=0,G0(∞)=0,H0(∞)=0.(3.104) 117October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems109TheChebyshevspectralcollocationmethodisthenappliedtosolvetheabovelinearequationsaboutF0,G0andH0.Thisgives,BΨ0=Φ,(3.105)subjecttotheboundaryconditionsXNXND0kF0(ξk)=0,DNkF0(ξk)=0,F0(ξN)=0,(3.106)k=0k=0G0(ξ0)=0,G0(ξN)=0,H0(ξ0)=0,H0(ξN)=0,(3.107)whereΨ=[F(ξ),...,F(ξ),G(ξ),...,G(ξ),H(ξ),...,H(ξ)]T,0000N000N000NΦ=[φ(η),...,φ(η),φ(η),...,φ(η),φ(η),...,φ(η)]T,101N202N303NF0=diag[F0(η0),F0(η1),...,F0(ηN)],H0=diag[H0(η0),H0(η1),...,H0(ηN)],G0=diag[G0(η0),G0(η1),...,G0(ηN)],(3.108)andB11B12B13B=B21B22B23(3.109)B31B32B33with30η22B11=D−KD−AD+diagD+diag(f0)D2000−2diag(f0)D+diag(f0),B=GrI,B=GcI,B=−diag(θ)D+diag(θ0),121321001210B22=D−AI+diag(η)D−diag(f0)G0+diag(f0)D,Pr2B=O,B=−diag(φ)D+diag(φ0),B=O,233100321210B33=D−AI+diag(η)D−γI−diag(f0)G0+diag(f0)D.Sc2ThelinearoperatortobeusedintheSHAMsolutionof(3.91)–(3.95) 118October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.3110S.MotsaandP.Sibandaisgivenby00000η000000L1[F,G,H]=F−KF−AF+F+f0F−2f0F2+f00F+GrG+GcH,(3.110)010010000L2[F,G,H]=G−AG+ηG−f0G+f0G−θ0FPr2+θ0F,(3.111)01001000L3[F,G,H]=H−AH+ηH−γH−f0H+f0HSc200−φ0F+φ0F.(3.112)Thecorrespondingzeroth-orderdeformationequationsaregivenby(1−q)L1[{F˜(η;q),G˜(η;q),H˜(η;q)}−{F0,G0,H0}]no=q~N1[F˜(η;q),G˜(η;q),H˜(η;q)]−φ1,(3.113)(1−q)L2[{F˜(η;q),G˜(η;q),H˜(η;q)}−{F0,G0,H0}]no=q~N2[F˜(η;q),G˜(η;q),H˜(η;q)]−φ2,(3.114)(1−q)L3[{F˜(η;q),G˜(η;q),H˜(η;q)}−{F0,G0,H0}]no=q~N3[F˜(η;q),G˜(η;q),H˜(η;q)]−φ3,(3.115)whereN[F,G,H]=L[F,G,H]+FF00−(F0)2,(3.116)11N[F,G,H]=L[F,G,H]−F0G+FG0,(3.117)2200N3[F,G,H]=L3[F,G,H]−FH+FH.(3.118)From(3.113)–(3.118),itcanbeshownthatthehigh-orderdeformation 119October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems111equationsreadL1[Fm,Gm,Hm]=(~+χm)L1[Fm−1,Gm−1,Hm−1]−φ1(η)~(1−χm)mX−1 +~FF00−F0F0,(3.119)nm−1−nnm−1−nn=0L2[Fm,Gm,Hm]=(~+χm)L2[Fm−1,Gm−1,Hm−1]−φ2(η)~(1−χm)mX−100+~(FnGm−1−n−FnGm−1−n),(3.120)n=0L3[Fm,Gm,Hm]=(~+χm)L3[Fm−1,Gm−1,Hm−1]−φ3(η)~(1−χm)mX−1+~(FH0−F0H),(3.121)nm−1−nnm−1−nn=0subjecttotheboundaryconditionsF(0)=F0(0)=F0(∞)=0,G(0)=G(∞)=0,mmmmm(3.122)Hm(0)=Hm(∞)=0.ApplyingtheChebyshevpseudo-spectraltransformationtoequations(3.119)–(3.122)givesBPm=(χm+~)BPm−1−~(1−χm)Φ+~Qm−1,(3.123)subjecttotheboundaryconditionsXNXND0kFm(ξk)=0,DNkFm(ξk)=0,Fm(ξN)=0,(3.124)k=0k=0Gm(ξ0)=0,Gm(ξN)=0,Hm(ξ0)=0,Hm(ξN)=0(3.125)whereBandΦaredefinedby(3.108)and(3.109),respectively,andP=[F(ξ),...,F(ξ),G(ξ),...,G(ξ),H(ξ),...,H(ξ),]T,mm0mNm0mNm0mNmX−12Fn(DFm−1−n)−(DFn)(DFm−1−n)n=0mX−1Qm−1=[(DGn)Fm−1−n−(DFn)Gm−1−n].n=0mX−1[(DHn)Fm−1−n−(DFn)Hm−1−n],n=0Thus,startingfromtheinitialapproximation,whichisobtainedasaso-lutionofequations(3.105)–(3.106),theSHAMapproximatesolutionsforf,θ,φareobtainedbyrecursivelysolvingequations(3.123)–(3.124). 120October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3112S.MotsaandP.SibandaInTable3.4wepresenttheSHAMcomputationsofflowproperties,namelytheskinfrictionf00(0),surfaceheattransferrateattheθ0(0)andmasstransferrateatthewallφ0(0).TheaccuracyoftheSHAMresultsareverifiedbycomparingwiththeMATLABin-builtroutinebvp4c.ItcanbeseenfromtheresultsthattheSHAMresultsconvergetothebvp4cresults.Table3.4.SHAMvaluesoff00(0),θ0(0),φ0(0)atdifferentordersofapproximationwhen~=−1,N=100andη∞=30.Orderf00(0)θ0(0)φ0(0)2−1.58973019−1.89527636−2.243051964−1.59345387−1.89839841−2.245405466−1.59362599−1.89853714−2.245489198−1.59363732−1.89854621−2.2454940710−1.59363819−1.89854691−2.2454944212−1.59363826−1.89854697−2.2454944514−1.59363827−1.89854697−2.2454944616−1.59363827−1.89854697−2.24549446bvp4c−1.59363827−1.89854697−2.245494463.4.Convergenceacceleration3.4.1.ConvergenceaccelerationthroughchoiceoflinearoperatorIntheframeworkoftheHAM,thereisgreatfreedomtochoosethelinearoperatorforthezeroth-orderandhigherorderdeformationequations.Inthissection,wesuggestageneralapproachthatcanbeusedtoacceleratetheconvergenceoftheSHAM.WeillustratehowusingadifferentlinearoperatorcansignificantlyimproveconvergenceoftheSHAMbyconsider-ingthefollowingDarcy–Brinkman–Forchheimerequationthatmodelsthesteadystatepressuredrivenfully-developedparallelflowthroughahori-zontalchannelthatisfilledwithporousmedia[2,20,28],d2y122−sy−Fsy+=0,y(−1)=0,y(1)=0,(3.126)dx2MwhereFisthedimensionlesstheForchheimernumberandsistheporousmediashapeparameter.Thisproblemwaspreviouslysolvedusingthe 121October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems113SHAMin[20]InthecontextoftheSHAM,theinitialguessischosentobe1cosh(sx)y0(x)=1−,(3.127)s2Mcosh(s)whichisthesolutionofthelinearpartof(3.126).Usingthelinearpartof(3.126)toformthelinearoperatorforthezeroth-orderdeformationequa-tions,itcanbeshownthatthehigh-orderdeformationscheme(see[20]fordetails)isgivenby002002~ym−sym=(χm+~)(ym−1−sym−1)+(1−χm)MmX−1−~Fsynym−1−n,(3.128)n=0subjecttotheboundaryconditionsym(−1)=ym(1)=0.(3.129)Startingfromtheinitialapproximation(3.127),thehigh-orderdeforma-tionequations(3.128)canbesolvediterativelyforym,m≥1andtheapproximatesolutiony(x)isgiven,inseriesform,asX+∞y(x)=ym(x).(3.130)m=0Themodifiedlinearoperatorisobtainedbyintroducingthetransfor-mationy(x)=y0(x)+u(x),(3.131)wherey0(x)istheinitialapproximationgivenby(3.127).Substituting(3.131)intothegoverningequation(3.126)gives,u00−s2u−2Fsyu−Fsu2+φ(x)=0,u(1)=u(−1)=0,(3.132)0where00212φ(x)=y0−sy0+−Fsy0.MThus,theinitialguessforsolvingthenonlinearequation(3.132)usingtheSHAMisu00−s2u−2Fsyu+φ(x)=0,u(1)=u(−1)=0.(3.133)000000Thelinearoperatorforischosenasfunctionofy0(x)asL(u)=u00−s2u−2Fsyu.(3.134)0 122October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3114S.MotsaandP.SibandaWeremarkthatthelinearoperatorischoseninsuchawaythatL(u0)+φ(x)=0.Basedonthegoverningequation(3.132),thenonlinearoperatorisde-finedas2N(u)=L(u)+N1(u),N1(u)=−Fsu.(3.135)Thus,intheframeworkoftheHAM,thezeroth-orderdeformationequationbecomes(1−q)L[U(x;q)−u0(x)]=q~{N[U(x;q)]+φ(x)},(3.136)whereq∈[0,1]istheembeddingparameter,U(x;q)isacontinuousmap-pingofu(x)and~istheconvergencecontrollingparameter.Themth-orderdeformationequationscorrespondingto(3.136)aregivenbyL[um(x)−(χm+~)um−1(x)]=~Rm−1[u0,u1,...,um−1],(3.137)where1∂m−1{N[U(x;q)]+φ(x)}R[u,u,...,u]=1.m−101m−1(m−1)!∂qm−1q=0Thus,thehomotopyseriessolutionthatapproximatesy(x)readsX+∞y(x)=y0(x)+u0(x)+uk(x).(3.138)k=1Usingthedefinitions(3.134)and(3.135),thehigh-orderdeformationequationsaregivenasu00−s2u−2Fsyu=(χ+~)(u00−s2u−2Fsyu)mm0mmm−1m−10m−1mX−1+~(1−χm)φ(x)−~Fsumum−1−n.n=0(3.139)Fromequation(3.139)itcaneasilybeshownthattheSHAMrecursiveschemeisobtainedas"#mX−1U=(χ+~)U+E−1~(1−χ)φ(x)−~FsUU,mm−1smmm−1−nn=0whereE=D2−s2I−2Fsdiag[y].s0 123October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems115Table3.5givesacomparisonbetweentheresultsobtainedusingequa-tions(3.130)and(3.138)forthemaximumresidualobtainedusingtheoptimal~,ineachcase,atdifferentordersofapproximation.TheresultsforthebasicSHAMimplementation(3.130)arelabelledasSHAM1andtheresultsfortheversionoftheSHAMthatincorporatestheinitialguessy0inthelinearoperator(3.138)arelabelledasSHAM2.ItcanbeseenfromtheTable3.5thattheconvergenceintheSHAM2resultsissignifi-cantlyfasterthanthatofSHAM1.Thisshowsthattheconvergencecanbeacceleratedbyincorporatingtheinitialguessy0andtheparameterFintothelinearoperator.Ingeneral,usingdifferentlinearoperatorsintheSHAMimplementationisexpectedtogiveresultswithdifferentlevelsofaccuracy.Table3.5.Maximumresidualatselectedoptimalvaluesof~whenM=s=1,F=2,N=101.SHAM1SHAM2OrderOptimal~Emax~Optimal~Emax~4−0.917.92E-04−13.60E-076−0.973.97E-04−11.04E-098−0.993.68E-04−11.05E-1010−1.006.75E-04−11.05E-1012−1.002.66E-04−11.05E-1014−1.001.09E-04−11.05E-1016−1.004.59E-05−11.05E-10001010−110−210−5order=410hmaxorder=6hmaxE−3E10order=10order=2order=14order=4−4order=610order=8−1010−1.1−1−0.9−0.8−0.7−0.6−1.1−1−0.9−0.8¯h¯h(a)ResidualcurveforSHAM1.(b)ResidualcurveforSHAM2.Fig.3.8.Maximumresidualcurvefors=M=1,F=2,N=101.Figures3.8showthemaximumresidualcurves,thatcanbeusedto 124October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3116S.MotsaandP.Sibandalocatetheoptimalvalueoftheconvergencecontrollingparameter~inboththeSHAM1andSHAM2approaches.3.4.2.ConvergenceaccelerationbyiterationInthissectionwediscussamodificationofthespectral(orcompactfinitedifference)homotopyanalysismethodthatisbasedonsuccessivelyup-datingtheinitialapproximationthroughiteration.Thealgorithmfortheproposedmethodisoutlinedbelowforasecondorderdifferentialequationfory(x)withknownboundaryconditionsaty(a)andy(b).Considerthenonlineardifferentialequationsy00+p(x)y0+q(x)y(x)+F(x,y,y0)=0,y(a)=y,y(b)=y,(3.140)abwhereF(x,y,y0)isanonlinearfunction,p(x)andq(x)areknownfunctionsofx,andyaandybareknownconstants.1.Startingfromagiveninitialguessy0,whichischosentosatisfytheboundarycondition,defineu(x)suchthaty(x)=u(x)+y0(x)(3.141)andsubstitutein(3.140)toobtainthefollowingequationforu(x),u00+˜p(x)u0+˜q(x)u(x)+F(x,u,u0)+φ(x)=0,(3.142)u(a)=0,u(b)=0,whereφ(x)=y00+p(x)y0+q(x)y(x)+F(x,y,y0),00000∂F0∂F0p˜(x)=p(x)+0(y0,y0),q˜(x)=q(x)+(y0,y0).∂y∂y2.Atthecurrentiteration,r,choosealinearoperatorfromthelinearpartof(3.142),L(u)=u00+˜p(x)u0+˜q(x)u(x).(3.143)rrrrThenonlinearoperatorisobtainedbyaddingthenonlinearpartof(3.142)tothelinearoperator,thatisN(u)=L(u)+F(x,u,u0).(3.144)rrrr3.Obtaintheinitialguessur,0(x)forsolving(3.142)asthesolutionofL(ur,0)+φ(x)=0.(3.145) 125October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems1174.Definethezeroth-orderdeformationequationas(1−q)L[Ur(x;q)−ur,0(x)]=q~{N[Ur(x;q)]+φ(x)},(3.146)whereq∈[0,1]istheembeddingparameter,Ur(x;q)isacontinuousmappingofur(x)and~istheconvergence-controlparameter.Themth-orderdeformationequationscorrespondingto(3.146)aregivenbyL[ur,m(x)−(χm+~)ur,m−1(x)]=~Rm−1[ur,0,ur,1,...,ur,m−1],(3.147)where1∂m−1{F[U(x;q)]+φ(x)}R=r.m−1(m−1)!∂qm−1q=0Thus,themth-orderhomotopyseriessolutionofur(x)readsXmur(x)=ur,0(x)+ur,k(x).(3.148)k=1Thecurrentestimateforthesolutiony(x)isyr(x)=y0(x)+ur(x).(3.149)Clearly,themth-orderapproximation,afterriterationsgivenbyequa-tion(3.149)satisfiestheproblem’sunderlyingboundaryconditionsandcanbeusedastheinitialapproximationatthenextiteration,r+1.5.Replacey0inStep1,bythecurrentestimateforyr(x),andrepeatSteps1–4.TheabovealgorithmprovidesuswithaniterationschemeintheframeworkoftheSHAM.Foranmhomotopyseriessolution,whenusingriterations,themethodiscalledthe[m,r]iteratedspectralhomotopyanalysismethod(iSHAM).InTable3.6wepresenttheiSHAMresultsforthesolutionoftheLane–Emdenequation.ThetabledemonstratesthesignificantimprovementintheconvergenceoftheSHAMwhentheiterationapproachisused.Usinganm=5SHAMseries,convergenceto13decimalplacesisachievedafteronly3iterations.Weremarkthattheresultscorrespondingtor=1aretheoriginalSHAMresults.Table3.7givestheresultsforthemaximumresidualoftheDarcy–Brinkman–Forchheimerproblem(3.126)usingthe[m,r]iSHAM.Again,it 126October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3118S.MotsaandP.SibandaTable3.6.FirstzeroαoftheLane–Emdenequationusingλ0=15,N=30and~=−1.OrderFirstzero,αm\r1214.39321209256224.352874964072824.33594117958774.352874595990334.35388782821284.352874595946144.35004729310294.352874595946154.35255878603754.3528745959461Ref.[6,25]4.3528745959461Table3.7.MaximumSHAMresidualfortheDar-cy–Brinkman–Forchheimerproblemwhen~=−1,M=1,s=1,F=2andN=30.m\r1214.6948e-0035.1289e-01421.6722e-0043.6195e-01437.3501e-0063.6195e-01443.6032e-0073.6195e-014canbeseenthattheiSHAMapproachresultsinacceleratedconvergenceoftheoriginalSHAMapproach.Tables3.8–3.10givetheresultsoftheiSHAMevaluationoftheskinfrictionf00(0)usingr=1,2fortheBlasius,PohlhausenandHomannflowrespectively.Again,itcanbeseenfromthetablesthattheiSHAMsig-nificantlyimprovestheconvergencetothebenchmarknumericalresultsof[12].Table3.8.Skinfrictionf00(0)forBlasiusflowusing~=−1.32.Orderf00(0)m\r1210.36124527500.332057336320.33121811980.332057336230.33297092910.332057336240.33200817570.332057336250.33211120100.332057336260.33205397850.3320573362Ref.[12]0.3320573362 127October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems119Table3.9.Skinfrictionf00(0)forPohlhausenflowusing~=−0.94.Orderf00(0)m\r1211.15819390471.154700538421.15471241621.154700538431.15470403401.1547005384Ref.[12]1.1547005384Table3.10.Skinfrictionf00(0)forHomannflowusing~=−0.79.Orderf00(0)m\r1211.33563391991.311937694021.31259285751.311937693931.31224054731.311937693841.31195293321.3119376938Ref.[12]1.31193769383.5.ConclusionInthischapter,thebasicideaoftheSpectralhomotopyanalysismethod(SHAM)forthesolutionofboundaryvalueproblemsisdescribedthroughthesolutionofFalkner–Skanboundarylayerequations.Inparticular,theBlasius,PohlhausenandHomannflowsarediscussed.TheSHAMusestheChebyshevspectralcollocationmethodtosolvethelinearizedhigherorderdeformationequationswhicharedevelopedusingtheconceptoftheoriginalHAM.Inthiswork,wepresentageneralapproachofimplementingtheSHAMandintroducetwomethodsofidentifyingtheoptimalconvergence-controlparameter~thatcontrolsandadjustsaccuracyandconvergenceoftheSHAM.Inthefirstapproach,itissuggestedthattheoptimal~isthevaluethatliesattheintersectionof~-curvesplottedatdifferentordersoftheSHAMapproximation.Inthesecondapproach,theoptimal~isidentifiedtobethevalueatwhichtheminimumofthemaximumresidualcurveislocated.TheapplicationoftheSHAMisalsoextendedtosolveanonlineareigenvalueproblemderivedfromtheLane–Emdenequation.ThechapteralsodemonstratesthattheconvergenceoftheSHAMcan 128October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3120S.MotsaandP.Sibandabeacceleratedbyusingalinearoperatorthatincludestheinitialguessandasmanyofthegoverningphysicalconstantsofthegoverningequationsaspossible.Thiscanbeachievedbyhomogenizationoftheboundarycondi-tionsusingafunctiondescribedintermsoftheinitialapproximationofthesolution.ThestudyalsoillustrateshowtheSHAMcanbeusedtosolvenonlin-earequationswithmultiple(dual)solutionsthatariseinfluidmechanicsapplications.Thisisachievedthroughtheintroductionofanunknownpa-rameterintotheinitialguesswhichwhenvariedchangestheconcavityoftheinitialguessprofile.AniteratedversionoftheSHAMisalsosuggested.Thisapproachinvolvessuccessivemodificationoftheinitialguessandlin-earoperatorusedintheSHAMalgorithm.Theiteratedversion,callediSHAM,isshowntobesignificantlymoreaccurateandrobustthanthestandardSHAM.Lastly,theextensionoftheSHAMtononlinearsystemsoftwoandthreeequationsisdemonstrated.Itisnotedthatforsystemsoftwoormoreequations,thelinearoperatorsusedintheSHAMimplementationmaybecoupled.Thus,theSHAMhavegreatpotentialtobeapplicableinmorecompli-catedproblemsofscienceandengineering,includingsomenonlinearpartialdifferentialequations.References[1]S.AbbasbandyandE.Shivanian,Multiplesolutionsofmixedconvectioninaporousmediumonsemi-infiniteintervalusingpseudo-spectralcollocationmethod,CommunicationsinNonlinearScienceandNumericalSimulation,16,2745–2752(2011).[2]S.M.Rassoulinejad-Mousavi,S.Abbasbandy.AnalysisofforcedconvectioninacirculartubefilledwithaDarcy-Brinkman-Forchheimerporousmediumusingspectralhomotopyanalysismethod.J.FluidEng.133(2011)101207.[3]Z.P.Atabakan,A.Kili¸cman,A.K.Nasab,OnspectralhomotopyanalysismethodforsolvinglinearVolterraandFredholmintegrodifferentialequa-tions,AbstractandAppliedAnalysis,Vol.2012,ArticleID960289,16pagesdoi:10.1155/2012/960289(2012).[4]W.Auzinger,E.Karner,O.KochandE.Weinm¨uller,Collocationmeth-odsforthesolutionofeigenvalueproblemsforsingularordinarydifferentialequations,OpusculaMath.26(2),229–241(2006).[5]R.E.BellmanandR.E.Kalaba,Quasilinearizationandnonlinearboundary-valueproblems,Elsevier,NewYork,(1965).[6]J.P.Boyd,ChebyshevSpectralMethodsandtheLane-EmdenProblem,Nu-mer.Math.Theor.Meth.Appl.4,142–157(2011). 129October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.3SpectralHomotopyAnalysisMethodforNonlinearBoundaryValueProblems121[7]C.Canuto,M.Y.Hussaini,A.Quarteroni,andT.A.Zang,SpectralMethodsinFluidDynamics,Springer-Verlag,Berlin,(1988).[8]A.J.Chamkha,A.M.Aly,M.A.Mansour,Similaritysolutionforunsteadyheatandmasstransferfromastretchingsurfaceembeddedinaporousmediumwithsuction/injectionandchemicalreactioneffects,Chem.Eng.Comm.,197,846–858(2010).[9]S.Chandrasekhar,AnIntroductiontotheStudyofStellarStructure,Dover,NewYork,(1958).[10]H.T.Davis,Introductiontononlineardifferentialandintegralequations,Dover,NewYork,371–394,(1962).[11]R.Ellahi,E.Shivanian,S.Abbasbandy,S.U.Rahman,T.Hayat,Analysisofsteadyflowsinviscousfluidwithheat/masstransferandslipeffects,Int.JofHeatandMassTrans,55,6384–6390(2012).[12]B.D.Ganapol,HighlyAccurateSolutionsoftheBlasiusandFalkner-SkanBoundaryLayerEquationsviaConvergenceAcceleration,arXiv:1006.3888(June2010).[13]S.Kazem,M.Shaban,Tau-homotopyanalysismethodforsolvingmicropolarflowduetoalinearlystretchingofporoussheet,CommunicationsinNumer-icalAnalysis,2012ArticleIDcna-00114,doi:10.5899/2012/cna-00114.[14]S.K.Lele,Compactfinitedifferenceschemeswithspectral-likeresolution,J.Comp.Phys.103,16–42(1992).[15]S.J.Liao,BeyondPerturbation:IntroductiontotheHomotopyAnalysisMethod,BocaRaton:Chapman&Hall/CRCPress(2003).[16]S.J.Liao,HomotopyAnalysisMethodinNonlinearDifferentialEquations,SpringerBerlinHeidelberg,(2012).[17]C.Yang,S.J.Liao,Ontheexplicit,purelyanalyticsolutionofVonKarmanswirlingviscousflow,CommunicationsinNonlinearScienceandNumericalSimulation.11,83–93(2006).[18]E.Magyari,I.PopandB.Keller,ExactdualsolutionsoccurringintheDarcymixedconvectionflow,Int.J.HeatMassTransfer,44,4563–6(2001).[19]Z.Makukula,P.SibandaandS.S.Motsa,ANoteontheSolutionoftheVonK´arm´anEquationsUsingSeriesandChebyshevSpectralMeth-ods,BoundaryValueProblems,Vol.2010,ArticleID471793,17pagesdoi:10.1155/2010/471793(2010)[20]S.S.Motsa,P.SibandaandS.Shateyi,Anewspectral-homotopyanaly-sismethodforsolvinganonlinearsecondorderBVP,CommunicationsinNonlinearScienceandNumericalSimulation15,2293–2302(2010).[21]S.S.Motsa,P.Sibanda,F.G.Awad,S.Shateyi,Anewspectral-homotopyanalysismethodfortheMHDJeffery-Hamelproblem,Computer&Fluids391219–1225(2010).[22]P.Sibanda,S.S.Motsa,Aspectral-homotopyanalysismethodforheattrans-ferflowofathirdgradefluidbetweenparallelplates,InternationalJournalofNumericalMethodsforHeat&FluidFlow,22(1)4–23(2012).[23]S.S.Motsa,Applicationofthenewspectralhomotopyanalysismethod(sham)inthenon-linearheatconductionandconvectivefinproblemwithvariablethermalconductivity,InternationalJournalofComputationalMeth- 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131October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4Chapter4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameterintheHomotopyAnalysisMethodRobertA.VanGorder∗DepartmentofMathematics,UniversityofCentralFloridaOrlando,Florida32816USArav@knights.ucf.eduWeconsiderthestabilityofthehomotopyanalysismethodunderthechoiceofbothlinearoperatorandconvergence-controlparameter.Inparticular,throughseveralexamples,wedeterminehowchangesinthelinearoperatorcaninfluencetheconvergencepropertiesofhomotopysolutions.Itisseenthatthereisoftenabestwaytopickthelinearoperator,butthiscanchangeforeachproblem.Weconsidervariouslin-earoperatorsforsomeordinarydifferentialoperators,andalsodiscussthemethodofselectionforsomenonlinearevolutionPDEs.Through-outthischapter,weconsidertheoptimalhomotopyanalysismethod,whichpermitsustoselectaconvergence-controlparameterthatmini-mizesresidualerrors.Itisnaturaltoaskwhethertheoptimalvalueoftheconvergence-controlparametervariesmuchaswechangethenumberofiterationstaken.Forcomputationalefficiency,wewouldliketotakeasfewtermsaspossibleinordertoguaranteealowerrorofapproximation,solearningwhentheoptimalconvergence-controlparameterstabilizescouldhelpusinknowingwhentotruncateourapproximation.Wethenturnourattentiontootherpropertiesofthehomotopyanalysismethod.Throughapplications,westudytheeffectofhomotopieswhicharenon-linearintheembeddingparameter,q.Inanotherapplication,weshowthattheauxiliaryfunctionH(x),whichisoftentakentounity,canbeusefulinamoregeneralform.Finally,wepresentanapplicationofthehomotopyanalysismethodtoahighlysingularproblem,andwedemon-stratehowtogetaccurateapproximatesolutionsforsuchproblems.∗ThisworksupportedinpartbyNSFgrantnumber1144246.123 132October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4124R.A.VanGorderContents4.1.Overview.......................................1244.2.Ordinaryauxiliarydifferentialoperators.....................1294.2.1.Painlev´eIequation.............................1304.2.2.Lane–Emdenequation............................1344.2.3.Flowoveranonlinearlystretchingsheet..................1384.3.TimeevolutionPDEsandauxiliarylinearoperators...............1414.3.1.Simplepolynomialevolution........................1414.3.2.Evolutionandexponentialtemporaldecay................1424.3.3.NonlinearKlein–Gordonequation.....................1434.3.4.Zakharovsystemwithdissipation.....................1464.4.Theconvergencecontrolparameter........................1504.4.1.Lane–EmdenequationunderL1=y00...................1504.4.2.Lane–EmdenequationunderL00203=y+y...............151x4.4.3.Flowoveranonlinearlystretchingsheet..................1534.5.Modifyingthehomotopy..............................1544.5.1.Thegeneralhomotopy...........................1544.5.2.Standardhomotopyanalysismethod...................1554.5.3.Ahomotopyquadraticinq.........................1574.6.WhatabouttheauxiliaryfunctionH(x)?....................1584.7.Avoidingsingularities................................1654.8.Conclusions.....................................169References.........................................1714.1.OverviewThehomotopyanalysismethod(HAM)[1–10]hasrecentlybeenappliedtothestudyofanumberofnon-trivialandtraditionallyhardtosolvenonlineardifferentialequations,forinstancenonlinearequationsarisinginheattransfer[11–14],fluidmechanics[15–22],solitonsandintegrablemod-els[23–27],nanofluids[28,29],theLane–Emdenequationwhichappearsinstellarastrophysics[30–33],andmodelsfrequentlyusedinmathematicalphysics[34–36],tonameafewareas.Forthoseunfamiliarwiththemethod,thehomotopyanalysismethodisananalyticaltechniquewhichmaybeusedtosolvecomplicatednonlinearproblemswhenotherapproaches,suchasperturbationornumericalanal-ysis,failtoprovidedesirableresults.Consider,forinstance,theperturbedoscillatorequation0030y+y+y=0,y(0)=a,y(0)=b,where,aandbareconstantswithphysicalmeanings.When||<<1,thenweareintheperturbativeregime,andwemayexpressthesolution 133October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter125asafunctionofinthestandardway:22y(x;)=y0(x)+y1(x)+y2(x)+···.Thereasonforsuchanexpansionisthat,when||issufficientlysmall,andthetermsyk(x)aresufficientlybounded,thisrepresentationexhibitsageometricrateofconvergence.Hence,intheperturbativeregime,thestandardperturbationsolutionisrathergood.However,whathappensifisnotsmallenough,orifthereisnosuchsmallphysicalparameterintheproblemwhatsoever?Notethatforperturbationproblemswithasmallphysicalparameter,weoftenhaveanequationoftheformA[u]+B[u]=0,(4.1)whereAisalinearoperatorandBiseitherlinearornonlinear.WhenBisnonlinear,theperturbationassumption2y(x;)=u0(x)+u1(x)+u2(x)+···(4.2)effectivelyreducesthisnonlinearequationintoinfinitelymanylinearequa-tions,indexedbypowersof,oftheformA[uk]=f[u0,...,uk−1].Inthisway,weobtaintheuk’ssuccessively.Oftenwethencalculatethefirstfewtermsofaperturbationexpansion,sayu(x;)=u(x)+u(x)+···`u(x).(4.3)01`Ifissmallenough,thenthisexpansionoftenisagoodapproximationtothetruesolution.Inmanynonlinearproblemsarisinginmathematics,physics,engineer-ing,economics,finance,biology,fluiddynamics,andthelike,thereisnosuchanaturalsmallparameter.Ifweintroducesuchaparameter,weriskobtainingsolutionswhicharenotinterestingorrelevant.However,withoutsuchsmallparameters,analyticalsolutionscanbechallengingtoobtain,andnumericalmethodscouldbeconsidered.Eventhen,wewouldpreferanalyticalsolutionsinmanycases,sincenumericalresults,whileuseful,donotgivethewholepicture.Inaway,thisisonereasontoconsiderthehomotopyanalysismethod(HAM)[1–10].Thismethodmaintainsonefundamentalaspectofpertur-bationtheory,thefactthatwemayiterativelysolvelinearequations,whilefeaturinganimprovementoverperturbationinthefactthatitdoesnot 134October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4126R.A.VanGorderrequireasmallparameter.IntheframeoftheHAM,welinktheoriginalnonlinearequation,sayN[u]=0,withalinearproblemL[u]=0(whichiseasiertosolve,sinceitisfreeforustoselect)bymeansofenforcingahomotopyH(q)[u]:(1−q)L[u]−hqN[u](4.4)tobezero,i.e.(1−q)L[u]=hqN[u],(4.5)whereq∈[0,1]istheembeddingparameter,h6=0iscalledtheconvergence-controlparameter,andLiscalledtheauxiliarylinearoperator,respectively.NotethatwehavelargefreedomtochoosetheauxiliarylinearoperatorLandthevalueoftheconvergence-controlparameterh.Besides,theconvergence-controlparameterhhasnophyicalmeaningatall.Observethatwhenq=0,L[u]=0while,whenq=1wehavetheoriginalnonlinearproblem,namelyN[u]=0.So,equation(4.5)constructsacontinuousvari-ation,ordeformation,fromthesolutionofthelinearequationL[u]=0tothatoftheoriginalnonlinearequationN[u]=0.Forthisreason,equation(4.5)iscalledthezeroth-orderdeformationequationintheframeoftheHAM.Ifwecansolvethisequation,andthenlocalizeatq=1,weshallindeedhaveasolutiontotheoriginalnonlinearequation.Ifweassumeu(x;q)=u(x)+u(x)q+u(x)q2+···012andthenmatchpowersofqinthezeroth-orderdeformationequation(4.5),wefindthatwemayiterativelysolveequationsoftheformL[uk]=F[u0,...,uk−1](4.6)whereFdependsonthelowerorderterms.Again,thisisjustlikeintheperturbationcase.Thezeroth-ordersolution,u0(x),satisfiesanequationoftheformL[u0]=0,andwecallitthezeroth-orderdeformationequation.Inthiscase,thezeroth-orderapproximationisthesolutiontothelinearizedproblem.Inordertodeterminethecontributionofthenonlinearterms, 135October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter127weneedhigherordercorrections.Bysolvingtheso-calledhigher-orderdeformationequationL[uk−χkuk−1]=hRk−1[u0,...,uk−1],(4.7)wedeterminethesecorrections,where1∂nN[φ(x;q)]Rn=n!∂qnq=0and0,whenk≤1,χk=1whenk>1.Comparing(4.7)with(4.6)givesL[uk]=F[u0,...,uk−1]=χkL[uk−1]+hRk−1[u0,...,uk−1].Theconvergence-controlparameterh6=0isselectedinawaythatwillimprovetheconvergencepropertiesofsolutions.Indeed,asinstandardperturbation,weassumeasolutionu(x;q)=u(x)+u(x)q+u(x)q2+···.(4.8)012However,unlikeinstandardperturbation,wemusthaveq=1(thatis,thesolutiontothenonlinearproblemN[u]isu(x;1)),soqisnotatallasmallparameter.Toremedythis,notethatthesolutionsdependimplicitlyontheconvergence-controlparameterh,sothatreallyu(x;1)=U(x;h).Then,wehavesomehopetofindapropervalueofhforwhichU(x;h)isasolutiontotheoriginalnonlinearproblem.DeterminingtheinfinitesumoftermsinU(x;h)=u0(x)+u1(x;h)+u2(x;h)+···(4.9)isoftendifficultorevenimpossibleinclosed-form,sowemusttruncatethisexpressiontoobtainasortofapproximateanalyticalsolution,sayafter`terms:Uˆ(x;h)=u0(x)+u1(x;h)+u2(x;h)+···+u`(x;h).(4.10)Thisintroducesanumberofquestions.First,howdowebestpicktheauxiliarylinearoperatorLinordertoiterativelysolveforthesolutionofthehigherorderdeformationequation(4.7)?Afterthis,howdowepicktheapproximatesolution(4.10)?Ifwepicktofewterms,wewilllikelynotgetagoodapproximation.Ifwepicktoomanyterms,thenthecomputationsmaynotbeefficient.Eventhen,howdowepicktheconvergence-control 136October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4128R.A.VanGorderparameterhsothattheconvergencerateofthesolutionsisgoodenough?Isitpossibletopicktheconvergence-controlparameterhsothattheerrorinherentinourapproximation(4.10)isthelowestpossible?And,howmuchdoesthesolutionmethoddependontheproblemathand?Mustweconsidereverythingonacase-by-casebasis,orcanwemakesomegeneralobservations?Inthepresentchapter,weshallconsidersomeofthesequestions.Inparticular,weshalltakealookatsomeofthefeaturesofthehomotopyanalysismethod,inordertodeterminetheinfluenceofselectingcertainquantities,suchastheauxiliarylinearoperatorLandtheconvergencecontrolparameterh.Whiletherehavebeenimprovementsonhowthehomotopyanalysismethodhasbeenappliedinrecentyears,thereisstillmuchworktobedoneinansweringtheabovequestions.Weshalloutlineanumberofissueswhichmustbeconsideredwhenapplyingthehomotopyanalysismethodtononlineardifferentialequations.Inthefirstsection,weshallstudytheeffectofchangingauxiliaryopera-torswhensolvinganonlinearordinarydifferentialequation.Weshowthatwhilemanyoptionsareavailable,thereisoftenonechoiceofauxiliarylinearoperatorwhichisbest.So,itappearsthatthehomotopyanalysismethodisnotalwaysstableunderthechoiceofauxiliarylinearoperator:rather,insomecases,certainauxiliarylinearoperatorscancontributetolargeer-ror.Foreachchoiceofanauxiliarylinearoperator,wemightobtainratherdifferentvaluesoftheoptimalconvergencecontrolparameter.Weshallrefertooptimalconvergencecontrolparameters,throughout.Recallthattheso-calledconvergencecontrolparameterh,appearsinthezeroth-orderdeformationequationlinkingthelinearandanoriginalnonlinearproblem,namely(1−q)L[u]=hqN[u],whereq∈[0,1]istheembeddingparameter,Nisthenonlinearoperatordescribingtheoriginalproblem,Listheauxiliarylinearoperator.Ifˆu(x;h)isanapproximatesolutionobtainedbythehomotopyanalysismethod,wemaydefinetheaccumulatedL2normoftheresidualerrorbyZ2E(h)=(N[ˆu(x;h)])dx,DwhereDistheproblemdomain.Byconstruction,E(h)ispositivedefinite,sothereexistsaglobalminimum,sayh∗.Then,suchanh∗isaminimizerfortheaccumulatedL2normoftheresidualerror,andwerefertoitas 137October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter129optimal.Then,werefertoˆu(x;h∗)asanoptimalhomotopyanalysissolu-tion.Thismethodhasbeenemployedtostudyoptimalapproximationsforanumberofnonlinearproblems[37–44].Inthesecondsection,wethenconsiderpartialdifferentialequations,inparticularnonlineartimeevolutionequations,andshowthatwhenselectingtheauxiliarylinearoperator,itcanbeimportanttochooseanoperatorwhichpermitsthepropertypeoftime-evolution.Weshowthatoperatorsinvolvingut+upermitsmall-timeconvergenceduetotheappearanceoftermsoftheforme−t,whereasoperatorssimplyoftheformucanleadtotblow-upinsolutions,owingtotermsoftheformtα.Inthethirdsection,westudythebehavioroftheconvergencecontrolparameterasafunctionofthenumberofiterations.Thevalueoftheoptimalconvergencecontrolparameterseemstostabilizeafterrepeatediterations,providedtheauxiliarylinearoperatorisappropriatelychosen.Inthenextthreesections,weconsideradditionalfeaturesofthehomo-topyanalysismethod.Inthefourthsection,weshowthatthehomotopyanalysismethodappearsstableundermodificationstotheformoftheho-motopy.Todemonstratethis,weconstructanonlinearhomotopyinq,anddemonstratethattheresidualerrorinthetwoapproachesarealmostequivalent.AnotherwaytomodifythehomotopyistoincludetheauxiliaryfunctionH(x)asamultiplierforthenonlinearoperatorinthehomotopy.Wehighlightacaseintheliteraturewherethisapproachwasuseful.Fi-nally,inthesixthsection,wedemonstratehowthemethodcanbeusedonnonlinearproblemswithstrongsingularitieswhenweappropriatelyselecttheauxiliarylinearoperator.Eachofthesetopicsdemonstratestheversatilityofthehomotopyanal-ysismethod,whichmakesitahighlyusefultoolinthestudyofnonlin-earphenomenon,inparticularforobtainingapproximatesolutionsforthenonlinearordinaryandpartialdifferentialequationsgoverningsuchphe-nomenon.4.2.OrdinaryauxiliarydifferentialoperatorsHereweshallstudytheinfluenceofselectingdifferentauxiliarylinearop-eratorsinordertoarriveatHAMsolutions.Inordertobestcomparethedifferentsolutions,weshallalwaystruncatetheinfiniteHAMexpansiontoarriveatanaccurateapproximation.Wepicktheconvergence-controlparameter,h,inordertominimizeresidualerrors.Foreachlinearoperatorgiven,weshalldeterminethecorresponding 138October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4130R.A.VanGordererror-minimizingvalueofh.Weshallthenbeabletocomparethechoicesoflinearoperators,todeterminewhichisthebestfitfortheproblem.Threedistinctproblemsaregiventodemonstratethemethod.Ineachoftheseproblems,weseethatthereisindeedabestchoiceforthelinearoperator.Hence,someoperatorscanprovidebetterconvergenceandcontroloferrorthanothers.4.2.1.Painlev´eIequationThePainlev´eIequationreadsy00=y2+x,(4.11)withassociatedinitialconditionsy(0)=1andy0(0)=0.(4.12)Hereprimedenotesdifferentiationwithrespecttox.Weshallrestrictourattentiontotheintervalx∈[0,1].FormoreinformationonthePainlev´eequations,referto[45]andreferencestherein.TherearesixPainlev´etran-scendents,correspondingtosixsecond-orderordinarydifferentialequationswhoseonlymovablesingularitiesareordinarypoles(thischaracteristicisknownasthePainlev´eproperty)andwhichcannotbeintegratedintermsofotherknownfunctionsortranscendents;seetheoriginalworksonPainlev´etranscendents[46–54],oranymoderntextbookcoveringthetheoryofnon-linearordinarydifferentialequations(e.g.,Ince[55]).AsthesolutionstothesixPainlev´eequationscannotbeobtainedexactly,onemayresorttoseriesorperturbationsolutions.Forthisproblem,wedefineanonlinearoperator002N[y]=y−y−x(4.13)andconstructthezeroth-orderdeformationequation(1−q)L[y]=hqN[y],(4.14)subjecttotheinitialcondition0y=1,y=0,atx=0,(4.15)whereLisalinearauxiliaryoperator,histheconvergence-controlparam-eter,andxistheinhomogeneitypresentintheoriginalequation. 139October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter1314.2.1.1.L[y]=y00−y1UsingthelinearoperatorL[y]=y00−y,1thezeroth-orderdeformationequationbecomes(1−q)L[y]=hqN[y],y(0)=1,y0(0)=0.(4.16)1Assumingathree-termsolution2F1(x)=y0(x)+y1(x)q+y2(x)q,andthenbalancingpowersofqin(4.16)(afterwhichwesetq=1,torecoverthethree-termapproximatesolutiontothenonlinearproblem),weobtainL[y]=0,y(0)=1,y0(0)=0,1000 L[y]=L[y]+hy00−y2−x,y(0)=0,y0(0)=0,11100011000L1[y2]=L1[y1]+h(y1−2y0y1),y2(0)=0,y2(0)=0,...Wefindsuccessivelythaty0(x)=cosh(x),hhh2hxh−xy1(x)=+hx−cosh(2x)+sinh(x)−e+e,26233andsoon.Inordertodeterminetheaccuracyofthisthree-termapproximation,wecomputetheaccumulatedL2normoftheresidualerrorofF(x)over1x∈[0,1]:Z1E(h)=(N[F(x;h)])2dx=µ1+µ1h+···+µ1h8,110180whereµ1throughµ1areconstants.E(h)ispositivedefinite,soaglobal081minimumexists.Wefindthath∗=argminE(h)=−1.09958,11h∈RwhichgivesminimalerrorofE(h∗)=1.15996×10−5.Thiserrorisvery11good,andwouldimprovewiththeadditionofhigherorderterms.Foroursake,itissufficient,sincewedesireonlytocomparetheerrorbetweensolutionsinvolvingdifferentlinearoperators. 140October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4132R.A.VanGorder4.2.1.2.L[y]=y00−y02UsingthelinearoperatorL[y]=y00−y0,2thezeroth-orderdeformationequationbecomes0(1−q)L2[y]=hqN[y],y(0)=1,y(0)=0.(4.17)Assumingathree-termsolutionF(x)=y(x)+y(x)q+y(x)q2,2012andthenbalancingpowersofqin(4.17),weobtainL[y]=0,y(0)=1,y0(0)=0,2000 L[y]=L[y]+hy00−y2−x,y(0)=0,y0(0)=0,21200011000L2[y2]=L2[y1]+h(y1−2y0y1),y2(0)=0,y2(0)=0,...Wefindsuccessivelythaty0(x)=1,1−x1xy1(x)=1+h+hx−e−+he,22andsoon.Inordertodeterminetheaccuracyofthisthree-termapproximation,wecomputetheaccumulatedL2normoftheresidualerrorofF(x)over2x∈[0,1]:Z1E(h)=(N[F(x;h)])2dx=µ2+µ2h+···+µ2h8,220180whereµ2throughµ2areconstants.E(h)ispositivedefinite,soaglobal082minimumexists.Wefindthath∗=argminE(h)=−1.0867,2h∈RwhichgivesminimalerrorofE(h∗)=1.32057.Theresidualerrorisvery22bad.So,theuseofL2hasresultedinmuchworseerror. 141October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter1334.2.1.3.L[y]=y003Forthefinalchoiceoflinearoperator,weselect00L3[y]=y.Thezeroth-orderdeformationequationthenbecomes(1−q)L[y]=hqN[y],y(0)=1,y0(0)=0.(4.18)3Assumingathree-termsolutionF(x)=y(x)+y(x)q+y(x)q2,3012andthenbalancingpowersofqin(4.18),weobtain0L3[y0]=0,y0(0)=1,y0(0)=0, 0020L3[y1]=L3[y0]+hy0−y0−x,y1(0)=0,y1(0)=0,L[y]=L[y]+h(y00−2yy),y(0)=0,y0(0)=0,323110122...Wefindsuccessivelythaty0(x)=1,h3h2y1(x)=−x−x,62andsoon.Notethattheseexpressionsaremoresimplethanthoseob-tainedpreviously,sothepresentchoiceofauxiliarylinearoperatorseemstomaintaincomputationalefficiency.Inordertodeterminetheaccuracyofthethree-termapproximationF(x),wecomputetheaccumulatedL2normoftheresidualerrorofF(x)33overx∈[0,1]:Z1E(h)=(N[F(x;h)])2dx=µ3+µ3h+···+µ3h8,330180whereµ3throughµ3areconstants.E(h)ispositivedefinite,soaglobal083minimumexists.Wefindthath∗=argminE(h)=−1.24296,3h∈RwhichgivesminimalerrorofE(h∗)=4.71645×10−3.Thisresidualerror3isgood,thoughnotnearlyasgoodaswefoundfortheoperatorL1.Insummary,theoperatorL1wasthebestchoice,intermsofbothcon-trolofresidualerrorsandcomputationalefficiency,withtheotherauxiliarylinearoperatorschosendisplayingdrawbacks. 142October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4134R.A.VanGorder4.2.2.Lane–EmdenequationTheLane–Emdenequationofthefirstkind(withpowerindexthree)reads00203y+y+y=0,(4.19)xwithassociatedinitialconditions0y(0)=1andy(0)=0.(4.20)Lane–Emdenproblemsareofgreatinterestintherecentliterature[32](andseealso[56–73]),duebothtotheirapplicationinstellarphysicsandbecauseoftheirnonlinearityandsingularityatx=0.4.2.2.1.L[y]=y001UsingtheauxiliarylinearoperatorL[y]=y00,1thezeroth-orderdeformationequationreads(1−q)L[y]=hqN[y],y(0)=1,y0(0)=0.(4.21)1Assumingafour-termsolutionF(x)=y(x)+y(x)q+y(x)q2+y(x)q3,10123andthenbalancingpowersofqin(4.21)(afterwhichwesetq=1,torecoverthethree-termapproximatesolutiontothenonlinearproblem),weobtainL[y]=0,y(0)=1,y0(0)=0,1000002030L1[y1]=L1[y0]+hy0+y0+y0,y1(0)=0,y1(0)=0,x002020L1[y2]=L1[y1]+hy1+y1+3y0y1,y2(0)=0,y2(0)=0,x2 L[y]=L[y]+hy00+y0+3yyy+y2,y(0)=0,y0(0)=0.131222002133x 143October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter135Wefindsuccessivelythaty0(x)=1,h2y1(x)=x,2h2h42y2(x)=x+(1+3h)x,823h2h 36422y3(x)=hx+(7h+3)x+9h+6h+1x.80122Inordertodeterminetheaccuracyofthisfour-termapproximationF(x;h),wecomputetheaccumulatedL2normoftheresidualerrorof1F1(x;h)overx∈[0,1]:Z1211118E1(h)=(N[F1(x;h)])dx=µ0+µ1h+···+µ18h,0whereµ1throughµ1areconstants.E(h)ispositivedefinite,soaglobal0181minimumexists.Wefindthath∗=argminE(h)=−0.373817,11h∈RwhichgivesminimalerrorofE(h∗)=2.364958×10−4.Thiserrorwill11improvewiththeadditionofhigherorderterms.4.2.2.2.L[y]=y00+y02UsingthelinearoperatorL[y]=y00+y0,2thezeroth-orderdeformationequationreads0(1−q)L2[y]=hqN[y],y(0)=1,y(0)=0.(4.22)Assumingafour-termsolutionF(x)=y(x)+y(x)q+y(x)q2+y(x)q3,20123andthenbalancingpowersofqin(4.22)(afterwhichwesetq=1,torecoverthethree-termapproximatesolutiontothenonlinearproblem),we 144October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4136R.A.VanGorderobtainL[y]=0,y(0)=1,y0(0)=0,2000002030L2[y1]=L2[y0]+hy0+y0+y0,y1(0)=0,y1(0)=0,x002020L2[y2]=L2[y1]+hy1+y1+3y0y1,y2(0)=0,y2(0)=0,x2 L[y]=L[y]+hy00+y0+3yyy+y2,y(0)=0,y0(0)=0.232222002133xWefindsuccessivelythaty(x)=1,y(x)=h(x−1+e−x),01andsoon.However,attheevaluationofthetermy2(x),weobtainaninte-gralinvolvingtheEifunction(anexponentialintegral)undertheintegral:Zxy(x)=−h2e−ξEi(1,−ξ)+(1−4ξh)e−ξ−1+(6−3ξ)hdξ.20Uponsuccessiveiterations,thisexpressionwouldonlybecomemorecom-plicated.Asaconsequence,theresidualerrorfunctioncannotbetabulatedinausefulway,hencewehavenogoodwaytocontroltheerrorthroughtheconvergence-controlparameter.Insummary,theoperatorL=y00+y0isnoteffectiveforuseasan2auxiliarylinearoperatorforthepresentproblem.4.2.2.3.L[y]=y00+2y03xForourfinalchoice,weusethelinearoperator0020L3[y]=y+y.xThezeroth-orderdeformationequationreads(1−q)L[y]=hqN[y],y(0)=1,y0(0)=0.(4.23)3Assumingafour-termsolution23F3(x)=y0(x)+y1(x)q+y2(x)q+y3(x)q,andthenbalancingpowersofqin(4.23)(afterwhichwesetq=1,torecoverthethree-termapproximatesolutiontothenonlinearproblem),we 145October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter137obtain0L3[y0]=0,y0(0)=1,y0(0)=0,002030L3[y1]=L3[y0]+hy0+y0+y0,y1(0)=0,y1(0)=0,x002020L3[y2]=L3[y1]+hy1+y1+3y0y1,y2(0)=0,y2(0)=0,x2 L[y]=L[y]+hy00+y0+3yyy+y2,y(0)=0,y0(0)=0.333222002133xWefindsuccessivelythathh2h242y0(x)=1,y1(x)=x,y2(x)=x+(1+h)x,6406and19h2h y(x)=h3x6+(1+h)x4+2h2+h+1x2.35040206Inordertodeterminetheaccuracyofthisfour-termapproximationF(x;h),wecomputetheaccumulatedL2normoftheresidualerrorof3F3(x;h)overx∈[0,1]:Z1233318E3(h)=(N[F3(x;h)])dx=µ0+µ1h+···+µ18h,0whereµ3throughµ3areconstants.E(h)ispositivedefinite,soaglobal0183minimumexists.Wefindthat∗h3=argminE3(h)=−0.892269,h∈RwhichgivesminimalerrorofE(h∗)=4.35017×10−7.Thiserrorwill33improvewiththeadditionofhigherorderterms.Notethattheobtainederrorforthefour-termhomotopyanalysismethodsolutionobtainedusingL3isdrasticallybetterthanthatobtainedusingL(theorder10−7errorofthesolutioncorrespondingtoLismuch13betterthantheorder10−4errorofthesolutioncorrespondingtoL).1Hence,selectinganoperatorwhichisabetterrepresentationoftheoriginalproblemishelpfulinordertoensurerapidconvergenceoftheapproximatesolutionsobtainedviathehomotopyanalysismethod. 146October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4138R.A.VanGorder4.2.3.FlowoveranonlinearlystretchingsheetTheboundaryvalueproblemgoverningtheflowofafluidoveranonlinearlystretchingsheetreadsf000+ff00−ρf02=0,(4.24)00f(0)=0,f(0)=1andlimf(η)=0.(4.25)η→∞Thisandanumberofrelatedproblemsappearasamajorareaofworkinfluidmechanics;indeed,therehavebeeninnumerablereferencesinthisareainrecentyears.Forsomeoftheauthor’sreferences,see[74–92],andformoredetailseethereferencestherein.4.2.3.1.L[f]=f000+f001WeselecttheauxiliarylinearoperatorL[u]=u000+u00,1withgeneralsolution−ηu(η)=α0+α1η+α2e.LetNrepresentthenonlineardifferentialinquestion,i.e.0000002N[f]=f+ff−ρf.AssumingahomotopyoftheformH(q)[f]=(1−q)L1[f]−hqN[f],andassumingafour-termapproximatesolutionoftheformF(η;h)=f(η)+f(η;h)q+f(η;h)q2+f(η;h)q3,(4.26)0123enforcingH(q)[F]≡0impliesL[f]=0,f(0)=0,f0(0)=1,limf0=0,10000η→∞L[f]=h(f000+ff00−ρf02),f(0)=0,f0(0)=0,limf0=0,110000111η→∞L[f]=L[f]+h(f000+ff00+ff00−2ρf0f0),1211110010100f2(0)=0,f2(0)=0,limf2=0,η→∞ 147October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter139andL[f]=L[f]+h(f000+ff00+ff00+ff00−2ρf0f0−ρf02),13122201102021f(0)=0,f0(0)=0,limf0=0.333η→∞Wemaythensolvethissystem,successivelyobtainingthehigherorderterms.Observethatf(η)=1−e−η,0whichhappenstobetheexactsolutiontotheρ=1problem.Theothertermsaregivenbyh −η2f1(η)=(ρ−1)1−e,(4.27)4h(5ρ+14)+9h(8ρ+11+6η)+12−ηf2(η)=h(ρ−1)−e3624hρ+1−2ηh(4ρ−5)−3η+e−e,(4.28)472andsoforth.Inordertodeterminetheaccuracyofthisfour-termap-proximation,wecomputetheaccumulatedL2normoftheresidualerrorofF(η),Z∞212E(h)=(N[F(η;h)])dη=µ0+µ1h+···+µ12h,0whereµ0throughµ12dependonρ.E(h)ispositivedefinite,soaglobalminimumexists.Ingeneral,thisshalldependonthevalueofρ.Inthecasewhereρ=0.5,wefindthath∗=argminE(h)=−0.72731,h∈RwhichgivesminimalerrorofE(h∗)=1.9528×10−6.4.2.3.2.L[f]=f000−f02Wenowselectthedifferentauxiliarylinearoperator0000L2[u]=u−u,withgeneralsolutionu(η)=α+αeη+αe−η.012 148October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4140R.A.VanGorderAssumingahomotopyoftheformH(q)[f]=(1−q)L2[f]−hqN[f],andassumingafour-termapproximatesolutionoftheform23G(η;h)=f0(η)+f1(η;h)q+f2(η;h)q+f3(η;h)q,enforcingH(q)[F]≡0impliesL[f]=0,f(0)=0,f0(0)=1,limf0=0,20000η→∞L[f]=h(f000+ff00−ρf02),f(0)=0,f0(0)=0,limf0=0,210000111η→∞000000000L2[f2]=L2[f1]+h(f1+f1f0+f0f1−2ρf0f1),f(0)=0,f0(0)=0,limf0=0,222η→∞andL[f]=L[f]+h(f000+ff00+ff00+ff00−2ρf0f0−ρf02),2322220110202100f3(0)=0,f3(0)=0,limf3=0.η→∞Wemaythensolvethissystem,successivelyobtainingthehigherorderterms.Observethattheorderzerotermisidenticaltothepreviousresult,asitsatisfiesbothlinearoperators.So,−ηf0(η)=1−e.Theothertermswilldiffer,andaregivenbyh −η2f1(η)=(ρ−1)1−e,6andsoforth.Notethedifferenceintheorderoneterm,withthefactor1/6asopposedto1/4whichwastrueforthepreviouslinearoperator.So,differencesappearatthefirstorderwhenweselecteddifferentlinearoperators.TheaccumulatedL2normoftheresidualerrorofG(η),Z∞E(h)=(N[G(η;h)])2dη=ν+νh+···+νh12,201120whereν0throughν12dependonρ.Ofcourse,νj6=µjingeneral,whereµjisfromthepreviousexample.E2(h)ispositivedefinite,soaglobalminimum 149October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter141exists.Ingeneral,thisshalldependonthevalueofρ.Consideringagainthecasewhereρ=1/2,wefindthat∗h=argminE(k1)=−1.1661,k1∈RwhichgivesminimalerrorofE(h∗)=2.0593×10−6.Afewthingsareclear.First,thevalueoftheresidualerrorgivenafour-termapproximationisnotdrasticallydifferentduetothechoiceofeitherL1orL2asourauxiliarylinearoperator.However,thevalueoftheconvergence-controlparameterisquitedifferentineachcase.Whatthisimpliesisthatwecanhavereasonablefreedomtoselecttheauxiliarylinearoperator,providedthatweareabletochoosetheconvergence-controlparameter,h,inanoptimalway.Inotherwords,itappearsthattheproperselectionoftheconvergence-controlparameterismoreimportantthantheselectionoftheauxiliarylinearoperator.However,whenselectingtheauxiliarylinearoperator,westillneedtoensurethatthezeroth-orderapproximationisinthekernel.4.3.TimeevolutionPDEsandauxiliarylinearoperatorsConsiderthetimeevolutionequationut=F(u,ux,uxx),(4.29)subjecttotheinitialdatau(x,0)=f(x).(4.30)ThenonlinearoperatoristhenN[u]=ut−F(u,ux,uxx).(4.31)Weshalldiscusssomewaysofchoosingtheauxiliarylinearoperatorandtheconvergence-controlparameterforsuchnonlinearPDEsandevensystemsofnonlinearPDEs.4.3.1.SimplepolynomialevolutionHereweconsiderthebasicauxiliarylinearoperatorL[u]=ut.(4.32)ThenN[u]=L[u]−F(u,ux,uxx).WiththisformofL,wewillconstructasolutionintermsofpolynomialsint.Suchasolutionmethodislogically 150October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4142R.A.VanGorderequivalenttoTaylorseriessolutionsint:oneachiteration,wegainapowerint.However,throughproperuseoftheconvergence-controlparameter,wecanobtainsolutionsmoreeffectivelythanthestandardmethodofobtainingaTaylorseriessolution.Thatsaid,theconvergencecanonlybeensuredonafinitedomainformostnonlinearPDEs,sincethetruesolutionisnotapowerseriesint.Forinstance,assumewehaveinitialdatau(x,0)=f(x).Thentheorderzerosolutionsatisfiesu0,t=0,whichimpliesthatu0isconstantint,i.e.u0(x,t)=f(x).Thenexttermisgovernedby000u1,t=hF(u0,u0,x,u0,xx)=hF(f(x),f(x),f(x)).Solvingsubjecttou1(x,0)=0,wewillhaveu(x,t)=htF(f(x),f0(x),f00(x)),1andingeneralhigherordertermswilltaketheformofpolynomialsintwithcoefficientsinx.Suchsolutionscanbeusefullocallyforsmallenought,butseldomcanbeusefulforlarget.Therefore,inordertogetglobalsolutionsforallt>0,thismethodisnotuseful.4.3.2.EvolutionandexponentialtemporaldecayInordertotakeadvantageofexponentialdecay,weconstructtheauxiliarylinearoperatorL[u]=ut+u.(4.33)ThenN[u]=L[u]−u−F(u,ux,uxx).ThisformofLwillallowdecayingexponentialbasefunctions.Theresultinghomotopysolutionswillthenallowthepossibilityofconvergenceast→∞.Forexample,assumeu(x,0)=f(x).Thentheorderzerosolutionsatisfiesu0,t+u0=0,whichgivesu(x,t)=e−tf(x).0 151October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter143Thenextordertermisgovernedbyu+u=hF(e−tf(x),e−tf0(x),e−tf00(x)),1,t1whichadmitsasolutionZtu(x,t)=he−(t−s)F(e−sf(x),e−sf0(x),e−sf00(x))ds,10whichoftenwillbemuchmorewell-behavedthanasimplepolynomialint.Inthefollowingsections,weshalldemonstratehowtoapplythisformofevolutionoperatorut+uinordertoobtainaccurateapproximationsforsomenonlinearPDEs.First,weconsideranonlinearKlein–GordonequationthatwasrecentlysolvedinRussoandVanGorder[93]usingoptimalhomotopyanalysis.Then,weshallturnourattentiontoasys-temofPDEs,theZakharovsystemwithdissipation,whichwasrecentlysolvedbyMalloryandVanGorder[43].Ineachoftheseexamples,optimalconvergence-controlparametersareselectedinordertoensureconvergenceofthesolutionsafterarelativelysmallnumberoftermsarecomputed.Suchanapproachiscomputationallyefficient.Inthelatterexample,threeconvergence-controlparametersareused,sincethereisasystemofthreenonlinearPDEs.Itisfoundthattheseparametersmaybejointlyopti-mized,inordertoobtainsolutionswithminimalresidualerror.AsimilarapproachwasalsorecentlyappliedbyVanGorder[37]fortheconstructionofsolutionstotheF¨oppl–vonK´arm´anequationsgoverningdeflectionsofathinflatplate(whichisasystemoftwononlinearPDEs).4.3.3.NonlinearKlein–GordonequationOnemayapplythehomotopyanalysismethodtostudysolutionsofnonlin-earevolutionequations.IntherecentpaperofRussoandVanGorder[93],weconsideredthenonlinearKlein–Gordonequationutt−uxx=F(u),(4.34)u(x,0)=f(x),ut(x,0)=g(x).(4.35)Here,F∈C∞(R)isanarbitraryanalyticfunctionwhichwillserveasthenonlinearterm,whilef,g∈C∞(R)aretheanalyticinitialdata.Weshallhighlightthesignificanceoftheresultsobtainedin[93]fornonlinearevolutionPDEs.Additionally,someauthorshaveconsideredthehomotopyanalysisofsomespecialcasesforF(u).Sun[94]consideredthequasilinearcubicKlein–Gordonequation,andusedhomotopyanalysismethodtoobtainasolution 152October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4144R.A.VanGorderformintermsofatrigonometricbasis.However,suchsolutionswerefortherestrictiontothetravellingwavecase,whichreduces(4.34)fromapar-tialdifferentialequationtoanordinarydifferentialequation.Morerecently,Iqbaletal.[95]consideredthesameequationfromthestandpointoftheop-timalhomotopyanalysismethod,andsimilarresultswereobtained.Noneofthesestudiesconsideredarbitraryinitialdata.Approximatehomotopyanalysissolutionsforthesin-GordonequationwererecentlydiscussedbyY¨ucel[96],howevernodiscussionoferrorwasprovided(theauthorsimplyappliedtheso-calledh-curveinordertodeducepossibleregionsofconver-genceofthesolutions).ParticularlyinterestingexamplesarethequasilinearKlein–Gordon[97–99]equation(F(u)=u3−αu),themodifiedLiouville[100,101]equation(F(u)=eβu),thesinh-Gordon[102–108]equation(F(u)=sinh(u))andthetanh-Gordon[109]equation(F(u)=tanh(u)).4.3.3.1.HAMforthequasilinearKlein–GordonequationThequasilinearKlein–GordonequationcorrespondstoF(u)=u3−αu,whereαisareal-valuedparameter.Letψ(x,t;h)=u0(x,t)+u1(x,t;h)+u2(x,t;h)+···bethehomotopysolutionevaluatedatq=1.WethenfindthatthefirstseveraltermsofF(ψ(x,t;h))aregivenby F(ψ(x,t;h))=u3−αu+3u2u−αuq00011 +3(u2u+uu2)−αuq2+···.02012Forthepresentproblems,weshallconsideralinearoperatorofthetypeL[ψ]=ψt+ψ,whichpermitsadecayingexponentialbasisint.Thezeroth-orderapproximationremainsu0,whilethehigherorderap-proximationsaregovernedby L[u]=h(u)−(u)−u3+αu,10tt0xx00 L[u]=(u)+2(u)+u+h(u)−(u)−3u2u+αu,21tt1t11tt1xx011 L[u]=(u)+2(u)+u+h(u)−(u)−3(u2u+uu2)+αu,32tt2t22tt2xx02012 153October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter145etc.Wefindthatu1isgivenbyh0000u1(x,t;h)={(f(x)+g(x)−f(x)−g(x))t6−3(f(x)+2g(x)+f00(x))}t2e−tZtn−h(t−s)e−(t−s)(f(x)+(f(x)+g(x))s)3e−3s0−α(f(x)+(f(x)+g(x))s)e−sds−t−3t=hP1,1(x,t)e+hP1,3(x,t)e,(4.36)whereP1,1andP1,3arepolynomialsintofdegreenogreaterthanthree.Higherorderinversionformulasmaybeobtainedinasimilarmanner.Ingeneral,wefindthatforn≥2u(x,t;h)=P(x,t;h)e−t+P(x,t;h)e−3tnn,1n,3−(2n+1)t+···+Pn,2n+1(x,t;h)e,wherethePn,k’sarepolynomialsintwithcoefficientsinvolvingf(x),g(x)andtheirderivatives.Thesecoefficientsalsodependonh,whichweshallusetoouradvantage.4.3.3.2.Initialdataf(x)=sech(x),g(x)=0Considertheinitialdataf(x)=sech(x),g(x)=0.Wefindthatu(x,t)=sech(x)(1+t)e−t,0and,uponplacingthisexpressioninto(4.36),wehaveα t322u1(x,t;h)=hsech(x)t(3+t)+tanh(x)+1632219t−232−t−tanh(x)t−sech(x)e8h −sech3(x)2t3+12t2+27t+23e−3t.8Inordertoobtainanaccurateapproximation,wemuststillselecttheconvergence-controlparameter,h.Duetothecomplicatedexpressionsob-tainedforthefirstandsecondorderterms,wecomputediscreteresiduals.Wetakeadoublesumoverafinitecollectionofpointsinordertoapproxi-matethesquaredresidualerroroverafinitesubsetoftheproblemdomain. 154October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4146R.A.VanGorderTakingT=5and`=10,wehaveZ5Z102Res(x,t;h,α)E(h,α)=dxdt0−10U2(x,t;h,α)X50X202Res(0.5j,0.1i;h,α)≈0.05,(4.37)U2(0.5j,0.1t;h,α)i=0j=−20wherewehavesampled(41)(51)=2091pointsintheset(x,t)∈[−10,10]×[0,5],andthenormalizationfactoriscomputedbytheproductofthetwostepsizes.4.3.3.3.Initialdataf(x)=e−|x|,g(x)=0Considernextthepeakedinitialdatawithexponentialdecaycorrespondingtof(x)=e−|x|,g(x)=0.Notethatf(x)isnotanalytic;allderivativesfailtoexistatx=0.Notethatwewillbeexclusivelyinterestedinoddorderderivativesoff(x).Then,notethatforallx6=0,2dfd−|x|=sgn(x)edx2dx =sgn2(x)−2δ(x)e−|x|=e−|x|=f(x),(4.38)hencef(2k)(x)=f(x)forallx6=0.Wethenfindthatu(x,t)=(1+t)e−te−|x|,0α2−t−|x|h−t−3|x|u1(x,t;h)=h(3+t)−1tee−(19t−23)ee68h −2t3+12t2+27t+23e−3te−3|x|,8andsoon.Inordertoanalyzeerror,weshallusethesamediscreteresidualerrorsasin(8.17),withtheexceptionthatwenolongerincludethepointsoftheform(0,t),asx=0isundefined.4.3.4.ZakharovsystemwithdissipationOnemayalsoapplythehomotopyanalysismethodtosystemsofnonlinearPDEswithcertaininitialdata.Intheveryrecentpaper[43]ofMalloryandVanGorder,weconsideredthecontroloferrorinthehomotopyanalysisofsolutionstotheZakharovsystemwithdissipation.Therelevantreal-valued 155October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter147systemis2ρt=σv−∇σ,2(4.39)σt=−ρv+∇ρ, v=∇2v−ρ2−σ2−v−v.tttInitialdatatakestheformρ(x,0)=R(x)andσ(x,0)=S(x).Inwhatfollows,weshallhighlightthesignificantresultsofthepaperofMalloryandVanGorder[43]whicharerelevantforsystemsofnonlinearevolutionPDEs.WeshouldnotethattheZakharovsystemdescribesthepropagationofLangmuirwavesinanionizedplasma[110–112].Solitonsolutionswereconsideredin[113–115].Homoclinictubesolutionsandchaoswerestudiedin[116].Numericalsimulationswererecentlypresentedin[117,118].Letusdefinetheoperators∂L=+1,∂t2N1[ρ,σ,v]=ρ+σv−∇σ,N[ρ,σ,v]=σ−ρv+∇2ρ,2 N[ρ,σ,v]=v+∇2v−ρ2−σ2.3tWeshallalsomakeuseofthesquareofL,L2,whichisoftheform∂2∂L2=+2+1.∂t2∂tThen,Zakharovsystemwithdissipation(4.39)takestheformL[ρ]=N1[ρ,σ,v],L[σ]=N2[ρ,σ,v],(4.40)2L[v]=N3[ρ,σ,v].Then,weconstructthezeroth-orderdeformationequationsH1(ˆρ(x,t,q),σˆ(x,t,q),vˆ(x,t,q);q)=(1−q)L[ˆρ(x,t,q)−ρ0(x,t)]−qh1(L[ˆρ(x,t,q)]−N1[ˆρ(x,t,q),σˆ(x,t,q),vˆ(x,t,q)]),H2(ˆρ(x,t,q),σˆ(x,t,q),vˆ(x,t,q);q)=(1−q)L[ˆσ(x,t,q)−σ0(x,t)]−qh2(L[ˆσ(x,t,q)]−N2[ˆρ(x,t,q),σˆ(x,t,q),vˆ(x,t,q)]),2H3(ˆρ(x,t,q),σˆ(x,t,q),vˆ(x,t,q);q)=(1−q)L[ˆv(x,t,q)−v0(x,t)]−qh(L2[ˆv(x,t,q)]−N[ˆρ(x,t,q),σˆ(x,t,q),vˆ(x,t,q)]),33(4.41) 156October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4148R.A.VanGorderwhereh1,h2,andh3denotetheuniqueconvergence-controlparametersandqistheembeddingparameter.Wetakeˆρ,ˆσ,andˆvtobethesolu-tionsdependentuponqandρ0,σ0,andv0tobetheapproximategeneralsolutionstotheequationsconstructedfromthelinearoperators.Usingtheembeddingparameter,wemayrepresentthelinearitiesbytakingq=0andtheoriginalnonlineardifferentialequationswithq=1.Weassumethesolutionsˆρ,ˆσ,andˆvmayberepresentedbytheinfiniteseriesρˆ(x,t,q)=ρ(x,t)+ρ(x,t)q+ρ(x,t)q2+...,(4.42)012σˆ(x,t,q)=σ(x,t)+σ(x,t)q+σ(x,t)q2+...,(4.43)012vˆ(x,t,q)=v(x,t)+v(x,t)q+v(x,t)q2+...,(4.44)012whichwemaysubstituteintothezeroth-orderdeformationequation(4.41)andcollectcommonpowersofq.Utilizingthelinearoperatorsgivenin(4.40),webeginbysolvingforthezeroth-orderfunctionsrequiredforoursolutionsˆρ,ˆσ,andˆv.Representingordinarydifferentialequationsint,thesearedefinedas∂ρ0+ρ0=0,∂t∂σ0+σ0=0,∂t∂2∂2v0+2v0+v0=0.∂t∂tSolvingtheselinearequationssubjecttoinitialdata,weobtaintheinitialapproximations−tρ0(x,t)=R(x)e,−tσ0(x,t)=S(x)e,−tv0(x,t)=(U(x)+(U(x)+W(x))t)e.Inaccordancewiththehomotopyanalysismethod,wenotethatallsubse-quentfunctionsoccurrecursively.Oursystem(4.40)mayberewrittenasL[ρ(x,t)]−N1[ρ(x,t),σ(x,t),v(x,t)]=0,L[σ(x,t)]−N2[ρ(x,t),σ(x,t),v(x,t)]=0,(4.45)2L[v(x,t)]−N3[ρ(x,t),σ(x,t),v(x,t)]=0. 157October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter149Inordertodeterminethedeviationofourapproximationsfromtheexactsolutions,weemployresidualerrortechniques.Accordingly,foranyx∈(−∞,∞)andt≥0,wemaycalculatetheresidualerroratsomepoint(x∗,t∗)by1(x,t)=L[ˆρ(x,t)]−N1[ˆρ(x,t),σˆ(x,t),vˆ(x,t)],x=x∗,t=t∗2(x,t)=L[ˆσ(x,t)]−N2[ˆρ(x,t),σˆ(x,t),vˆ(x,t)],x=x∗,t=t∗(x,t)=L2[ˆv(x,t)]−N[ˆρ(x,t),σˆ(x,t),vˆ(x,t)].33x=x∗,t=t∗Therefore,tosumtheresidualerroroverthedomainx∈(−∞,∞),t∈(0,T),withT>0assomestoppingtime,wetakeZTZ∞R1(x,t)=L[ˆρ(x,t)]−N1[ˆρ(x,t),σˆ(x,t),vˆ(x,t)]dxdt,0−∞ZTZ∞R2(x,t)=L[ˆσ(x,t)]−N2[ˆρ(x,t),σˆ(x,t),vˆ(x,t)]dxdt,0−∞ZTZ∞R(x,t)=L2[ˆv(x,t)]−N[ˆρ(x,t),σˆ(x,t),vˆ(x,t)]dxdt.330−∞However,suchintegrationistypicallytoodifficulttoperform,soinordertoavoidthedifficultyposedbytheabsolutevalue,weevaluatethesumofthesquaredresidualerroranddefineE1(h1,h2,h3)ZTZ`=(L[ˆρ(x,t)]−N[ˆρ(x,t),σˆ(x,t),vˆ(x,t)])2dxdt,(4.46)100E2(h1,h2,h3)ZTZ`=(L[ˆσ(x,t)]−N[ˆρ(x,t),σˆ(x,t),vˆ(x,t)])2dxdt,(4.47)200E3(h1,h2,h3)ZTZ`=(L2[ˆv(x,t)]−N[ˆρ(x,t),σˆ(x,t),vˆ(x,t)])2dxdt.(4.48)300Choosingdesiredstoppingpoints,denoted`andTforxandt,respec-tively,wecompute(4.46)–(4.48)forgiveninitialdataasfunctionsofh1, 158October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4150R.A.VanGorderh2,andh3.Thuswemayutilizetheseembeddedconvergence-controlpa-rameterstominimizetheerrorofourcalculatedsolutions,therebyfinding(h∗,h∗,h∗)∈R3whichoffersthebestapproximationofthetruesolu-123tionsˆρ,ˆσ,andˆv.4.4.TheconvergencecontrolparameterInthepresentsection,werevisitthepreviousexamples,inordertodeter-minethebehavioroftheconvergence-controlparameter,h,afterafinitenumberofiterations.Whilethenumberofiterationswerefixedinthepre-viousexamples,herewefixalinearoperatoranddeterminetheoptimalvalueh∗oftheconvergence-controlparameterafterkiterations.Asmorekiterationsofthemethodaretaken,weexpectthattheresidualerrorwilldecrease.However,itisnaturaltowonderifthevalueofh∗willappeartokconvergetoafixedvalueoncewemakeklargeenough.4.4.1.Lane–EmdenequationunderL=y001LetusrevisittheLane–Emdenproblem(4.19)–(4.20).ConsideragaintheauxiliarylinearoperatorL=y00.Thetwo-termapproximationtakesthe1formF2(x)=y0(x)+y1(x;h),andreadsh2F2(x,h)=1+x.2TheaccumulatedL2normoftheerrorisgivenbyZ1E(h)=(N[F(x,h)])2dη=µ2+µ2h+···+µ2h6,220160whereµ2isaconstantforallj=0,1,...,6.ThefunctionEis,byj2construction,positivedefinite,sothereexistsaglobalminimum.Wedefineh∗=argminE(h)=−0.2858422htobethisminimizingvalueoftheconvergence-controlparametercorre-spondingtoF2(x,h).TheminimalvalueofthefunctionE2(h)readsE(−0.28584)=1.2871×10−2.2Thethree-termapproximationtakestheformF3(x;h)=y0(x)+y1(x;h)+y2(x;h),sothathh2hF(x;h)=1+x2+x4+(1+3h)x2.3282 159October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter151TheaccumulatedL2normoftheerrorisgivenbyZ∞E(h)=(N[F(x,h)])2dη=µ3+µ3h+···+µ3h12,3301120whereµ3isaconstantforallj=0,1,...,12.ThefunctionEispositivej3definite,sothereexistsaglobalminimum.Wedefine∗h3=argminE3(h)=−0.23975htobethisminimizingvalueoftheconvergence-controlparametercorre-spondingtoF3(x,h).TheminimalvalueofthefunctionE3(h)readsE(−0.23975)=5.4550×10−3.3Thefour-termapproximationtakestheformF4(x;h)=y0(x)+y1(x;h)+y2(x;h)+y3(x;h).Thenhh2hF(x;h)=1+x2+x4+(1+3h)x242823h2h +h3x6+(7h+3)x4+9h2+6h+1x2.80122TheaccumulatedL2normoftheresidualerrorofF(x;h)overx∈[0,1]is4Z1244118E4(h)=(N[F4(x;h)])dx=µ0+µ1h+···+µ18h,0whereµ4throughµ4areconstants.E(h)ispositivedefinite,soaglobal0184minimumexists.Wefindthath∗=argminE(h)=−0.373817,41h∈RwhichgivesminimalerrorofE(h∗)=2.364958×10−4.14Whatweseeisthattheoptimalvalueoftheconvergence-controlpa-rametertendstooscillatenearh=−0.3.Ifthereisconvergenceofthesequenceofhk’s,itmaynotbemonotone.4.4.2.Lane–EmdenequationunderL=y00+2y03xConsideragaintheLane–Emdenproblem(4.19)–(4.20),thistimewiththeauxiliarylinearoperatorL=y00+2y0.Weshallnowcompareoptimal3xvaluesfortheconvergence-controlparameterundermultipleiterationsofthesolutionprocess.Thetwo-termapproximationtakestheformF2(x)=y0(x)+y1(x;h),andreadsh2F2(x,h)=1+x.6 160October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4152R.A.VanGorderTheaccumulatedL2normoftheerrorisgivenbyZ1E(h)=(N[F(x,h)])2dη=µ2+µ2h+···+µ2h6,220160whereµ2isaconstantforallj=0,1,...,6.ThefunctionEis,byj2construction,positivedefinite,sothereexistsaglobalminimum.Wedefineh∗=argminE(h)=−0.85752.22hTheminimalvalueofthefunctionE2(h)readsE2(−0.85752)=1.28713×10−2.Thethree-termapproximationtakestheformF3(x;h)=y0(x)+y1(x;h)+y2(x;h),sothathh2hF(x;h)=1+x2+x4+(1+h)x2.36406TheaccumulatedL2normoftheerrorisgivenbyZ∞233312E3(h)=(N[F3(x,h)])dη=µ0+µ1h+···+µ12h,0whereµ3isaconstantforallj=0,1,...,12.ThefunctionEispositivej3definite,sothereexistsaglobalminimum.Wedefineh∗=argminE(h)=−0.8786933htobethisminimizingvalueoftheconvergence-controlparametercorre-spondingtoF3(x,h).TheminimalvalueofthefunctionE3(h)readsE(−0.87869)=8.67763×10−5.3Thefour-termapproximationtakestheformF4(x;h)=y0(x)+y1(x;h)+y2(x;h)+y3(x;h).Thenhh2hF(x;h)=1+x2+x4+(1+h)x24640619h2h +h3x6+(1+h)x4+2h2+h+1x2.5040206Z1E(h)=(N[F(x;h)])2dx=µ4+µ4h+···+µ4h18,4401180whereµ4throughµ4areconstants.E(h)ispositivedefinite,soaglobal0184minimumexists.Wefindthath∗=argminE(h)=−0.892269,43h∈R 161October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter153whichgivesminimalerrorofE(−0.892269)=4.35017×10−7.Inthisex-4ample,weseethattheoptimalvalueoftheconvergence-controlparameterappearstoconvergetonearh∗=−0.9asthenumberofiterationsincreases.Aswasshowninaprevioussection,thechoiceofL=L3wassuperiortothechoiceL=L1,intermsofobtainingminimalerrorapproximations.So,itappearsthatwhenanauxiliarylinearoperatorisselectedproperly,wemayexpecttheoptimalvalueoftheconvergence-controlparametertoconvergemorerapidlytosomefixedvalue.Suchbehaviorcanbeasignthatweareontherighttracktoselectingtheauxiliarylinearoperator.4.4.3.FlowoveranonlinearlystretchingsheetReturningtotheproblemofaflowoveranonlinearlystretchingsheet(4.24)–(4.25),andselectingoperatorLgivenbyL[f]=f000+f00,we11maygivethefirstfewlow-orderapproximations.Thetwo-termapproximationtakestheformF2(η)=f0(η)+f1(η,h),andreads−ηh −η2F2(η,h)=1−e+(ρ−1)1−e.4TheaccumulatedL2normoftheerrorisgivenbyZ∞E(h)=(N[F(η,h)])2dη=µ+µh+···+µh4,220140whereµjisingeneralafunctionofρforallj=0,1,2,3,4.ThefunctionE2is,byconstruction,positivedefinite,sothereexistsaglobalminimum.Wedefine∗h2=argminE2(h)htobethisminimizingvalueoftheconvergence-controlparametercorre-spondingtoF2(η,h).Forthesakeofdemonstration,atthevalueρ=0.5,wefindthath∗=−0.69447.Atthisvalue,weseethattheminimalvalue2ofthefunctionE(h)readsE(−0.69447)=1.4553×10−4.22Thethree-termapproximationtakestheformF3(η)=f0(η)+f1(η,h)+f2(η,h).Weshallomitthelengthyexpression,butthefunctionsfkarelistedelsewhereinthischapter.TheaccumulatedL2normoftheerrorisgivenbyZ∞28E3(h)=(N[F3(η,h)])dη=µ0+µ1h+···+µ8h,0 162October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4154R.A.VanGorderwhereµjisingeneral,afunctionofρforallj=0,1,...,8.ThefunctionE3ispositivedefinite,sothereexistsaglobalminimum.Wedefine∗h3=argminE3(h)htobethisminimizingvalueoftheconvergence-controlparametercorre-spondingtoF(η,h).Atthevalueρ=0.5,wefindthath∗=−0.69116.33Atthisvalue,weseethattheminimalvalueofthefunctionE3(h)readsE(−0.69116)=4.1367×10−5.3Thefour-termapproximationtakestheformF4(η)=f0(η)+f1(η,h)+f2(η,h)+f3(η,h).Weshallomitthelengthyexpression,butthefunctionsfarelistedelsewhereinthischapter.TheaccumulatedL2normofthekerrorisgivenbyZ∞E(h)=(N[F(η,h)])2dη=µ+µh+···+µh12,4401120whereµjisingeneral,afunctionofρforallj=0,1,...,12.ThefunctionE4ispositivedefinite,sothereexistsaglobalminimum.Wedefine∗h4=argminE4(h)htobethisminimizingvalueoftheconvergence-controlparametercorre-spondingtoF(η,h).Atthevalueρ=0.5,wefindthath∗=−0.72732.44Atthisvalue,weseethattheminimalvalueofthefunctionE3(h)readsE(−0.72732)=1.95074×10−6.34.5.ModifyingthehomotopyIncontrasttothestandardlinearhomotopyconnectingLandN,onemayconsidermoreexoticformsofthehomotopy.Toillustratethispoint,wereturntotheproblemofaflowoveranonlinearlystretchingsheet.4.5.1.ThegeneralhomotopyConsiderthehomotopyH(q)[f]=A(q)L[f]−B(q)N[f]=0.(4.49)HereLisalineardifferentialoperatorandNisthenonlineardifferentialoperatorofinterest.AandBarefunctionsofthehomotopyparameterq,andhavethepropertythatA(1)=0=B(0)andB(1)6=06=A(0).Then,whenq=1,theaboveequationreducestothenonlinearoperatorof 163October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter155interest,whilewhenq=0,theequationislinear.WewishtoapproximatesolutionstoNaspartialsumsofpowerseriesinq.Withthisinmind,weattempttoselectAandBinsuchawaytominimizeerror.Inmanyrecentpapers,thishasbeendonebylettingA(q)=1−qandB(q)=cq,wherecisaconvergence-controlparameter,usuallyaconstant.Now,weconsideranewstrategy:thatAandBarenotlinearfunctionsofq.WeletA(q)=(1−q)a(q),B(q)=qb(q).Thesefunctionsmeetthecriteriaoutlinedabove,aslongasb(1)6=0.Ingeneral,aswewishtoexpand(4.49)inpowersofq,weleta(q)andb(q)bepolynomialsinq:a(q)=c+cq+cq2+...+cqn,(4.50)012nb(q)=k+kq+kq2+...+kqn.(4.51)123n+1Here,nisonelessthanthedesireddegreeoftheapproximationtobecomputedvia(4.49),andciandkiareconvergence-controlparameters.Ingeneral,theseextraconvergence-controlparameterscanonlyimprovetheerrorofthemethod.However,havingsomanyofthemincreasescomputa-tionalcomplexityiftrulyoptimalsolutionsaresought.4.5.2.StandardhomotopyanalysismethodWeagainconsidertheequationfornonlinearflowoveranonlinearlystretch-ingsheet:f000+ff00−ρf02=0,(4.52)subjecttotheconditionsf(0)=0,f0(0)=1,andf0(∞)=0.Hereρ∈(0,2)describesthepowerlawstretchingofthesheet;ρ=1correspondstolinearstretching.Weseekapproximatesolutionsto(4.52)bythehomotopyanalysismethod.WeselectthelinearoperatorL[u]=u000+u00,withgeneralsolutionu(η)=α+αη+αe−η.012Nshallrepresentthenonlineardifferentialinquestion:N[f]=f000+ff00−ρf02. 164October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4156R.A.VanGorderAssumingahomotopyoftheformH(q)[f]=(1−q)L[f]−k1qN[f],andassumingafour-termapproximatesolutionoftheformF(η;k)=f(η)+f(η;k)q+f(η;k)q2+f(η;k)q3,(4.53)10112131theconditionH(q)[F]≡0impliesL[f]=0,f(0)=0,f0(0)=1,limf0=0,0000η→∞L[f]=k(f000+ff00−ρf02),110000f(0)=0,f0(0)=0,limf0=0,11η→∞1L[f]=L[f]+k(f000+ff00+ff00−2ρf0f0),2111100101f(0)=0,f0(0)=0,limf0=0,22η→∞2L[f]=L[f]+k(f000+ff00+ff00+ff00−2ρf0f0−ρf02),3212201102021f(0)=0,f0(0)=0,limf0=0.33η→∞3Wemaythensolvethissystem,successivelyobtainingthehigherorderterms.Observethatf(η)=1−e−η,0whichhappenstobetheexactsolutiontotheρ=1problem.Theothertermsaregivenbyk1 −η2f1(η)=(ρ−1)1−e,4k1(5ρ+14)+9k1(8ρ+11+6η)+12−ηf2(η)=k1(ρ−1)−e3624k1ρ+1−2ηk1(4ρ−5)−3η+e−e,472andsoon.WethencomputetheaccumulatedL2normoftheresidualerrorofF(η)asgivenin(4.53),Z∞212E(k1)=(N[F(η;k1)])dη=µ0+µ1k1+···+µ12k1,0 165October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter157whereµ0throughµ12dependonρ.E(k1)ispositivedefinite,soaglobalminimumexists.Ingeneral,thisshalldependonthevalueofρ.Inthecasewhereρ=0.5,wefindthatk∗=argminE(k)=−0.72731,11k1∈RwhichgivesminimalerrorofE(k∗)=1.9528×10−6.14.5.3.AhomotopyquadraticinqWiththesamelinearandnonlinearoperatorsasthosegiveninthepreced-ingsection,definethemodifiedhomotopyoftheformH(q)[f]=(1−q)L[f]−(k1+k2q)qN[f],sothatB(q)isnowquadratic,withtwofreeconvergence-controlparame-ters.Whenk2=0,werecoverthepreviousexample.Hence,weexpectthattheaddedfreedomofhavinganadditionalfreeparameterwillallowustobettercontroltheresidualerror.Assumingafour-termapproximatesolutionoftheformF(η;k,k)=f(η)+f(η;k)q+f(η;k,k)q2+f(η;k,k)q3,12011212312theconditionH(q)[F]≡0implies00L[f0]=0,f0(0)=0,f0(0)=1,limf0=0,η→∞L[f]=k(f000+ff00−ρf02),110000f(0)=0,f0(0)=0,limf0=0,11η→∞1L[f]=L[f]+k(f000+ff00+ff00−2ρf0f0)2111100101+k(f000+ff00−ρf02),20000f(0)=0,f0(0)=0,limf0=0,22η→∞22L[f]=L[f]+k(f000+ff00+ff00+ff00−2ρf0f0−ρf0)3212201102021+k(f000+ff00+ff00−2ρf0f0),21100101f(0)=0,f0(0)=0,limf0=0.33η→∞3Notethatthek2parameterbeginstoappearintheordertwoequation,withtheeffectbeingtheadditionofthepreviousorderinhomogeneityduetothenonlinearityinlowerorderterms.Forthisreason,f1(η;k1)isexactlythesameaswasgivenbefore(whilef0(η)willalwaysbethesame,withthepresentchoiceoflinearoperator). 166October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4158R.A.VanGorderCalculatingZ∞E(k,k)=(N[F(η;k,k)])2dη,12120weobtainacomplicatedexpressioninvolvingpowersofk1andk2,withcoefficientsgivenintermsofρ.Consideragainthecaseofρ=0.5.Wefindthat∗∗(k1,k2)=argminE(k1,k2)=(−0.77075,−0.016403),(k1,k2)∈R2whichgivesminimalerrorE(k∗,k∗)=1.7836×10−6.12Thisshowsthattheaddedcomplexityhasnotgainedusmuchinthewayofimprovederror.Ontheotherhand,thenumberofcalculationshasalmostdoubledforthosetermsofordertwoorhigher.Hence,fortheverymodestimprovementinerrorseen,themethodisnotefficient.Meanwhile,addingadditionaltermsappearstoimproveerrorbyanorderofonetotwomagnitudeseachiteration.Therefore,itislikelymoreefficienttoobtainhigherorderapproximationswiththestandardlinearhomotopythanitistomodifythestructureofthehomotopy,atleastinthisexample.Notethattheseresultsagreewithsimilarconclusionsobtainedin[42]fortheBlasiusproblem.4.6.WhatabouttheauxiliaryfunctionH(x)?Inthepresentsection,weshallrevisittheso-calledauxiliaryfunctionH(x)whichsometimesappearsinthehomotopylinkingLandN.Whilethisfunctionisoftentakentoone,arecentpaperbythepresentauthor[38]demonstratedaclearadvantagetotakingthefunctionH(x)tobemorecomplicatedincertaincircumstances.Weshallhighlightthemainresultsofthatpaperhere.LetusconsidertheFitzhugh–Nagumoequation∂u∂2u=+u(u−α)(1−u).(4.54)∂t∂x2Thisequation,andrelatedvariants,havebeenstudiedrecentlythroughavarietyoftechniques;sees[119–123]andreferencestherein.TheFitzhugh–Nagumoequationhasvariousapplicationsinthefieldsoflogisticpopulationgrowth,flamepropagation,neurophysiology,autocat-alyticchemicalreaction,branchingBrownianmotionprocessandnuclearreactortheory;see,e.g.,[124–126].Thereisalsoadensitydependentdif-fusionNagumoequation,wherethediffusioncoefficientisasimplepower 167October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter159function.Thisequationisusedinmodelingelectricalpulsepropagationinnerveaxonsandinpopulationgenetics;see[127–132]andthereferencestherein.Undertheassumptionoftravellingwavesolutions,wesetz=x−ctandconsidersolutionsofthetypeu(x,t)=w(z),wherecdenotesthewavespeed.From(4.54),wehavew00+cw0+w(w−α)(1−w)=0.(4.55)Inthecaseofasolitarywaveoverthewholerealline,naturalboundaryconditionsarew→0asz→+∞andw→0asz→−∞.(4.56)Asisstandardinthemethodofhomotopyanalysis,weconstructahomotopyH[φ(z,q)]=(1−q)L[φ(z,q)−w0(z)]−qhH(z)N[φ(z,q)],whereHdenotesthehomotopybetweenanonlinearoperatorN(whichistheoperatordescribingthenonlineardifferentialequationwewishtosolve)andanauxiliarylinearoperatorL.Hereq∈[0,1]istheembeddingpa-rameter,histheconvergence-controlparameter,andH(z)istheauxiliaryfunction.Itisclearthatthenonlineardifferentialoperatorshouldtaketheform∂2φ∂φN[φ(z,q)]=+c+φ(φ−α)(1−φ),∂z2∂zwhilethelinearoperatorshouldbechoseninordertopermittheinitialapproximationtaken.Notethatwhenq=0wehavetheinitialapproximation,i.e.φ(z,0)=w0(z),whereaswhenq=1weseethatφ(z,1)isasolutiontothenonlineardifferentialequationofinterest,i.e.N[φ(z,1)]=0.Consideringaseriesexpansioninq(treatingqasa“smallparameter”),wehaveX∞w(z)=w0(z)+wn(z)(4.57)n=1where1∂nφ(z,q)wn(z)=n|q=0n!∂q 168October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4160R.A.VanGorderisthesolutionto(4.55)providedthattheseriesforφ(z,q)convergesatq=1.Amoreusefulrecursiveformulaforthewn’sisgivenbythenth-orderdeformationequationsL[wn(z)−χnwn−1(z)]=hH(z)Rn(z,h)(4.58)wherenX−1R(z,h)=w00+cw0−αw+(1+α)wwnn−1n−1n−1in−1−ii=0nX−1Xi−wn−1−iwjwi−ji=0j=0and(0,n=0,1,χn=1,n≥2.Hence,oncewehaveselectedanauxiliarylinearoperatorL,wecanrecoverthetermsintheexpansion(4.57).Thetermsintheexpansion(4.57)shouldbesolvedfrom(4.58)subjecttoboundaryconditionswn→0asz→+∞andwn→0asz→−∞.(4.59)Inpractice,wewilltruncatetheseries(4.57)tosomedesirednumberofterms.Thus,weshallbeconcernedwithanapproximatesolution˜w(z)withn∗+1terms:∗Xnw˜(z)=w0(z)+wn(z).(4.60)n=1Uptothispointthechoiceoftheconvergence-controlparameterh,theauxiliaryfunctionH(z)andthelinearoperatorLhaveallbeenkeptarbitrary.OnceH(z)andLareselected,wecancomputeapproximationsoftheform(4.60)forfixedn∗.Wewillthenattempttominimizetheerrorinsuchanapproximationbywayofchoosingtheconvergence-controlparameter,h,inanappropriatemanner.Weseekasolutionoverthewholerealline,(−∞,∞),henceweshallassumethattheinitialapproximationtothesolutionprofiletakestheform2ofaGaussian,e−αz.Notethatsuchaninitialapproximationsatisfiesbothw→0asz→+∞andw→0asz→−∞,whereasthedecayingexponentialwillonlysatisfythefirstboundarycondition. 169October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter161Letusassumeaninitialapproximationoftheform w(z)=exp−(g(z))2.(4.61)0Wheng(z)=azweobtaintheGaussianapproximation,whilewheng(z)=azmweobtainageneralizedGaussianapproximation.Hereaeffectsthewidthandmeffectstheshapeofthewave.Forgreatestgenerality,weshallconsiderthelattercaseinourcomputations,andtakem=1forthespecialcaseofaGaussianapproximation.Inordertoconstructanappropriatelinearoperator,weshouldselectanoperatorLsatisfyingtheproperties(1):L[C1w0(z)+C2]=0forconstantsC1andC2,(2):deg(L)=deg(N).Thefirstconditionpermitstheinitialapproximationtosatisfythelinearoperator,whereasthesecondconditionmandatesthattheorderoftheoperatorsNandLmustmatch(whichisreasonable,asitcorrespondstoanon-degeneratelinearizationofN).Letussearchforanoperatord2dL=+ζ(z)(4.62)dz2dzassuchanoperatorsatisfiesthesecondcondition,aswellaspartofthefirstcondition(L[C2]=0).Placingtheinitialapproximation(4.61)into(4.62)wefindthat2g2g02−g02−gg00ζ(z)=gg0givesalinearoperatord22g2g02−g02−gg00dL=+dz2gg0dzwhichsatisfiesthefirstcondition.Giventhechoiceg(z)=azm,thisreducestod22m(a2z2m−1)+1dL=+.(4.63)dz2zdzNowthatwehavefoundanappropriatelinearoperator,wemayreturntothehigherorderdeformationequations(4.58).Inordertosolvetheseforthewn’s,wemustinvertL.NotethatasolutiontotheinhomogeneousdifferentialequationL[U(z)]=F(z)isgivenbyZzZs22m22mU(z)=F(t)eatt−2m|t|dt+Ae−ass2m|s|−1ds+B 170October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4162R.A.VanGorderwhereAandBareconstantsofintegration.Applyingthisgeneralformulatothehigherorderdeformationequations(4.58),wefindthatwn(z)=χnwn−1(z)+BnZzZs22m+hH(t)R(t,h)eatt−2m|t|dt+Ann(4.64)0022m×e−ass2m|s|−1ds.Enforcingtheboundaryconditions(4.59),weobtainexpressionsfortheAn’sandBn’s:21An=ma(In(−∞)−In(∞)),Bn=−(In(−∞)+In(∞)),2whereZzZs22m22mI(z)=hH(t)R(t,h)eat|t|t−2mdte−ass2m|s|−1ds.nn00Fromhere,itisclearthattheamplitudesatisfiesX∞w(0)=1+Bn.n=1Physically,weexpectauniquevalueoftheamplitude.Hence,whenplot-tingtheh-curvefortheproblemathand,itmakessensetousetheampli-tudew(0)asaninvariantinordertodetermineaproperdomainfortheconvergence-controlparameter,h.Duetothesingularnatureoftheintegrandin(4.64)att=0,weneedtoexercisecarewhenselectingtheauxiliaryfunctionH(z).Inmanycases,thisfunctionissettounity,whiletheconvergence-controlparameterischosensothatconvergentsolutionsareobtained.Inthepresentcase,weseethattakingH(z)=1wouldleadtoasingularintegrandatt=0forallm≥1.InadditiontoselectingH(z)topermitregularityofsolutions,wecanselectH(z)suchthattheintegrationin(4.64)issimplified.Notethatifafactoroft2m−1ispresentinsidethedoubleintegral(4.64)thenthesubstitutionµ=t2manddµ=2mt2m−1canbetakeninordertospeedtheintegrationprocesswhencomputingtheapproximatesolutions.WefindthatwemaypickafunctionalformofH(z)whichtakesbothconcerns,namelyregularityofsolutionsandefficiencyofcalculation,intoaccount.Todothis,wesetH(z)=z2m−1|z|2m−1.NotethatH(0)=0whereasH(z)6=0forallz6=0.Hence,thestructureofthenonlinear 171October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter163differentialequationisnotlostoveranyopenintervalinR.WiththischoiceofH(z),(4.64)isreducedtown(z)=χnwn−1(z)+BnZzZs22m+hR(t,h)eatt2m−1dt+Ann0022m×e−ass2m|s|−1ds,whileZzZs22m22mat2m−1−as2m−1In(z)=hRn(t,h)etdtes|s|ds.00InthecaseofaGaussianinitialapproximationtotheprofile,thesequantitiesreducetown(z)=χnwn−1(z)+BnZzZs2222+hR(t,h)eattdt+Ae−as|s|ds(4.65)nn00andZzZs2222I(z)=hR(t,h)eattdte−as|s|ds.nn00Withtheformulationgivenintheprevioussections,wearenowinapositiontocomputeapproximateanalyticalsolutionsfortheFitzhugh-Nagumoequationbyuseofthehomotopyanalysismethod.Whiletheparameteracanbeusedtoinfluencethebasisfunctionsselected,weshallseta=1asthisissufficienttostudythebehaviorofthesolutionswe2seek.Startingwithw=e−z,weiterativelycomputethew’sfromthe0krelationgivenin(4.65).Weobtainlengthyalgebraicexpressionsdependingonx,c,αandtheconvergence-controlparameter,h.Weomitthelengthyexpressions,andfocusonthesolutionsforspecificvaluesofthephysicalparameters.Hence,oursolutionsarenumericalinnature.However,inordertodemonstratetheformofthesolutionsgiven,wegivethefirstandsecondordersolutions,whichwerecomputedrecursivelyinthecomputeralgebraprogramMaple13.Wefindthat√hcπh−3z2h(1+α)−2z2w1(z)=(1−erf(z))−e+e4248h(α−2)hch(α−2)2hc3h4−z2++z+z+z−ze42432 172October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4164R.A.VanGorderandsoon.HereZz2−t2erf(z)=√edt.π0Wetaketheresidualerror(x,c,α,h)=N[˜w]forthen∗+1termapproximation.Integratingtheresidualerrorsoverthedomain(theentirerealline)iscomputationallydifficult,soweconsideradiscreteform(seeLiao[10])X1012ˆ(c,α,h)=((k,c,α,h)).21k=−10Thesolutionsalwaysdecayrapidly,sotheprofilesbecomenegligiblepastx=±10.Forfixedcandα,weareabletofindhwhichminimizethesquareoftheseerrors.Wefindthattherearetypicallytwocriticalvaluesofh,onepositiveandonenegative.Thepositivevalueofhleadstosolutionswhichblow-up,andhenceisnotreasonable.Thenegativevalueofhleadstothephysicallymeaningfulsolutions.Fig.4.1.ProfilesoftheGaussianwaveprofiles,w(z),forvariousvaluesofthewavespeed,c.Herewehavetakenα=0.5.WeplottheHAMsolutionsoforder6withvarioustypesoflines,whereasthenumericalsolutiontotheproblem(thenumericalsolutionissetupasaninitialvalueproblem,andhenceisvalidoverthepositiveregion)isgivenbydots.Theerrorminimizingvaluesoftheconvergencecontrolparameter,h,aregivenbyh=−0.90026(c=0.1),h=−0.88407(c=0.2),h=−0.86139(c=0.5),h=−0.79109(c=1.0).Withthesevalues,errorintheapproximatesolutionswascontrolledtowithin10−4.Thenumericalsolutionswerecontrolledtoanerrortoleranceof10−5. 173October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter165Fig.4.2.ProfilesoftheGaussianwaveprofiles,w(z),forvariousvaluesofthewavespeed,c.Herewehavetakenα=0.2.WeplottheHAMsolutionsoforder6withvariouslines,whereasthenumericalsolutiontotheproblem(thenumericalsolutionissetupasaninitialvalueproblem,andhenceisvalidoverthepositiveregion)isgivenbydots.Theerrorminimizingvaluesoftheconvergencecontrolparameter,h,aregivenbyh=−0.98612(c=0.35),h=−0.96980(c=0.4),h=−0.94635(c=0.5),h=−0.88060(c=1.0).Withthesevalues,errorintheapproximatesolutionswascontrolledtowithin10−4.Thenumericalsolutionswerecontrolledtoanerrortoleranceof10−5.InFig.4.1,forthesakeofdemonstrationwefixw(0)=1,α=0.5,andallowthewavespeedctochange.Weseethatthederivativeofthesolutionasoneapproachesfromeithersideoftheoriginstronglydependsonthe00choiceofc.Bysymmetry,limz→0−w(z)=limz→0+|w(z)|.Weareabletocomparethesolutionbranchonthepositivehalf-linetoarelevantinitialvalueproblemsolvednumericallybyRKF-45method,andtheagreementisverygood.InFig.4.2,weprovideasimilarplot,onlytakingα=0.2.Hereweobservethat,forsmallervaluesofthewavespeed,thesolutionsappeartohavecontinuousderivativesatz=0.Thismakessensebecause,aswehavediscussedabove,theconditionw0(0)=0impliesthatw(0)isanonlinearfunctionofthewavespeed,theparameterα,andthephysicalR∞02invariant(w(z))dz.So,forappropriatevaluesoftheparameters,we0canexpecttofindsolutionspermittingw0(0)=0,whereasforothervalues,solutionsmaybecontinuousbutnotdifferentiableatz=0.4.7.AvoidingsingularitiesOften,singularitiesariseinmathematicalproblems.Eithertheyarere-movablesingularities,likeinthecaseoftheLane–Emdenproblem,orthey 174October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4166R.A.VanGordermaybeessential.Inthelattercase,theycanseriouslyimpedethesolutionprocessforsuchproblems.However,thehomotopyanalysismethodgivesusawaytodealwithsuchproblems,giventhatweareabletoadequatelychoosetheauxiliarylinearoperator.ThestronglysingularproblemfortheErnstequationgoverningaxiallysymmetricstationaryvacuumgravitationalfieldswasrecentlyconsideredbyBaxterandVanGorder[44].Weshallhighlightthefindingsofthisstudyrelevanttorecoveringsolutionsnearsingularities.Physically,theErnstequationservesasamodelofaxiallysymmetricstationaryvacuumgravitationalfields[133–136].Harrison[137]showsthattheErnstequationadmitsaB¨achlundtransform.Perturbationtheoryhasbeenconsidered[138].In[139],theErnstequationisusedtocompletelyseparatethevacuumEinsteinequationsforanarbitrarystationaryaxisym-metricspace-time.In[140],theVirasoroalgebraisshowntoexistinthesolutionspaceoftheErnstequation.Rationalapproximationsoftheflipan-gledependenceofanMRIsignalarederivedusinghalf-angletrigonometricsubstitutionsintheErnstequationin[141].In[142],theclassofhyper-ellipticsolutionstotheErnstequationarederivedusingRiemann–Hilberttechniques.Theinversescatteringmethod[143]andthealgebra-geometricideology[144]havealsobeendiscussed.Thetransformedequationreads002s020u+u−uu=0,(4.66)s2−1s2−1whereprimedenotesdifferentiationwithrespecttos.Wemayrenormalizethedomainsothats∈[1+,∞),where0<isaparameter.Meaningfulboundaryconditionsareu(1+)=Aandlimu(s)=0,(4.67)s→∞whereAisaconstant.WemayapplythehomotopyanalysismethodtothereducedErnstequation(4.66)withinitialdata(4.67).Weshallselectthelinearoperator002s0L[u]=u+u.(4.68)s2−1Thezeroth-orderapproximationisthengovernedbytheboundaryvalueproblemL[u0]=0,u0(1+)=A,limu(s)=0.s→∞ 175October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter167Thezeroth-orderapproximationisthengivenbyln(s+1)−ln(s−1)u0(s)=A.(4.69)ln(2+)−ln()Thefollowingwillbeusefulincomputingthehigherorderterms.Con-sidertheinitialvalueproblemL[U(s)]=Y(s),U(1+)=0,limU(s)=0.s→∞LetusdefinethefunctionZsZζ12I(s;)=(ξ−1)Y(ξ)dξdζ.1+ζ2−11+Thesolutionobeyingtheconditionats=1+takestheformZsdξU(s)=I(s;)+C1+ξ2−1Cs−1=I(s;)+ln−ln,(4.70)2s+12+whereCisaconstanttobedetermined.Evaluatingthisass→∞andusingtheremainingboundarycondition,wehaveC0=limI(s;)−ln,s→∞22+whichgives2I(∞;)C=,ln2+wherethenumeratorisdefinedinthelimitingsense.Withthis,wehavethesolutionI(∞;)s−1U(s)=I(s;)+ln−lnlns+12+2+I(∞;)s−1=I(s;)−I(∞;)+ln.(4.71)lns+12+TheHomotopybetweenthereducedErnstequationandtheauxiliarylinearoperatorL,givenin(4.68),is0≡H[u,u0;q]=(1−q)L[U−u0]−hqN[U],(4.72) 176October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4168R.A.VanGorderwhereu0(s)isaninitialapproximatesolution,histheso-calledconvergence-controlparameter,and2dUN[U]=L[U]−U.(4.73)s2−1dsTheinitialapproximationis(4.69).Thefirst-orderdeformationequationreads2h0L[u1]=−2u0(s)u0(s),s−1whichsimplifiesto4hA2s+11L[u1]=ln(2+)2lns−1(s2−1)2uponusing(4.69).Thesolution,accordingto(4.71),is(22)hA2s−1s−1u1(s)=−2lnln−ln.6ln()s+1s+12+2+Thesecondorderdeformationequationis2h00L[u2]=(1+h)L[u1]−2(u0(s)u1(s)+u1(s)u0(s)).s−1Using(4.69),thesolutionis(2hA2+2+u2(s)=−2+3ln15(1+h)−2Ahln90ln()2)s−1s−1+3Ahlnlns+1s+1(22)s−1×ln−ln.(4.74)s+12+ThesumofthefirstthreetermsgivesustheapproximatesolutionUb(s,h,A,)=u0(s)+u1(s)+u2(s).WerunUbthroughtheoriginalnonlinearoperatorcorrespondingto(4.66),thenweset=1anddefinetheresidualerrorN[Ub(s,h,A,1)].Togettheerrorovertheinterval[2,∞),wecomputethesumofsquaredresidualerrorsZ∞2E(h,A)=(N[Ub(s,h,A,1)])ds.(4.75)2 177October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter169WhatwegetisthatE(h,A)isaneighth-degreepolynomialinh.Togetnu-mericalestimatesoftheerror,caremustbetakeninusingapproximationsinthecoefficientsofE(h,A)whenactuallyplottingtheerrornumerically.Whenusing5or10digitsofaccuracyforthecoefficientsofh,roundingerrorscanbeverylarge.Uponexplicitevaluation,(4.75)becomesE(h,A)=µ(A)h8+µ(A)h7+···+µ(A).(4.76)870WhenA=1wehaveaminimumath=−0.982.ThisgivestheapproximatesolutionofUb(s,−0.982,1,1)withasumofsquaresresidualerrorofE(−0.982,1)=3.19×10−5.TheerrorincreaseswiththevalueinA,however.WhenA=0.1,theerroris3.46×10−13,butwhenA=5,theerroris0.567.Thismakessense:givenalargervalueofA,theadjustableparameterisclosertothesingularity,hencethesingularity’seffectisstronger.ToseetheroleincreasingAhasonmakingthesolutionmoresingu-lar,notethatthesolutiontoourchoiceoflinearoperatorLhasLaurentexpansionnears=1oftheformCγU(s)=+higherorderterms,(s−1)γforarbitraryγwhereCγistheleadingordercoefficient.Then,applyingA=u(1+),wehaveCγA=γ(neglectinghigherorderterms,sincethesingularitydominatesnears=1).Ifissmallyetfixed,theparameterγscalesasγ∼ln(A).So,asAbecomeslarge,thestrengthofthesingularityincreases.Forsuchstronglysingularcases,theapproximationmethodbreaksdown.However,forsmallormoderatevaluesofA,thethree-termexpansionsapproximatethesolutionsremarkablywell.TheconnectionwiththesingularityandthevalueAwasexploredingreaterdetailinthepaper;see[44]formoredetails.4.8.ConclusionsUsingoptimalhomotopyanalysismethodthroughadequateselectionoftheauxiliarylinearoperatorandconvergence-controlparameter,wehavebeenabletostudyanumberoffeaturesofhomotopyanalysis.Whenchoosingdifferentlinearoperators,wefindthattherateofconvergenceisstronglytiedtoourchoice.Indeed,forsomeauxiliarylinearoperators,the 178October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4170R.A.VanGorderrateofconvergencemaybequiteslow,whereforothersitcanberatherfast.So,itisimportanttoproperlyselectthelinearoperator.Similarcommentsholdinthecaseofnonlinearpartialdifferentialequations.Therateofconvergenceisimportantifwewishonlytocomputefewterms.Forcomplicatedequations,particularlyPDEs,eachiterationofthemethodcanbecomeverycomputationallydemanding.Hence,perhapsonlyafewtermscanevenbecalculatedinminimaltime.Therefore,thechoiceoftheauxiliarylinearoperatorisvitalforcomputationalefficiency.Whenapplyinganoptimalformofthehomotopyanalysis,wetreattheconvergence-controlparameterasanoptimizationparameter,andattempttominimizesomefunctionoftheerrororresidualerror.Doingso,wearriveatanerrorminimizingvalueoftheconvergence-controlparameter.Itisnaturaltoaskwhathappenstothisparameterasweincreasethenumberoftermsinourhomotopyapproximations.Whatwefindfromsolvingsomenonlinearordinarydifferentialequationsisthatwhentheauxiliarylinearoperatorisproperlychosen,thevaluesoftheoptimalconvergence-controlparameterappeartostabilizearoundsomefixedvalueasweincreasethenumberofiterations.Ontheotherhand,forpoorchoicesofoperators,itseemsthattheoptimalvalueoftheconvergence-controlparameterismorevariableoneachiteration,owingtothefactthatapoorchoiceoflinearoperatorresultsinslowerconvergence.Thereareanumberofotherwaysonecanthinkofmodifyingtheho-motopyanalysismethod.Forinstance,insteadofthestandardhomotopyH(q)[U]=(1−q)L[U]−hqN[U],itispossibletoconsidermoregeneralformsofthehomotopyH(q)[U]=(1−q)a(q)L[U]−hqb(q)N[U].However,suchageneralizationisnotalwaysuseful.Weshowthatwhenwehaveaquadratichomotopyintheembeddingparameterq,thereisnosignificantimprovementinthecontroloferrorofthesolution(eventhroughweintroducetwo,asopposedtoone,convergence-controlparameters),yetthecomputationsarefarmorecomplicated.Again,whethersuchanap-proachisusefulwillusuallydependonthespecificproblem.So,forsomeproblems,thisapproachmightbeworthwhileanduseful.AnotherformofthehomotopyisH(q)[U]=(1−q)L[U]−hH(x)qN[U].ThiswasthetraditionaltypeintroducedbyLiao,howevermanyauthorsomitthegeneralfunctionH(x).Fromwhatwenowknowofoptimalho- 179October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.4StabilityofAuxiliaryLinearOperatorandConvergence-ControlParameter171motopy,wecanviewhH(x)asatypeofvariableconvergence-controlterm.Usingthisterminit’smostgeneralform,itispossibletogreatlysimplifythesolutionprocessinvolvedinobtaininghigherorderapproximations.Weillustratethisbyconstructingsolutionstoanonlinearmodeloverthereal2line,withbasefunctionsofthetypee−αx.Whichsuchbasefunctionsareusuallytoohardandcomputationallydemandingtoworkwith,weshowthatitispossibletopickH(x)insuchawayastosimplifytheprocess.Hence,thechoiceofH(x)canbeusefulforcertainproblems.Finally,wepresentahighlysingularproblem,andwedemonstratethattheoptimalselectionoftheconvergence-controlparameterstillcanbeuse-fulforsuchsituations.Themethodalsogivesanindicationofthestrengthofthesingularity.Insummary,thehomotopyanalysismethodgivesusgreatfreedominsolvingnonlinearordinaryandpartialdifferentialequations.Byappropri-atelyselectingtheauxiliarylinearoperator,theconvergence-controlparam-eter,andsomeotherfeatureswhenneeded,wecanobtaincomputationallyefficientapproximatesolutionswithlowresidualerrorsforanumberofnonlinearproblems.References[1]S.J.Liao,Ontheproposedhomotopyanalysistechniquesfornonlinearprob-lemsanditsapplication,Ph.D.dissertation.(ShanghaiJiaoTongUniver-sity,1992).[2]S.J.Liao,BeyondPerturbation:IntroductiontotheHomotopyAnalysisMethod.(Chapman&Hall/CRCPress,BocaRaton,2003).[3]S.J.Liao,Anexplicit,totallyanalyticapproximationofBlasiusviscousflowproblems,InternationalJournalofNon-LinearMechanics.34,759–778,(1999).[4]S.J.Liao,Onthehomotopyanalysismethodfornonlinearproblems,AppliedMathematicsandComputation.147,499–513,(2004).[5]S.J.LiaoandY.Tan,Ageneralapproachtoobtainseriessolutionsofnonlineardifferentialequations,StudiesinAppliedMathematics.119,297–354,(2007).[6]S.J.Liao,Notesonthehomotopyanalysismethod:somedefinitionsandtheorems,CommunicationsinNonlinearScienceandNumericalSimula-tion.14,983–997,(2009).[7]S.J.Liao,HomotopyAnalysisMethodinNonlinearDifferentialEquations.(Springer&HigherEducationPress,Heidelberg,2012).[8]R.A.VanGorderandK.Vajravelu,Ontheselectionofauxiliaryfunctions,operators,andconvergence-controlparametersintheapplicationoftheHomotopyAnalysisMethodtononlineardifferentialequations:Ageneral 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189October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5Chapter5AConvergenceConditionoftheHomotopyAnalysisMethodMustafaTurkyilmazogluMathematicsDepartment,UniversityofHacettepe06532-Beytepe,Ankara,Turkeyturkyilm@hotmail.comInthischapter,wepresentaconditionenablingthehomotopyanalysismethod(HAM)toconvergetotheexactsolutionofthesoughtsolutionofalgebraic,highlynonlineardifferential-difference,integro-differential,fractionaldifferentialandordinaryorpartialdifferentialequationsorsystems.Thepreviousnotionsofconvergencecontrolparameterarecarefullyreviewedandanoveldescriptionisproposedtofindoutanop-timalvaluefortheconvergencecontrolparameter,which,althoughitiscompletelydifferentfromtheclassicaldefinitionbymeansofthesquaredresidualerrorasoftenusedintheliterature,yieldsnearlythesameinter-valofconvergenceandoptimalconvergenceparametersasthosefoundfromthesquaredresidualerror.Whenanunknownparameterisem-beddedintothegoverningequations,theconvergenceoftheHAMisbetterpursuedbytheratiorelevanttothisparameterratherthantheratioofotherfunctionsinvolvingmuchharderintegrations.AnerrorestimatefortheHAMisalsoprovided.Physicalandmechanicalexam-ples,includingtheVolterradifferential-differenceequation,theFredholmintegro-differentialequationforthestaticbeam,theAiryequation,theundampedanddumpedDuffingoscillators,theThomas–Fermiequation,theGelfandproblem,thefractionaldifferentialequation,therotatingsphere,andmore,clearlyillustratethevalidityofthenewapproachandfurtherprovideknowledgeonwhythecorrespondinghomotopyse-riesgeneratedbytheHAMshouldconvergetotheexactsolutioninthedomainofinterest.181 190October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5182M.TurkyilmazogluContents5.1.Introduction.....................................1825.2.Abriefdescriptionofthehomotopyanalysismethod..............1845.3.Aconvergencetheoremanditserrorestimate..................1865.3.1.Importantconsequencesandremarks...................1895.4.Convergencecontrolparameterinthehomotopyanalysismethod.......1905.4.1.Intervalofconvergence...........................1915.4.2.Optimumvaluefromanappropriateresidual...............1925.4.3.Optimumvaluefromtheratio.......................1945.5.Illustrativeexamples................................1965.5.1.Analgebraicequation............................1965.5.2.AnonlinearVolterradifferential-differenceequation...........2005.5.3.Anonlinearhigh-orderFredholmintegro-differentialequation......2035.5.4.Someordinarydifferentialequations....................2065.5.5.Anonlinearfractionaldifferentialequation................2335.5.6.Couplednonlineardifferentialequations..................2375.5.7.Partialdifferentialequations........................2445.6.Concludingremarks.................................252References.........................................2545.1.IntroductionSincethenonlinearordinary/partialdifferentialequationsorsystemsforinitialandboundaryvalueproblemsaremostchallenginginfindingtheirexactsolutions,besidestheclassicalperturbationmethods,somenewper-turbationoranalytical-naturemethodshavebeenintroducedanddevelopedbyresearchersintheliterature.Amongvarietyofmethodsproposedtofindanalyticapproximatesolutionsofagivennonlinearmathematicalmodel,themostrecentpopularandpowerfultechniqueisthehomotopyanalysismethod(HAM).Inthismethod,whichrequiresneitherasmallparameternoralinearterm,ahomotopywithanembeddingparameterp∈[0,1]isconstructed[1].Thesolutionisconsideredasthesumofaninfiniteseriesconvergingrapidlyandaccuratelytotheexactsolutionsbymeansofen-joyingtheso-calledconvergencecontrolparameter,amissingtoolinmostoftheothertechniques.FundamentalcharacteristicsandsmartadvantagesoftheHAMovertheexistinganalyticaltechniqueswereclearlylaidoutbyLiaointherecentbook[2].Inadditiontoitsearlysuccessinseveralnonlinearproblemsassummarizedinthebook[3],furthernumerousnon-linearproblemsinscience,financeandengineeringweresuccessfullytreatedbythemethod,seeamongthem[4–12].Particularly,afewnewsolutionsofsomenonlinearproblemswerediscoveredbymeansofthemethod[13], 191October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod183whichwereunfortunatelyneglectedbyotheranalyticmethodsandevenbynumericaltechniques.Themethodwassuccessivelyappliedrecentlytosomeseriesofstronglynonlinearproblems,suchastheBlasiusequationfortheflowoveraflatplate[14],thesystemofdifferentialequationsconcern-ingtheflowoverarotatingcone[15],thesystemofdifferentialequationsrelatedtotherotatingdisk[16–19],thesystemofdifferentialequationsre-latedtotherotatingsphere[20],thesingularlyperturbedboundarylayerproblems[21,22],theundampedanddumpedDuffingoscillators[23,24],thelimitcycleofDuffing-vanderPolequation[25],thenonlinearpendu-lumproblem[26],theThomas–Fermiequation[27]andtheAirydifferentialequation[28].AnanalyticshootingapproachcombinedwiththeHAMwasalsoproposedin[29].Aftertheworkof[30],theuseoftheHAMismoresafenowsinceanoptimalparametercontrollingthefastconvergencecanalwaysbepickedfromthesquaredresidualerrorensuringtogainthemostaccurateresults,asalsoimplementedintheabovecitations.Despitethefactthatallthesedemonstratethevalidityandhighpotentialofthehomo-topyanalysismethodforstronglynonlinearproblemsofreallife,apartfromsomegeneralapproachesaspresentedin[2],thequestionofconvergenceofthemethodisyettobeanswered.Thepresentchapterisdevotedtotheinvestigationofthehomotopyanalysistechniquefromamathematicalpointofviewtoservetoitsconver-genceissue.TheaimisthustoanalyzethemethodandtoshowthatunderagivenconstrainttheHAMconvergestotheexactsolutiondesired,withanerrorestimate,withoutapriorknowledgeoftheexactsolution.Inaddi-tiontothewell-knownintervalofconvergencecontrolparameterbymeansofconstanth-curvesandoptimalvaluefortheconvergencecontrolparam-eterviathesquaredresidualerror,anewconceptualdefinitionisoffered,whichmakesuseoftheratiosofthehomotopyseriesbasedonaproperlychosennorm.Itisshownthroughexamplesthatbothyieldapproximatelythesamevaluesregardingtheconvergencecontrolparameter,thoughthenewlyintroducedschemeseemsmoreadvantageousinsomeaspectsatleastintermsofcomputationalefforts.Embeddinganunknownparameterintothestudieddifferentialequationfromtheboundariesisdemonstratedtofacilitateagreatadvantageforunderstandingtheconvergencethroughthisratio,thatavoidstheuseofratiosconcerningtheotherphysicalfunctionsinvolvingheavierintegrations.Thegivenconvergencecriterionisjustifiedexemplifyingitbybasiccommonly-knownexamplesfromnonlinearalge-braic,differential-difference,integro-differential,thefractionaldifferential,ordinaryandpartialdifferentialequationsandalsosystemsoftenstudied 192October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5184M.Turkyilmazogluintheliterature.TheconvergenceoftheHAMfortheconsideredprob-lemsisnotonlyguaranteed,buttheintervalofconvergenceandfurthertheoptimumvaluefortheconvergencecanalsobedeterminedbythepre-sentedtheory.Thenewapproachalsoenlightensusabouttheinevitablefailureoftheso-calledhomotopyperturbationmethodblindlyusedbysomeinvestigators.5.2.AbriefdescriptionofthehomotopyanalysismethodAsystematicdescriptionofthehomotopyanalysismethodisoutlinedinthissection.Withinthispurpose,letusconsiderthefollowinggeneralnonlinearequationN[u(t)]=0,(5.1)whereNiseitheranonlinearfunction(concerningthealgebraicequations),oranonlinearoperator(concerningthedifferentialordifferenceequations),tdenotesanindependentvariable,u(t)isanunknownfunction,respec-tively.Inmoregeneralcase,equation(8.1)mayrepresentafullsetofnonlinearequations.Forsimplicity,weignoreallboundaryorinitialcon-ditionsinthecaseofadifferentialequation(orasystem),whichcanbetreatedinthesimilarway.AftertheearlyHAMdescribedbyLiaoinhisPhDdissertation[1],thefirstthingandthusthekeypointistoconstructtheso-calledzeroth-orderdeformationequation[3,31](1−p)L[ϕ(t,p)−u0(t)]−phH(t)N[ϕ(t,p)]=0,(5.2)wherep∈[0,1]iscalledthehomotopyembeddingparameter,hisanon-zeroauxiliaryparameterwhichiscalledtheconvergencecontrolparameter,Lisanauxiliarylinearoperator,u0(t)isaninitialguessforu(t),H(t)isanauxiliaryfunctiontoadjustthesoughtsolution,andϕ(t,p)isanun-knownfunction,respectively.Infact,thesuccessoftheHAMsubstantiallyreliesuponthezeroth-orderdeformationequation(8.2),whichcanbefur-thergeneralized[2].Itisimportant,thatonehasgreatfreedomtochooseauxiliaryparametersintheHAM.Obviously,whenp=0andp=1,itholdsϕ(t,0)=u0(t),ϕ(t,1)=u(t).(5.3)Thus,aspincreasesfrom0to1,thesolutionϕ(t,p)variesfromtheinitialguessu0(t)tothefinalsolutionu(t)oftheoriginalnonlinearequation(8.1). 193October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod185Expandingϕ(t,p)inMaclaurinserieswithrespecttopatp=0,itreadsX∞ϕ(t,p)=u(t)+u(t)pk,(5.4)0kk=1wheretheseriescoefficientsukaredefinedby1∂kϕ(t,p)uk(t)=.(5.5)k!∂pkp=0Here,theseries(8.4)iscalledthehomotopyseriesandtheexpression(8.5)iscalledthekth-orderhomotopy-derivativeofϕ,see[31].IftheauxiliarylinearoperatorL,theinitialguessu0(t),theconvergencecontrolparameterhandtheauxiliaryfunctionH(t)aresoproperlychosen,thehomotopy-series(8.4)convergesatp=1,thenusingtherelationshipϕ(t,1)=u(t),onehastheso-calledhomotopyseriessolutionX∞u(t)=u0(t)+uk(t),(5.6)k=1whichmustbeoneofthesolutionsoforiginalnonlinearequation(8.1),asprovedbyLiao[3].Basedonthedefinition(8.5),thegoverningequationforthehomotopyseries(5.6)canbedeductedfromthezeroth-orderdeformationequation(8.2).Differentiatingthezeroth-orderdeformationequation(8.2)ktimeswithrespecttothehomotopyparameterp,settingp=0andfinallydividingthembyk!,wehavetheso-calledkth-orderdeformationequationL[uk(t)−χkuk−1(t)]=hH(t)Dk−1[ϕ(t,p)],(5.7)whereDnistheso-callednth-orderhomotopyderivativeoperatorgivenby1∂nN[ϕ(t,p)]Dn[ϕ(t,p)]=,n!∂pnp=0andχk=0fork≤1,χk=1fork>1.(5.8)Notethattheright-handsidetermDk−1[ϕ(t,p)]of(5.7)isdependentonlyuponu0(t),u1(t),u2(t),···,uk−1(t),whichareknownforthekth-orderdeformationequationdescribedabove.Becauseofthefactthatthehigher-orderdeformationequation(5.7)islinearinnature,anappropriatelinearoperatorLwilleasilygeneratethe 194October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5186M.Turkyilmazogluhomotopytermsukinhomotopyseries(5.6)bymeansofcomputeralgebrasystemssuchasMathematica,Mapleandsoon.Finally,anMth-orderapproximateanalyticsolutionofpracticalinterestisgivenbytruncatingthehomotopyseries(5.6)XMuM(t)=u0(t)+uk(t),(5.9)k=1andtheexactsolutionisgivenbythelimitu(t)=limuM(t).M→∞Itshouldberemindedthatthehomotopytermsu1(t),u2(t),···,uk(t)in(8.27)stronglydependonboththephysicalvariabletandtheconver-gencecontrolparameterh.Inessence,hisanartificialparameterwithoutphysicalmeaningsbutitcanadjustandcontroltheconvergenceregionofthehomotopyseriessolution(8.27).Infact,theuseofsuchanauxiliaryparameterdistinguishestheHAMfromotherperturbation-likeanalyticaltechniques.5.3.AconvergencetheoremanditserrorestimatePerformingthemethodologyunderlinedabovein§5.2,thenumberofprob-lemstreatedbythehomotopyanalysismethodapproachesacoupleofthousandsnow.Existingtheoremsabouttheconvergenceoftheresultinghomotopyseriesofagivennonlinearproblemunfortunatelyhavegeneralmeanings[2,3]ortheconvergenceofsolutionseriesisbelievedtotakeplacesincetheHAMlogicallycontainsthefamousEulerTheoremasprovedbyLiao[2].However,arigorousandfirmanswertotheverybasicquestionofwhytheseries(5.6)obtainedbysettingp=1in(8.4)shouldbeconvergentremainsunansweredtilltoday.Tomakesureoftheconvergence,theana-lyticityofsolutionsisgenerallypresumed,otherwise,aMaclaurinseriesofafunctionmaynotnecessarilyconvergetothatfunction,seepages23and24in[2].Thismaylimitthehomotopymethodleadingtodivergenthomotopyseriessolutionsespeciallyfornonlinearproblemswithstrongnonlinearity.Moreover,althoughitisfortunatetoknowthattheconvergencecontrolparameterhcangreatlymodifytheconvergenceofthehomotopyseriessolution,theguaranteeofconvergencestillneedsamathematicalexpla-nation.Toremedythisissueuptoapoint,weprovidethesubsequent 195October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod187theoremsandtheresultingcorollarieshere.Itshouldbenotedthateventhoughtheproofsrequireprescriptionofconvergencecontrolparameterh,howtofindapropervalue,orevenbetter,togetafastestconvergentone,willbediscussedlater.Sincethehomotopyanalysismethodologyasdescribedin§5.2isaveryuser-friendlytoolamongthescientistsfrequentlyusedforsolvingcom-plicatedhighlynonlinearproblems,theconvergencecriteriontobegivenshouldalsobeeasy-to-usebeyondthegeneralityandintheabsenceofexactsolutiontothenonlinearequationunderconsideration.ThisisessentialindeeperunderstandingofwhethertheHAMperformedforaspecificproblemwillconvergetothetrueexactsolutionornot.Suchaconvergencecriterionwasmadeuseofinseveralphysicalproblems,forexamplethereadermayreferto[20,32]and[24,27,28].Inwhatfollowswestatethecriterion,thatisbasedonthefixedpointtheoremwellknowninthefunctionalanalysis.Theorem5.1.SupposethatA⊂RbeaBanachspacedonatedwithasuitablenormkk(dependingonthephysicalproblemunderconsideration),overwhichthefunctionalsequenceuk(t)of(8.4)isdefinedforaprescribedvalueofh.Assumealsothattheinitialapproximationu0(t)remainsinsidetheballofthesolutionu(t)of(8.1).Takingr∈R+beaconstant,thefollowingstatementsholdtrue:(i)Foraprescribedconvergencecontrolparameterh,ifkvk+1(t)k≤rkvk(t)kforallk,providedthat0 196October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5188M.Turkyilmazogluinmakinguseof(5.10)andthetriangleinequalitysuccessively,kSn(t)−Sm(t)k=k(Sn(t)−Sn−1(t))+···+(Sm+1(t)−Sm(t))k1−rn−m≤rm+1ku(t)k.(5.11)01−rSincebythehypothesis0 197October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod1895.3.1.ImportantconsequencesandremarksAfewimportantconsequencesoftheaforementionedtheoremswillbepre-sentednowtogetherwithsomerelevantremarks.Corollary5.1.Sincethefinitenumberoftermsdoesnotaffectthecon-vergence,Theorem5.1isequallyvalidiftheinequalitiesstatedin(i-ii)aretrueforsufficientlylargek0s.Thus,forapreassignedvalueofh,itissufficienttokeeptrackofmagnitudesoftheratioβdefinedbykvk+1(t)kβ=,(5.15)kvk(t)kandwhetheritremainslessthanunityforincreasingvaluesofk.Anopti-malvaluefortheconvergencecontrolparameterhcouldalsobedeterminedfrom(5.15)byrequiringtheratioβtobeasclosetozeroaspossible,sothatforsuchavaluetherateofconvergenceofhomotopyseries(8.27)willbethefastest,sincethentheremainderoftheserieswillmostrapidlydecay.Corollary5.2.Ontheconditionthatthenormistakeninthesenseofabsolutevalue,byenforcingtheratioin(i)toholdtrueintheinfinitelimitor,atleastforlargek,thevaliditydomainfortofthesoughtsolutioncanalsobeconstructedhavingprovidedavaluefortheconvergencecontrolvalueh,thatis,|vk+1(t)|lim<1.(5.16)k→∞|vk(t)|Corollary5.3.Thegraphicalconstanth-curvesideaofLiao[3]canalsobeapprovedbytheratiogivenin(5.15),insuchawaythatananalyticalinter-valofconvergenceforhcanbedeterminedbyapplicationofTheorem5.1tosomecertainphysicalquantities,sayforinstanceu(m)(t),m∈N,t∈R00andbysolvingtheinequality(m)|uk+1(t0)|<1,(5.17)(m)|uk(t0)|(m)sinceu(t)isafunctionofh.kCorollary5.4.Inthecasethatthenonlinearequation(8.1)comprisesoffindingtherootofanequationf(x)=0,solutionoftheinequalityforlargek|xk+1|<1,(5.18)|xk|directlyyieldstheintervalofconvergencecontrolparameterh. 198October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5190M.TurkyilmazogluRemark5.1.Ifexactornumericalsolutionue(t)isavailable,thenonecanalwaysusetheabsoluteerrordefinedbyZerr=|ue(t)−u(t)|dt,(5.19)Γwhereu(t)isthehomotopysolutionof(8.1)definedoverthedomainΓ.Remark5.2.ReadersshouldbewarnedthattheconditiongiveninThe-orem5.1andinthesubsequentcorollariesisonlyasufficientconditionfortheconvergenceofthehomotopyanalysismethod.Thisstronglyimpliesthatinthecaseswherethelimitfortheratioin(5.15)cannotbereachedortendstounity,themethodmaystillconvergeorfailtodoso.Remark5.3.Inthecasethattheremayexistanunknownparametertobesolvedtogetherwiththedifferentialequation,suchastheparameterizeddifferentialequations,itwouldsufficetokeeptrackoftheratioregardingthisparameterfromtheratio(5.18),thatwillluckilyavoidtheuseofresid-ualsandratioswhichinvolvecomplicatedintegrations.Suchexampleswillbeprovidedlateron.Notethateveryconsideredsystemalwaysvirtuallypossessesthissortofunknownparameters,namedhereafterasthepseudoparameters,atleastinsertedfromthephysicalboundaryconditionswhicharethemselvesunknown.Remark5.4.ThehomotopyPad´etechniqueintroducedin[3]canalwaysbeappliedtoenlargetheintervalofconvergenceandalsotoacceleratetherateofconvergenceforagivennonlinearproblem.Inadditiontothis,bymeansofthefreedomontheselectionofinitialguess,eitheranoptimalinitialapproximationissought,orahomotopyiterationapproachcanbedevisedintheframeoftheHAM,whichcangreatlyspeeduptheconver-genceofthehomotopyseries,see[2]forillustrations.5.4.ConvergencecontrolparameterinthehomotopyanalysismethodItiswidelyknownthatinthegeneralhomotopymethodimprovedbyLiao[2,3],theconvergencecanbecontrolledbytheconvergencecontrolparameterh,aconceptthatplaysakeyroleintheHAM.Theprimeroleofhinahomotopyseriesistoadjusttheconvergenceatareasonablerate,fortunatelyatthequickestrate.Therefore,inrecentapplicationsoptimal 199October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod191convergencecontrolparameterisgenerallymadeusetogainsufficientlyac-curateapproximationswiththesmallestnumberofhomotopytermsinthehomotopyseries(5.6).Inthissection,wefirstrecallthepreviouslyusedschemesforfindinganappropriatevalueoftheconvergencecontrolparam-eter,lateron,anewandnovelapproachforfindinganoptimumvalueofhispresented,basedontheratiopresentedinequation(5.15).Itshouldbestatedthatifasingleparameterhisinvolvedin(8.2),thebelowproce-dureoffindinghisrecentlytermedasbasicoptimalHAM,see[2].Besides,althoughtwo-parameter,three-parameterandinfinite-parameteroptimalHAMs,seeforinstancetheso-calledoptimalhomotopyasymptoticmethodin[33],canalsobetreatedwithoutanydifficulty,forthesakeofbrevityandconciseness,weonlypreferthebasicoptimalHAMinwhatfollows.Thisissatisfactoryenough,sincetheoptimalHAMswithmoreconvergencecon-trolparametersdonotalwaysgivebetterhomotopyapproximationsthanthebasicoptimalHAMingeneral,thusthebasicoptimalHAMisstronglysuggestedtouseinpractice[2].Howcanoneidentifyanappropriateconvergencecontrolparameterhthatresultsinaconvergenthomotopyseriessolution?Thereareinfacttwodifferentup-to-datewaysofdeterminingtheconvergencecontrolparame-terhmostfrequentlyusedamongtheHAMsociety;eitherbyplottingitsintervalofconvergenceorso-calledeffectiveregionviaaphysicalquantitydependingonhorbysearchingforaglobaloptimalvaluebymeansofaresidualerror,asformulatedbelow,whichisimplementedbyastraightfor-wardsubstitutionofhomotopyseriessolutionintotheoriginalgoverningequation.5.4.1.IntervalofconvergenceOneoftheremarkablepropertyoftheHAMisthatthevalueoftheauxil-iaryparameterhcanbefreelychosentoensuretheconvergence,andevenmore,toincreasetheconvergencerateofthesolutionseries(8.27).How-ever,thefreedomofselectinghissubjecttotheso-calledvalidregionsofh.ThisnotionisfirstintroducedbyLiao[3],thecreatoroftheHAM,andhasbeencommonlyusedinmanyHAMapplications,eventoday.Aphysi-calvariablefromtheproblemunderconsideration,whichisunknown,butevaluatedanalyticallyatthenumberoftruncatedhomotopyseriesMfromthehomotopyseries(8.27);sayu(m)(t),m∈N(likeu0(0)oru00(0))ofthe0nonlineardifferentialequation(8.1),isusuallyplottedversush(inthecaseofanalgebraicequationf(x)=0,thehomotopysolutionsofxaredrawn 200October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5192M.Turkyilmazogluagainsth).Thesecurvesareso-calledastheconstanth-curvesorcurvesfortheconvergencecontrolparameter,whichhavebeensuccessfullyhandledinmanynonlinearproblems,seeatleastthecitedreferencesherein.IthasbeenfoundbytheHAMresearchersthatthereoftenexistssuchaneffectiveregionthatcertainvaluesofhobtainedfromsomephysicalquantitiesresultinaconvergenthomotopyseriessolution.Eventhoughsucharegioncanalwaysbefound,withlesscomputationaleffortascom-paredtothesquaredresidual,byplottingthecurvesoftheseunknownquantitiesversush,theinformationabouttheintervalofconvergencecanbegainedonlyapproximatelyfromtheplot.Besides,nooptimumvaluecanbeingeneralrealizablefromtheapproach,thatcorrespondstothequick-estconvergenthomotopyseries.JustrandomvaluesfromthisintervalarepickedoverandoveragaintocarryouttheHAMsolutionsinnonlinearproblems.Inprinciple,thevalidregionofhwillexpandwithoutboundastheorderofhomotopyapproximationapproachesinfinity.However,infiniteorderoccasions,itisthecasethatthesolutionseriescorrespondingtoafixedvalueofhmaynotgiveagoodapproximationevenwhenhischosenwithinthevalidregion.Thisisduetothefactthattheh-curveisjustaplotofaphysicalquantityversushataspecifict=t0.Therefore,oneshouldexpectthesolutionseriestoconvergetotheexactsolutiononlylocallyaboutt=t0whenoneselectsahvaluefromsuchaplottedregion.Neverthelessbychance,therearecaseswhereavalidhvaluechosenthiswaydoesgivegoodapproximationforalargerangeoftheindependentvariablet.InspiteofthefactthatsomeinvestigatorsstillinsisttousethisideaintheirrecentHAMpublications,itisnowabolishedandhenceoutofdate.ItisworthyofemphasizingthattheinequalitiesinvolvingtheratiospresentedinCorollaries5.3and5.4effectivelycovertheaboveidea,beingcapableofreproducingtheconstanth-curveintervals.Besides,themostcrucialadvantageusingsuchratiosisthatexactintervalofconvergenceforconvergencecontrolparameterhcanbeobtainedbysolvingtheseinequal-ities,asclearlydemonstratedinthebelowexamples.5.4.2.OptimumvaluefromanappropriateresidualInsteadofapproximatelylocatingtheintervalofconvergenceasabove,itisbetterandhopefullymoreeconomicaltotrytogetabestvalueofh,fromwhichthehomotopysolutionsfromtheseries(8.27)aretobegenerated.Ingeneralspeaking,anoptimalvaluefortheconvergencecontrolparameterh 201October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod193attheMth-orderhomotopyapproximationmaybefoundfromthefollowingnormrepresentingtheresidualofthegoverningequation(8.1)XMRes(h)=kN[uk(t)]k,k=0withthenormbeingunderstoodasLp.Theoreticallyspeaking,wheneverthisresidualgoestozero,thentheobtainedhomotopyseriesisasolutionoftheoriginalgoverningequation(8.1).Researchers,intime,wereadaptedthemselvestoL2only,andfurther,sincebeingmoreconciseaftertheworkof[34],thesubsequentsquaredresidualerrorisoftenemployedtodetermineanoptimalvaluefortheconvergencecontrolparameterhZ("#)2XMRes(h)=Nuk(r)dr,(5.20)Γk=0whereΓistheregionofinterestfortheproblemunderconsideration.Ontheotherhand,ifitisknownthattheintegrandN[u(t)]ispositive,thenitisbettertousethesubsequentresidualerrorbasedonL1Z"#XMRes(h)=Nuk(r)dr,(5.21)Γk=0inordertogreatlysavefromthecomputationtime.Itshouldberemindedthatviatheresiduals(5.20)and(5.21),theoptimalauxiliaryoperatorL,theoptimalinitialapproximationu0andtheoptimalauxiliaryfunctionHcanalsobedetermined,see[2].Obviously,themorequicklyRes(h)in(5.20)or(5.21)decreasestozero,thefasterthecorrespondinghomotopyseriessolution(8.27)convergesandtheaccuracyofthehomotopyapproximationsincreases.So,atthetruncatedorderMofapproximation,thecorrespondingso-calledoptimalvalueoftheconvergencecontrolparameterhisgivenbytheminimumofRes(h),correspondinggenerallytoanonlinearalgebraicequationtobesolvedfromdRes(h)=0.(5.22)dhFromthisapproach,theintervalofconvergenceforhwillbemoreevidentthantheconstanth-curveanalysis,besidesityieldsanoptimalvalue.Asaconsequence,theconvergencecontrolparameterobtainedvia(5.20–5.22)canbesafelysuppliedintotheTheorem5.1tocomputetheratioβintheconvergenceanalysisoftheHAM. 202October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5194M.TurkyilmazogluHowever,sincetheexactintegrationin(5.20)or(5.21)ishardlypossibletoperformespeciallyforstrongnonlinearproblems,oritistimeconsumingparticularlyforlargeM,asmentionedonpage105in[30],adiscreteformofaveragedvalueof(5.20)canbesubstitutedviathenumericalintegrationaccountingforthesimplerectanglerule,asagainfirstsuggestedbyLiao[30].Hence,thediscretesquaredresidualerrorisexpressedby("#)21XNXMRes(h)≈Nuk(tj),(5.23)N+1j=0k=0seealsoequation(3.29)onpage105anddefinition(3.44)onpage119in[30],whereNisthenumberofdiscretepointswithtj=j∆tandlengthofΓ∆t=.NAlternatively,providedthattheintegrandispositive,theresidualgivenby(5.21)canbesimilarlydiscretizedintheform"#1XNXMRes(h)≈Nuk(tj).(5.24)N+1j=0k=0Theaboveresidualsmayalsobeadjustedtothesystemofdifferentialequa-tionsasdemonstratedlaterintheexamples.Afewshortcomingsoftheaboveresidualapproachshouldbementionedhere.Analyticalintegrationsmaynotalwaysbepossibleiftranscendentalfunctionsareinvolved.Evendiscretizationmaynothelp,intermsofCPUtime,duetotherequirementofevaluationofstronglynonlinearoperatorin(5.23)and(5.24).Moreover,ifthephysicalproblemisdefinedoverasemiinfinitedomain,eitherthecomputationaldomainiscuttoafiniteintervaloraplentyofgridpointsmayberequiredtoresolvetheinfiniteregion,whichbecomesatediouscomputationaltaskfromtheaboveresiduals.5.4.3.OptimumvaluefromtheratioTheaforementionedapproachofsquaredresidualerroristheoreticallyrig-orous,butitsminimizationtaskinpracticemaynotalwaysbeefficientincomputationalterms.Takingthisintoaccount,betterandmoreeffectivewaysshouldalwaysbetargeted.Toservetothispurpose,inthepresentsection,bymeansoftheratiogiveninequation(5.15),anovelandeasywayofidentifyingtheoptimumvalueofconvergencecontrolparameterhwillbeintroduced.Asunderlinedbytheabovetheoremsandbyitscorollaries, 203October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod195foraprescribedh,iftheratioislessthanunity,thentheconvergenceofHAMisguaranteed.Inadditiontothis,togetafasterrateofconvergencetowardstheexactsolutionof(8.1),itissensibletorequirethattheratioβisassmallasdesired,possiblygettingclosetozero,sothattheconvergencehappensatthefastestrate.Keepingthisinmind,itisrationaltosearchforavalueofhthatgivesrisetoassmallratioaspossiblefromequation(5.15).Thiscanbesimultaneouslyachievedviapracticallyplottingβversushin(5.15),aswellastheoreticallycheckingwhetherdβ=0,dhifexists,asaresult,itproducestheoptimumvaluefortheconvergenceparameterh.Infact,thefollowingexamplesclearlydemonstratethatthisprocedureresultsinveryclosevaluesofhtothoseobtainedfromtheresid-ualsvia(5.22).ConsideringforinstanceLp(p=1orp=2),sinceintegralsintheratioRpu(r)drΓRk+1β=p,(5.25)u(r)drΓkdemandlesslaboriousworkascomparedto(5.20)and(5.21)(noticethatexactintegrationisalwayspossiblefrom(5.25)unliketheresidualsfrom(5.20)and(5.21)),asalsorevealedinbelowexamples,equation(5.25)oritsdiscretecounterpart(5.26)givenasPNpj=0[uk+1(tj)]β≈P,(5.26)N[u(t)]pj=0kjbringsamoreconvenientwayofevaluatingtheconvergencecontrolparam-eterh.Thisisapparent,sinceβin(5.25)involvesonlythehomotopytermsofthehomotopyseries(8.27)thatareusuallyintegrable,whereasequations(5.20)and(5.21)incorporatethenonlinearoperatorfromequation(8.1).Similartothesquaredresidualidea,atthegivenorderofapproximation,thecurvesofratioβversushindicatenotonlytheeffectiveregionfortheconvergencecontrolparameterh,butalsotheoptimalvalueofhthatcorrespondstotheminimumofβ.Anemphasizeisdeservedsuchthatthepresentapproachseemsmorepromisingintermsofcomputationalef-ficiencyandhencecanbeusedinplaceofthetwoexistingaforementionedmethods.Weshouldfurtherremindthattheabovedefinitionsforratiosmayalsobeadjustedtothesystemofdifferentialequationsasrevealedlater.Finally,anemphasisshouldbemadethatfortheconvergenceoftheHAMitisnotnecessaryfortheratiotoreachaminimumasdescribed 204October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5196M.Turkyilmazogluabove.Inthecircumstancesthatnominimumisattainablefortheratio,theneitheravaluethatmakestheratiolessthanunityischosen,oritisenoughtosupplyintotheratiotheoptimumhvalueworkedoutfromtheminimumresidualerrortocheckouttheconvergenceoftheHAM.5.5.IllustrativeexamplesToillustratethevalidityandaccuracyofthenewapproachoutlinedinSec-tion5.4throughtheutilities(5.20–5.26),wetakeintoaccountthefollowingexamplestakenfromthehomotopyanalysisstudiesintheliterature,whicharealgebraic,nonlineardifferential-differenceequation,linearorstronglynonlinearordinaryandpartialdifferentialequationsandsystems.AllthenormsarebasedonL1,unlessotherwiseismentionedinthebelowex-amples.WeshouldnotethatthecomputationalefficiencyoftheoptimalHAMdependsstronglyonthemethodofsearchingfortheminimumofeitherresidualorratio.AsmentionedinLiao’sbook[2],thecommandNMinimize(orsometimesFindRoot)withreasonableWorkingPreci-sionisusedinthecomputeralgebrasystemMathematicatogettheresultsprovidedhere.5.5.1.AnalgebraicequationLetusfirstconsiderthequadraticalgebraicequationf(u)=u2−2=0,(5.27)√whosenumericalsolutionuptotheninesignificantdigitissimplyu=2=1.414213562.Inaccordancewiththehomotopyconceptgivenin§5.2,theinitialguessandtheauxiliaryoperatorarechoseninthefromu0=1,L(u)=f(u)−f(u0)(seeforinstancepages19and20in[2]),sothatthehomotopyseriessolu-tionviathehomotopyapproach(8.2)canbestraightforwardlyconstructed.Weshouldremindthatanyrealnumbercouldbeselectedastheinitialap-proximation,thatwillcertainlyhaveimpactontheregionofconvergencyoftheHAM.Infact,forthealgebraicproblemslike(5.27),theresidualandabsoluteerrorscanbeimmediatelycomputedfromRes(h)=u2−2,(5.28)√err=u−2,(5.29) 205October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod197forwhich,udenotesthehomotopyseriessolution(8.27)dependingupononlytheconvergencecontrolparameterh.Hence,makinguseof(5.28),theminimumresidualoccursath=−0.69thatcorrespondstotheoptimalvalueofconvergencecontrolparameteratthe22th-orderapproximation.1.4201.415M=2216u1.4101061.4051.400-1.0-0.8-0.6-0.4-0.20.0hFig.5.1.Constanth-curvesforequation(5.27).AttheordersM=6,10,16and22ofapproximation,thecon-stanth-curvesaredisplayedinFigure5.1,whichshowsthatastheorderofapproximationincreasestheconvergenceisguaranteedintheintervalh∈[−1,0),whichisalsowellestimatedanalyticallybysolvingthein-equality|u22/u21|<1from(5.18)thatexactlyyields−1.0366 206October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5198M.TurkyilmazogluTable5.1.Valuesoftherootof(5.27)andabsoluteerrorsatsomeselectedordersofiterationMgeneratedwithh=−0.69.M5101520ua1.4138170501.4142125411.4142135611.414213562erra3.9651×10−41.0218×10−61.7317×10−97.9015×10−12ub1.4136244741.4142118771.4142135581.414213562errb5.8909×10−41.6854×10−64.8251×10−91.3814×10−11aSolutionsfromhomotopy(8.27)bSolutionsfromNewtoniteration(5.30)Table5.2.Optimumvaluesforhfromboththeratio(5.18)andresidual(5.28).M10203050100200300350h−0.7100−0.6919−0.6849−0.6786−0.6734−0.6704−0.6693−0.6686letusfindsomeoptimumvaluesfortheconvergencecontrolparameterh,bothfrom(5.28)andfrom(5.18)usingashighoddorderhomotopytermsaspossible.Withinthisrespect,Table5.3showstheoptimumsandcorrespondingvaluesofβatsufficientlyhighlevelofiterations.Itisindicatedthatboththeresidualusing(5.28)andtheratiowith(5.18)yieldnearlythesamevaluesfortheoptimumofh,asalsojustifiedfromFigures5.2(a–b),evaluatedtakingM=301.Thus,thereisnodoubtthatintheinfinitelimitofM,bothoptimumswillcollideontoasinglevalue.Thisprovesthevalidityoftheratioapproachasproposedin§5.4.WeshouldremarkherethatthelimitingvalueofβtowardszeroforevennumberofapproximationsMandthelimitingvalueofβtowardsthevalueaspresentedinTable5.3foroddnumberofapproximationsMdonotcontradictwiththeTheorem5.1,sinceintotalthesequenceofpartialsumforthehomotopyseries(5.6)willeventuallyconvergetoafinitelimit.Table5.3.Optimumvaluesforhandtheresultingratioβ.M2151101151201251301351ha−0.6936−0.6798−0.6740−0.6719−0.6702−0.6700−0.6695−0.6686hb−0.6644−0.6658−0.6662−0.6664−0.6664−0.6665−0.6665−0.6665β0.332810.333120.333230.333260.333280.333290.333300.33330aEquation(5.28)bEquation(5.18) 207October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod199-1451.´100.3608.´10-1460.3550.350-146È6.´10Β0.345ÈRes-1464.´100.340-1462.´100.33500.330-0.671-0.670-0.669-0.668-0.667-0.668-0.666-0.664-0.662-0.660hh(a)|Res|versush(b)βversushFig.5.2.Residualerrorandratioforequation(5.27)attheorderofapproximationM=301.Eventhoughtheconvergenceofthehomotopyseriessolutionisappar-entusingtheoptimalvaluesofhshowninTables5.2,5.3andFigures5.2(a–b),theconvergenceofthehomotopyserieswiththeoptimumconver-gencecontrolparameterh=−0.666andhaving300thtermscanbefurtherassessedbycheckingtheratio(5.18)fromFigure5.3andevidentlyobserv-ingthedevelopmentoftheratioβfromTable5.4.Itappearsthat,afteraninitialoscillatorycharacter,theratioβisrapidlysettlingdowntoalimitingvalue0.33353,remaininglessthanunity,thatfurtherensurestheconvergenceoftheproducedhomotopyseriesforthealgebraicequation(8.27).Itfurtherdeservestomentionthatwhilefindingtheoptimums,theCPUtimetoevaluatetheminimumofβfrom(5.18)ismuchlessthantheCPUtimetoevaluatetheminimumofResfrom(5.28).Forexample,itneeds9.938seconds,11.466seconds,21.7secondsand39.234secondstocalculatetheminimumsof(5.28)forM=151,201,251and351,respec-tively.Ontheotherhand,ittakesonly1.279seconds,2.075seconds,3.697secondsand5.367secondsfortheminimumsof(5.18)atthesameordersofapproximations.Table5.4.Theratioβevaluatedwith300homotopytermsandh=−0.666.M100150200250280290296300β0.333250.333340.333420.333480.333520.333530.333530.33353 208October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5200M.Turkyilmazoglu1.00.80.6Β0.40.20.0050100150200250300kFig.5.3.AlistplotoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.27).5.5.2.AnonlinearVolterradifferential-differenceequationThedifferential-differenceequationsplayacrucialroleinmodelingcom-plicatedphysicalphenomenasuchasparticlevibrationsinlattices,cur-rentflowinelectricalnetworksandpulsesinbiologicalchains.OnesuchequationisthefamousnonlinearVolterradifferential-differenceinitialvalueproblemu0(t)=u(t)(u(t)−u(t)),u(0)=n,(5.31)nnn−1n+1nwhichpossessestheexactsolutionu(t)=n,asmentionedbyWangn1+2tetal.[35].Withoutlossofgenerality,theabovesystemisconfinedtothetimedomain[0,1].0-50L0’H-100100uM=30,20,10-150-200-2.0-1.5-1.0-0.50.0hFig.5.4.Constanth-curvesforequation(5.31).TheapproximatesolutionoftheVolterraequation(5.31)isobtainedbymeansoftheHAMusingthefollowingauxiliarylinearoperator,theinitial 209October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod201guessandtheauxiliaryfunctiondL=,un,0(t)=n−t,H(t)=1,dtrespectively.Usingthem,akindofhomotopyisconstructedasdescribedin§5.2,butmoredetailsoftheHAMregardingsuchdifferenceequationscanbefoundin[35].Bymeansoftheexactsolutionoftheinequality(5.18)foru0(0),itisderivedthattheintervalofconvergencecontrolparameter100hisexactly[−2,0]forallnatallordersofapproximation.Theconstanth-curveplotsshowninFigure5.4alsoverifythisoutcome.0.000300.260.000250.240.000200.22ResΒ0.200.000150.180.000100.160.000050.14-0.50-0.49-0.48-0.47-0.46-0.45-0.50-0.49-0.48-0.47-0.46-0.45hh√(a)Resversush(b)βversushFig.5.5.Residualerrorandratioforequation(5.31)atthe20th-orderofHAMapprox-imation.Carryingoutexactintegrationovert∈[0,1]forthefixedn=100,theresidual(5.20)andtheratio(5.25)(withp=2)aredepictedinFig-ures5.5(a–b)atthe20th-orderofHAMapproximation.Itisfascinatingtoobservethattheregionofconvergenceisalmostthesamefromthetwomethods,andmoreimportantlytheoptimumvaluefortheconvergencecon-trolparameterhapproachesthesamelimitingoptimalvalueh=−0.479,asalsoverifiedinTable5.5.Italsoindicateshowfasttheminimizationisperformedfromtheratioascomparedtothesquaredresidual,especiallyforincreasingorderofapproximations.Thus,thepresentratioapproachindeedconstitutesastrongalternativetotheclassicalsquaredresidualmethod.Although,Figure5.4itselfsuggeststheconvergenceofthehomo-√topymethodforh=−1,theresidualResandratioβreceivethevalues1.0691×1013and3.8139,respectively.Thisobviouslymeansthattheso-calledhomotopyperturbationmethodwillabsolutelyfailfortheVolterradifferential-differenceequation(5.31).EventhoughtheaccuracyoftheHAMusedforthepresentproblemisclearlyundoubtedfromTable5.5andFigures5.4and5.5,listplotsforthe 210October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5202M.TurkyilmazogluratioβarefurthergiveninFigures5.6(a–c)atsomedifferentconvergencecontrolparametersh.Itisclearthattheinitialbehaviorendsupwiththesettlementasthenumberofhomotopytermsisincreased.Hence,theaccuracyissupportedbytheconvergenceoftheHAMwhichcanbesafelyusedtogainhigherorderapproximationswithbetterperformance.Table5.5.Theoptimumvaluesofhandβevaluatedfromtheminimumofresidualandratioforequation(5.31)(withCPUtimesinparenthesis).M26101420ha−0.4131(2.77)−0.4522(6.56)−0.4654(19.41)−0.4724(26.91)−0.4785(100)hb−0.4221(0.70)−0.4567(2.14)−0.4674(4.20)−0.4733(6.34)−0.4791(8.97)β0.061190.120780.135990.139040.13797aFromequation(5.20)bFromequation(5.25)1.01.00.80.80.60.6ΒΒ0.40.40.20.20.00.00102030405001020304050kk(a)h=−0.50(b)h=−0.481.00.80.6Β0.40.20.001020304050k(c)h=−0.46Fig.5.6.ListplotsoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.31). 211October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod2035.5.3.Anonlinearhigh-orderFredholmintegro-differentialequationTheKirchhofftypeequationmathematicallymodelingthedeflectionofanextensiblebeamwithhingedendsisgivenbythestronglynonlinearnon-dimensionalFredholmintegro-differentialequationZ1(4)000002u(t)−u(t)−u(t)(u(x))dx=1,0 212October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5204M.Turkyilmazoglutheconvergenceintervalis[α,0],whereαiscloseto−2,butitisuncertainhowclosetoorfarfrom−2.Tocompletelysortoutthismatter,itisnec-essarytosolvetheinequality(5.17)foru0(0)analytically,thatyieldstheconvergenceintervalsas[−1.8293,0],[−1.8214,0],[−1.8180,0],respectivelyfortheapproximationordersasshowninFigure5.7.-292.´100.010-291.5´100.008Res-29Β0.0061.´100.004-305.´100.002-0.960-0.958-0.956-0.954-0.952-0.950-1.00-0.98-0.96-0.94-0.92-0.90hh√(a)Resversush(b)βversushFig.5.8.Residualerrorandratioforequation(5.32)atthe21st-orderofHAMapprox-imation.Exactintegrationofthesquaredresidual(5.20)isonlypracticaluptoM=12with68.17secondsspent,afterwhichtheCPUtimeincreasesenormously.Therefore,wepreferminimizationthroughthediscreteversion(5.23)fortheresidualusing500equallyspacedpoints.However,nodiscreteintegrationisneededfortheratio,thustheexactintegrationfrom(5.25)(withp=2)isimplemented.Atthe21st-orderofHAMapproximation,thebestintervalofconvergenceandalsotheoptimumvaluesarerevealedinFigures5.8(a–b).Itisseenthattheoptimalvaluesaregettingclosertoh=−0.95fromboththeclassicalsquaredresidualandtheproposedratiomethods,whicharealsosummarizedinTable5.6togetherwiththecon-sumedCPUtimes.Againforthisnonlinearproblemthenewly-introducedratioapproachevenwiththeexactintegrationissuperiortotheclassicalsquaredresidualmethod.TheconvergenceoftheHAMinCase1canalsobeconceivedfromthelistplotfortheratioβexhibitedinFigure5.9computedbymeansoftheconvergencecontrolparameterh=−0.95.Thefastlimitingvalueofβ,towards0.00213,isthebestindicatorfortherapidconvergenceoftheHAM,too.AsforCase2,thecomputationalcosttoevaluatethehomotopyterms 213October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod205Table5.6.Theoptimumvaluesofhandβevaluatedfromtheminimumofresidualandratioforequation(5.32)(withCPUtimesinparenthesis).M15111521ha−0.9938(1.75)−0.9636(11.98)−0.9566(27.20)−0.9551(39.42)−0.9541(72.64)hb−0.9091(0.85)−0.9394(5.19)−0.9442(9.64)−0.9452(16.61)−0.9464(35.25)β0.000120.001400.001670.001740.00183aFromequation(5.23)bFromequation(5.25)0.0100.0080.006Β0.0040.0020.00005101520kFig.5.9.AlistplotoftheratioβtorevealtheconvergenceoftheHAMsolutionforequation(5.32).istooexpensive.Toillustrate,althoughapproximately20minutesaresufficientforevaluationofthewholetermsuptoM=21inCase1,only4homotopyseriestermsarecalculatedoverhoursinCase2.Nevertheless,itisveryaccurateasalsoconcludedin[36].-76.´100.0001-75.´100.00008-74.´100.00006-7Res3.´10Β-70.000042.´101.´10-70.0000200.0000-1.010-1.005-1.000-0.995-0.990-1.010-1.005-1.000-0.995-0.990hh√(a)Resversush(b)βversushFig.5.10.Residualerrorandratioforequation(5.32)atthe1st-orderofHAMapprox-imation.Onlytwohomotopyterms,i.e.thefirst-orderhomotopyapproxima- 214October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5206M.Turkyilmazoglu√tion,giveanexactresidualRes=1.34707×10−10andanexactoptimalconvergencecontrolparameterh=−0.999808,seeFigure5.10(a),whicharethesamevaluesasthosefoundin[36].Theminimizationoftheresid-ualtakes47.69seconds.Ontheotherhand,theminimizationoftheratiorequiresjust11.27secondstogainthesameh,butwithapracticallyvan-ishingβ,seeFigure5.10(b).This,ofcourse,impliesthattheconvergenceoftheHAMviatheCase2ismuchacceleratedascomparedtotheCase1,whichisalsoverifiedfromthefirstfourconsecutiveratiosofβtabulatedinTable5.7.Table5.7.Firstfourratioswhenh=−1.k0123β4.09119×10−93.68135×10−86.54474×10−88.59284×10−8Finally,theconstanth-curvecorrespondingtotheCase2isfurthershowninFigure5.11.Theintervalofconvergenceasdepictedinthisfigureisresolvedexactlybymeansoftheratio(5.17),whichleadstotheexactinterval[−1.9996,0].0.0378830.0378820.037881LH0u’0.0378800.0378790.037878-2.0-1.5-1.0-0.50.0hFig.5.11.Constanth-curveforequation(5.32).5.5.4.Someordinarydifferentialequations5.5.4.1.AirydifferentialequationConsidernowthesecond-orderlinearAirydifferentialequationrecentlytreatedwiththeHAMin[28]u00−tu=0,u(0)=Ai,u(∞)=0,(5.34) 215October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod207whereAiisthevalueofAiryfunctionatt=0.ForfurtherinformationabouttheAiryequation(5.34)anditsusageinscience,pleasereferto[28]andreferencestherein.Also,weareinterestedinthedecayingsolution,althoughtheblowingAirysolutionwasalsofoundusingHAMin[28].-0.20-0.22-0.24LH0u’-0.264121-0.28M=11-0.30-2.0-1.5-1.0-0.50.0hFig.5.12.Constanth-curvesforequation(5.34).Thecorrespondingauxiliarylinearoperator,theinitialguessandtheauxiliaryfunctionintheframeoftheHAMarechosenasfollows:d2L=−c2,u(t)=Aie−ct,H(t)=1,dt20wherec=3wasfoundtobeoptimumbyminimizingtheresidualwithinafeworderofapproximations.TogetherwiththeseemployingtheHAMin§5.2,thevalidintervalfortheconvergencecontrolparameterhusingu0(0)isshownapproximatelyas[−2,0]inFigure5.12atdifferentordersoftheHAMapproximations.Thisintervalisbetterpredictedusingtheinequality(5.17)correspondingtotheexactintervals[−2.048,0],[−2.025,0]and[−2.0124,0],respectively,fromthe11th-order,21st-orderand41st-orderhomotopyseriesapproximations.BasedonthenormL2,theintervalofintegrationisdividedinto100equaldiscretepointsintheinterval[0,10].Afterwards,thecurvesoftheresidual(5.23)andtheratio(5.26)versushareplottedatdifferentorderofapproximationsM=6,8and10,respectively,toperceivetheregionofconvergenceandalsotheoptimumvaluesofhinFigure5.13.Itisfoundthattheminimumsofdiscretesquaredresidualsdecreaseintheregion−1.65≤h≤−1.3shiftingtowardsh=−1.6astheorderofapproximationincreases,whichindicatesthatthehomotopyseriesmostquicklyconvergesforsuchvalueofh.Thestrongindicationisthattheresidualattainsitsminimumath=−1.6.Therefore,thecurvesofthediscreteresidualsversus 216October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5208M.Turkyilmazoglu0.0300.70.0250.6M=6,8,100.50.0200.4Res0.015Β0.30.010M=60.20.00580.1100.0000.0-1.65-1.60-1.55-1.50-1.45-1.40-1.35-1.30-1.65-1.60-1.55-1.50-1.45-1.40-1.35-1.30hh√(a)Resversush(b)βversushFig.5.13.Residualerrorandratioforequation(5.34)atdifferentorderMoftheHAMapproximation.hprovideboththeeffectiveregionoftheconvergencecontrolparameteraswellastheoptimalvalueofhthatgivestheoptimalhomotopyseriesthatconvergesfastest,asalsoemphasizedbyLiao[2].Forthosevaluesofhlead-ingtooptimumvaluesfromtheresiduals,theratiosremainalmostconstantsignifyingtotheconvergenceforthehintervalshown.Inadditiontothis,-90.71.´100.6-108.´100.5-106.´100.4ResΒ4.´10-100.30.2-102.´100.100.0-1.64-1.62-1.60-1.58-1.56-1.54-1.64-1.62-1.60-1.58-1.56-1.54hh√(a)Resversush(b)βversushFig.5.14.Residualerrorandratioforequation(5.34)atthe41st-orderoftheHAMapproximation.atthe41st-orderofapproximation,assupportedbyFigures5.14(a–b),thevaluesoftheconvergencecontrolparametercalculatedfromequations(5.23)and(5.26)almostcoincidentallyresidewithintheintervalshown,sothattheoptimalhcouldbechosenash=−1.6,asalsoindicatedbyFigure5.12.Hence,withthisvalueinmind,thehomotopy(8.2)generatesahomo-topyseries(8.27)whoseu0(0),absoluteerrorsandratioβaretabulatedinTable5.8atseveralorderofapproximations.AsobservedfromtheTa- 217October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod209Table5.8.Valuesofu0(0),absoluteerrorerrandratioβcalculatedfromthehomotopyseriescorrespondingtoequation(5.34).M110203040u0(0)−0.228795857−0.260777885−0.258833058−0.258819509−0.258819405erra1.309×10−15.084×10−34.014×10−53.279×10−72.715×10−9β0.196310.384810.384660.384670.38467aEquation(5.19)ble5.8,theconvergenceofthehomotopyseriestotheexactsolutiontakesplaceatareasonablyfastrate.Noticethattheexactvalueofu0(0)is−0.2588194038.Justforthepurposeofcomparison,Table5.9showstheminimumoftheresidualsobtainedfromtheexactintegration(5.20)andthediscreteone(5.23),thecorrespondingoptimalvaluesofhandalsotheusedCPUtimefortheseevaluations.AccordingtotheTable5.9,bothexactanddiscreteresidualsresultinverycloseresults,butmuchlessCPUtimeisneededbymeansofthediscreteintegration,especiallyforhigh-orderapproximations.Therefore,thediscretesquaredresidual(5.23)canbereliablyusedtoobtaintheoptimumvaluesandtheintervalofconvergenceforh.Table5.9.MinimumoftheexactanddiscreteresidualerrorsandthecorrespondingoptimalvaluesofhwiththeusedCPUtime(inseconds)for(5.34).√a√bMReshCPUtimeReshCPUtime20.18604−1.32122.730.06209−1.25492.3250.03915−1.48354.490.01344−1.43445.32100.00341−1.555810.620.00121−1.523011.55130.00079−1.572230.340.00028−1.544115.99150.00030−1.579244.070.00011−1.553719.18aEquation(5.20)aEquation(5.23)Theconvergenceofthehomotopyseriesincaseofh=−1.6canbefurtherconfirmedbycheckingtheratioβ,asshowninFigure5.15andTable5.8.Itismostlikelythattheratiotendsto0.38467,remaininglessthanunity.ThisensurestheconvergenceoftheHAMapproximationforthecurrentphysicalproblem.Figure5.16clearlydemonstrateshowtheconvergencetakesplacetowardstheexactsolutionofAiryfunctionby 218October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5210M.Turkyilmazoglumeansofh=−1.6(seealsoFigure1in[28],whichwasobtainedusingh=−1.1).1.00.80.6Β0.40.20.0010203040kFig.5.15.AlistplotoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.34).0.350.300.250.20u0.150.100.050.0002468tFig.5.16.SolutionofAiryequation(5.34).Solidline:theexactsolution;Dashed-line:the20th-orderHAMapproximation;Dashed-dottedline:the5th-orderHAMapproxi-mation;Dottedline:theinitialapproximation.Finally,Figure5.17demonstratesthatthedivergenceoftheHAMap-proximationbecomesinevitableifthevalueofh=−2.1isassignedwronglyoutsidetheintervalofconvergence,asworkedoutforthisexample.Whensuchachoiceismade,inconsistentwiththeTheorem5.1,alimitingvalueofratiogreaterthanonecausesthedivergenceoftheseries.Thisonceagainindicatestheimportanceoftheso-calledconvergencecontrolparameter,whichdifferstheHAMfromallotheranalyticapproximationmethods. 219October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod2111.00.80.6Β0.40.20.0010203040kFig.5.17.AlistplotoftheratioβtorevealthedivergenceoftheHAMapproximationsincaseofh=−2.1forequation(5.34).5.5.4.2.AfirstordernonlineardifferentialequationLetusconsiderthefirst-ordernonlineardifferentialequation02u+u=1,u(0)=0,(5.35)whichwasusedbyLiao[3]todescribethebasicideasoftheHAM.Equation(5.35)isknowntogovernthefree-fallingbodyproblems.Itsexactsolutionue(t)=tanhtwillbeusedforcomparisonpurposeswiththesolutionsobtainedbymeansoftheHAM.Weactuallytakeintoaccountthetwodifferentcases.Inthefirstcase,equation(5.35)issolvedoverthefinitedomaint∈[0,1].Inthesecondcase,weconsiderthephysicalproblem(5.35)overthesemi-infiniteregiont∈[0,∞).Inbothsituations,weemploythefollowinginitialguess,auxiliarylinearoperatorandauxiliaryfunctionauxiliaryparameters−2tdu0(t)=1−e,L=+2,H(t)=1,dtrespectively.Weinitiallyusethequantityu0(0)toevaluatetheconstanth-curvesattheordersofapproximationsM=11,21and41,respectively.AccordingtoFigure5.18,theregionofconvergenceisbestsuitedtoh∈[−2,0].Thisisindeedthecasefromtherelation(5.17),whichfortunatelyleadstotheexactformulaβ=|h+1|.Table5.10presentsevolutionoftheratio(5.15),usingtheexactformula(5.25)withL1normandphysicaldomainas[0,∞)foravarietyofthepreassignedvaluesoftheconvergencecontrolparametershfortheproblem(5.35).Additionally,thelastcolumnofTable5.10givestheerrordefinedby(5.19),takingintoaccounttheexactsolutionue(t)=tanhtfortheproblem(5.35).DatadisplayedinTable5.10clearlyexplainswhytheHAMgeneratescompletelyconvergent 220October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5212M.Turkyilmazoglu1.101.05M=1121LH01.0041u’0.950.90-2.0-1.5-1.0-0.50.0hFig.5.18.Constanth-curvesforequation(5.35).Table5.10.Theevolutionoftheratioβ(5.15)forequation(5.35).hM=1M=10M=20M=30M=40erra−1.50.000010.426900.454570.468750.476192.8411×10−17−1.40.0666714.7950.447360.381420.381432.2129×10−21−1.30.133330.324980.334360.338990.341591.2674×10−21−1.20.200000.365610.381520.387360.390403.7223×10−19−1.00.333330.458330.477270.484380.488105.1726×10−15−0.50.666670.701890.719690.728240.733082.3609×10−7aEquation(5.19)homotopyseriessolutiontotheproblem(5.35)forthechosenparametersh.AccordingtotheTable5.10,thehomotopyseriesgivenbydifferentvaluesofhhasdifferentrateofconvergence.Itisfoundthatthehomotopyseriesconvergesfastestforthevaluesofhbetween−1.4and−1.3,andthusanoptimumconvergencecontrolparameterexistswithinthisinterval.Notethat,inpractice,itisunnecessarytolocatetheexactvalueoftheoptimalconvergencecontrolparameter,insteaditisenoughtouseavalueclosetoit.Theoptimalvaluesofconvergencecontrolparameterharefurthercal-culatedfromtheexactformulae,respectivelyforresidual(5.21)andforratio(5.25)(withp=1),takingM=22ash=−1.3567,seeFigure5.19(a),andh=−1.2678,seeFigure5.19(b)forthefinitedomaincase.More-over,takingM=30,theinfinitedomaincaseresultsintheoptimumswhichareevaluatedash=−1.3947,seeFigure5.20(a)andfromthera-tioash=−1.3253,seeFigure5.20(b).Whenthenumberofhomotopytermsincreases,itistimeconsumingtoevaluatetheintegralsforbothfiniteandinfinitedomaincasesfrom(5.21),butastraightforwardevalua- 221October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod2131.´10-140.30-150.298.´10-150.286.´10ΒRes-154.´100.27-152.´100.2600.25-1.40-1.38-1.36-1.34-1.32-1.30-1.30-1.28-1.26-1.24-1.22-1.20hh(a)Resversush(b)βversushFig.5.19.Residualerrorandratioforequation(5.35)withfinitedomainatthe22nd-orderofHAMapproximation.1.´10-160.40-170.388.´10-170.366.´10ΒRes-174.´100.34-172.´100.3200.30-1.44-1.42-1.40-1.38-1.36-1.34-1.34-1.32-1.30-1.28-1.26-1.24-1.22-1.20hh(a)Resversush(b)βversushFig.5.20.Residualerrorandratioforequation(5.35)withinfinitedomainatthe30th-orderofHAMapproximation.tionfrom(5.25)stillyieldedtheoptimumsh=−1.2714forthefiniteandh=−1.3273fortheinfinitedomaincases,evenforM=40,whichstronglyindicatesthereasonofthefasterconvergeofthehomotopyseriesnearthisvalue,seeTable5.10.Table5.11furtherrevealstheoptimumvaluesofhevaluatedatdifferentordersofapproximationM.Notethat,duetoabovereason,numericalintegrationispreferredwiththeresidual(5.24),whereastheratioβisexactlycomputedfrom(5.25).Duringthenumericalscheme,theinfinityistruncatedat5andtheregionisdividedinto400equally-spacedpoints.Itisanticipatedthatastheorderofapproximationisincreased,boththeresidualandratioconvergetothesameoptimumvalueforh.Tobetteradmiretheadvantageofratioβintermsofcomputationalefficiency,Table5.12demonstratestheCPUtimestoevaluatethemini-mumsofresidualsandratioscomputedexactlyfromequations(5.21)and 222October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5214M.TurkyilmazogluTable5.11.Theoptimumvalues(togetherwithCPUtimes)ofhfromtheresidualandfromtheratioforequation(5.35)atdifferentordersofapproximationM.M10203040ha−1.3598(57.45)−1.3373(150.01)−1.3323(411.70)−1.3305(1069.34)hb−1.3101(12.98)−1.3214(8.721)−1.3253(31.761)−1.3273(97.766)aEquation(5.21)bEquation(5.25)(5.25),respectively,fortheinfinitedomaincase.SpecialattentionshouldbegiventoCPUtimes,whichclearlyshowsthattheapproachintroducedinthischapterisessentialintermsofcomputationaltime,ifexactinte-grationisdemanded.EventheCPUtimefromthediscreteresidualtakeslongerthantheexactevaluationoftheintegralsfromtheratio,asclearlyshowninTable5.11.So,itismoreeconomicaltofindtheoptimumhfromtheratio(5.25)thanfromtheresidual(5.21).Table5.12.CPUtimestoobtainoptimumvaluesofhfromtheresidualandfromtheratioforequation(5.35)atdifferentordersofapproximationM.M26102030CPUtimea0.62410.204.867227.04724.56CPUtimeb0.6243.91512.988.72131.761aEquation(5.21)bEquation(5.25)Moreover,theuniformvalidityregionofthehomotopyseriessolution(8.27)canalsobeanalyticallyresolvedforthespecialvalueofh=−1.Inthisparticularcase,consideringtheabsolutevaluenorm,thehomotopytermsinthehomotopyseries(8.27)satisfytheratioun+1(t)1−2tu(t)=2(1−e).(5.36)nHence,concerningtheequation(5.35),Theorem1assurestheconvergenceofthecorrespondinghomotopyseries(8.27)forallvaluesoftvalidintheintervalt>−ln3,thatcoverstheentirephysicaldomain.Withthefixed2valueh=−1.27forthefiniteandh=−1.3fortheinfinitedomaincases,listplotsofβfortheconvergencehistoryaredemonstratedinFigures5.21(a–b).Itseemsthattheevenandoddhomotopyseriestermsapproachthesamelimitatdifferentrates,whichcausestheoscillationsintheratiosin 223October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod215thefigures.Thelimitingvaluesturnouttobe0.28and0.34,respectivelywhicharesufficienttoexplaintheconvergencyoftheHAMapproximationsappliedtothephysicalproblemdefinedin(5.35).1.01.00.80.80.60.6ΒΒ0.40.40.20.20.00.0010203040010203040kk(a)Forfinitedomain(b)ForinfinitedomainFig.5.21.ListplotsoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.35).5.5.4.3.AparameterizednonlineardifferentialequationLetusreconsidernowthefirst-ordernonlineardifferentialequationmen-tionedintheaboveexampleu0+u2=1,u(0)=0,(5.37)tobetreatedoverthefiniteinterval[0,α],whereα∈Rwithα=1,withoutlossofgenerality.Althoughsuchafiniteintervalproblemcanbesolvedasinthepreviousexample,wedifferitfromthatbyassumingthat,attheendpointα=1,thesolutiontakesthevalueλ,i.e.u(1)=λ,whereλisunknowninpriori,calledthepseudoparameter.Usingthetransfor-mationu(t)=λv(t),equation(5.37)istransformedintoaparameterizeddifferentialequationλv0+λ2v2=1,v(0)=0,v(1)=1.(5.38)Inthesystemdescribedby(5.38),bothvandλareexpandedintoho-motopyseries(8.27)viathehomotopybuiltin(8.2).Eachhomotopytermforthepseudoparameterλisthenevaluatedusingeitheroftheboundaryconditionsin(5.37).Itisremarkedthatonlythezeroth-ordertermλ0forλsatisfiesaquadraticalgebraicequation,otherwisetherestofthetermssat-isfylinearequations.Thus,thehomotopysolutionfor(5.38)isobtainedbyvirtueofthefollowinginitialguess,auxiliarylinearoperatorandauxiliary 224October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5216M.Turkyilmazoglufunctiondv0(t)=t,L=,H(t)=1.dt1.51.001.40.950.90M=10,20,301.3LH0Λ0.85v’1.2M=10,20,300.801.10.751.00.70-2.5-2.0-1.5-1.0-0.50.0-2.5-2.0-1.5-1.0-0.50.0hh(a)v0(0)versush(b)λversushFig.5.22.Constanth-curvesforequation(5.38).Theconstanthcurvesfortheunknownvariablev0(0)andfortheun-knownparameterλattheordersofapproximationsM=10,20and30,respectively,aredisplayedinFigures5.22(a–b).Notethattheexactvalueofλistanh(1)=0.761594,whichistrulyestimatedbytheHAMapprox-imations,asshowninFigure5.22(b).Theintervalsofhforconvergenceshownbythegraphsextendtothepointh=−2,suggestingthatthecon-vergenceintervalisclosetoh∈[−2,0].Ontheotherhand,theexactintervalcanonlybefiguredoutbysolvinganalyticallytheinequalitiesin(5.17)and(5.18),thatyieldh∈[−2.024,0]at30th-orderofapproximation.1.´10-90.20-108.´100.15-106.´10ResΒ0.10-104.´100.05-102.´1000.00-1.34-1.32-1.30-1.28-1.26-1.24-1.22-1.20-1.34-1.32-1.30-1.28-1.26-1.24-1.22-1.20hh√(a)Resversush(b)βversushFig.5.23.Residualerrorandratioforequation(5.38)atthe41st-orderofHAMap-proximation.Employingtheexactresidual(5.20)andratio(5.25)(withp=2),theminimizationofthemresultsintheoptimumsdemonstratedinFigures5.23 225October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod217(a–b)atthe41st-orderofapproximation.Moreover,severaloptimumval-uesaretabulatedinTable5.13atvariousordersofapproximations.Itisapparentagainthatboththeformerly-usedsquaredresidualapproachintheliteratureandthepresentratioapproachgivenearlythesameopti-mumvaluesofhevenatsmallordersofapproximation,unliketheHAMtechniqueinthepreviousexample.Theoptimumsseemtoconvergetoh=−1.27inthelargetruncationlimit.Table5.13.Theoptimumvaluesofhandtheratioβevaluatedfromtheminimumofresidualandratioforequation(5.38).M26101214ha−1.2955−1.2818−1.2753−1.2733−1.2720hb−1.1607−1.2817−1.2769−1.2764−1.2754β0.060960.041540.027050.030970.03912aEquation(5.20)bEquation(5.25)TheconvergenceoftheHAMisensuredbyTheorem5.1followingtheratiosrelatedtothevariables(5.15)orparameters(5.18)existingintheconsideredproblem.Forthispurpose,listplotsofβwhenh=−1.31aredemonstratedinFigures5.24(a–b)forbothvandλ.Itisclearthatbothratiostendtofinitelimits,lessthanunity.AlthoughbotharegoodinexplainingtheconvergencyoftheHAMapproximationsofthephysicalproblemdefinedby(5.37)and(5.38),itneedsmuchmoreCPUtimetogaintheratioaboutv.Forexample,2400seconds(usingexactintegration)areneededtocalculatetheratiosinFigure5.24(a),whereasonly0.031secondsforFigure5.24(b).Thus,consistentwiththestatementsmadeinRemark3,theproposedratioapproachtogetherwithintroducinganunknownpa-rameterintotheconsideredphysicalproblemworksveryefficientlyforthepresentproblem.Finally,itisfoundthatkeepingtheoriginalequation(5.37)overtheinterval[0,1]withthesameauxiliaryparametersdoesnotgiveconvergingratiosasinFigures5.24(a–b),butinstead,ityieldswildlyoscillatoryratiosforthewholeconvergencecontrolparametersh.Thisil-lustratesasubstantialadvantageofplugginganunknownparameterintotheequationsgoverningthesystem. 226October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5218M.Turkyilmazoglu1.01.00.80.80.60.6ΒΒ0.40.40.20.20.00.00510152025300510152025kk(a)βforv(b)βforλFig.5.24.ListplotsoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.38).5.5.4.4.UndampedDuffingequationThesecond-ordernonlinearundampedDuffingequationu00+u+u3=0,u(t=0)=A,u0(t=0)=0,(5.39)governstheoscillatorsystems[23],whereArepresentstheamplitudeoftheoscillationsanddenotesthestrengthofthenonlinearity.Usingasuitabletransformationτ=ωt,whereωisthefrequencyoftheoscillations,equation(5.39)becomesω2u00+u+u3=0,u(0)=A,u0(0)=0.(5.40)Notethatωistreatedasunknownhere,whichdependsontheembeddingparameterp,asmentionedbyLiao[2].Togaintheapproximationsofthesolutionof(5.40)intheframeoftheHAM,wechoosetheinitialguess,theauxiliarylinearoperatorandtheauxiliaryfunctioninthefollowingmannerd2u(τ)=Acosτ,L=ω2(+1),H(τ)=1,00dτ2whereω0andotherapproximationsωiregardingX∞ω=ωii=0aredeterminedbyavoidingtheappearanceofseculartermsduringthesolutionprocessofhomotopytermsin(8.27).Formoredetails,pleasereferto[24].FortheparticularchoicesoftheparametersA==1,theintervalofconvergenceatthe10th-order,20th-orderand30th-orderofapproximationiscomputedusingu00(0)andωfortheh-curve,asshowninFigures5.25 227October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod219-1.001.330-1.05M=10,20,301.325-1.10M=10,20,30LH0-1.15Ω1.320u’’-1.201.315-1.25-1.301.310-2.0-1.5-1.0-0.50.0-2.0-1.5-1.0-0.50.0hh(a)u00(0)versush(b)ωversushFig.5.25.Constanth-curvesforequation(5.40).(a–b),respectively.Itseemsthattheintervalofconvergenceis[−1.8,0],whichappearstonarrowdownfromtheleftboundastheorderofapproxi-mationincreases.Infact,ajustificationismadeusing(5.17)foru00(0)thatresultsinanexactinterval[−1.8041,0]andalsousing(5.18)forωthatgivestheinterval[−1.7903,0],wherebothareevaluatedatthe31st-orderofap-proximation.Itisfoundthattheoptimalvalueofhcalculatedfromdirectnumericaltreatmentofequations(5.20–5.22)(takingtheintegrationinter-valas[0,500])atthe31st-orderofapproximationisabouth=−1,whoseeffectsonthefrequencyandtheabsoluteresidualerroratdifferentordersofapproximationsaresummarizedinTable5.14.Itisclearthatthehighertheorderofapproximation,thebettertheapproximationoffrequency,andthefastertheabsoluteresidualerrordecays.Table5.14.TheHAMapproximationsofωandthecorrespondingResofequation(5.40)incaseofA==1whenh=−1.M2102030ω1.3178039461.3177760651.3177760651.317776065Resa4.7575×10−12.0460×10−83.9128×10−171.0825×10−26aEquation(5.21)Afurtherassessmentontheconvergenceofthehomotopyseriescanbeimplementedbycheckingtheratios(5.15)foruand(5.18)forωwhenh=−0.98,asshowninFigures5.26(a–b).Bothratiosofβobviouslytendtoafinitevalueofapproximately0.15,whichindicatesthattheho-motopyapproximationsconvergetothesolutionoftheDuffingoscillatorproblem.Infact,Figures5.26(a–b)suggestthatkeepingtrackoftheratioofωissufficient.Moreover,itisunnecessarytoevaluatetheintegralsfor 228October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5220M.Turkyilmazogluthesquaredresiduals.Thisexamplehenceapprovestheadvantageofthepresentapproachofusingratio,insteadofthesquaredresidualerror.1.01.00.80.80.60.6ΒΒ0.40.40.20.20.00.00510152025051015202530kk(a)βforω(b)βforuFig.5.26.ListplotsoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.40)bymeansofh=−0.98.Infact,makinguseofωatdifferentordersofapproximation,optimumvaluesoftheconvergencecontrolparametershcanbegainedeasilybyminimizingtheratioβinthehomotopyexpansionofωin(8.27).ThisisimplementedinTable5.15.Thisstronglyindicatesthattheratioapproachpointstoanoptimalvalueclosetoh=−1.Table5.15.Theoptimumvaluesofhevaluatedbyminimizingratio(5.18)forω.M61216202430h−1.00096−1.00056−1.00044−1.00036−1.00031−1.00025Apropervalueofhcanfurtherbedeterminedbyrestrictingthedomainofintegrationin(5.40)onlyto[0,2π](whichmightbesufficientforthede-siredphysicalsolution),assupposedlytakeninLiao’sresidualexpressions(2.70)and(2.72)onpage41in[2].Toillustratethesimilaroccurrenceofoptimumsfromboththeresidualandtheratioapproachforthevariableu,Figures5.27(a–b)aredrawnonlywiththe20th-orderofHAMapproxima-tion,with100equidistantpoints.Theoptimumsarewellpredictedagainbymeansofbothmethods.Makinguseofthediscreteversionsforboththeresidualandtheratioapproachinvolvingu,theoptimumvaluesofcon-vergencecontrolparameterh,theminimumratioβandthecorrespondingvaluesofCPUtimeareeventuallysketchedonTable5.16.Consistentwiththeabovefindings,theoptimalvalueslimitsto−1forlargeM,theorderof 229October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod221approximation.Furthermore,itisfoundonceagainthattheratioapproachevaluatestheoptimumsfaster.1.´10-180.030-190.0258.´100.020-196.´10ResΒ0.015-194.´100.010-192.´100.00500.000-1.04-1.02-1.00-0.98-0.96-1.04-1.02-1.00-0.98-0.96hh√(a)Resversush(b)βversushFig.5.27.Residualerrorandratioforequation(5.40)atthe20th-orderofHAMap-proaximation.Table5.16.Theoptimumvaluesofh,theratiosβandtheCPUtimes(seconds)fromtheresidualandratioapproachforequation(5.40)atdifferentordersofapproximationM.MhaCPUtimeshbβCPUtimes2−1.00214.025−1.00870.00021.3575−1.001015.12−1.00410.00187.0410−1.000845.04−1.00280.002923.3420−1.000478.20−1.00150.004654.4325−1.000379.33−1.00120.005361.8530−1.000382.90−1.00110.005774.76aEquation(5.23)bEquation(5.26)Finally,Figures5.28(a–d)showtheuniformconvergenceofthehomo-topyapproximationsemployedhere,withtheconvergencecontrolparame-terh=−1evenforverymoderatenumbersofhomotopyseriestruncation,comparedtothenumericalsolution.Itshouldbenoticedthatalthoughtheaboveanalysiswasimplementedbyrestrictingthedomaintosmallerintervals,theHAMisstillabletoextractuniformlyvalidsolutionsoverreasonablylargerintervalsdemonstratedinFigures5.28(a–d),suchast∈[0,30000]. 230October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5222M.Turkyilmazoglu1.01.00.50.5u0.0u0.0-0.5-0.5-1.0-1.0808590951009809859909951000tt(a)Fort∈[80,100](b)Fort∈[980,1000]1.01.00.50.5u0.0u0.0-0.5-0.5-1.0-1.09980998599909995100002998029985299902999530000tt(c)Fort∈[9980,10000](d)Fort∈[29980,30000]Fig.5.28.SnapshotsfromthesolutionofundampedDuffingequation(5.39).Solidline:theexactsolution;Dashed-line:5th-orderhomotopyapproximation;Dashed-dottedline:2nd-orderhomotopyapproximation;Dottedline:theinitialapproximation.5.5.4.5.DampedDuffingequationLetusconsiderthedampedDuffingequationu00+αu0+βu+u3=0,u(0)=A,u0(0)=0,(5.41)whereαisthedampingcoefficient.Withoutlossofgenerality,letuscon-siderthecaseα=2,β=5,A==1inthefollowinganalysis.Differentfromtheundampedcasementionedabove,thereisnoneedtotransformtheoriginalvariables.Theauxiliaryinitialguess,theauxiliarylinearop-eratorandtheauxiliaryfunctioncorrespondingtothehomotopy(8.2)areasfollows:1d2du(t)=e−t[2cos(2t)+sin(2t)],L=+2+5,H(t)=1.02dt2dtEvaluatingtheratioβin(5.17)foru00(0)resultsintheexactformulaβ=|1+h|,whichmeansthattheconvergencecontrolparameterhcan 231October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod223-5.0-5.5LH0-6.0u’’-6.5-7.0-2.0-1.5-1.0-0.50.0hFig.5.29.Constanth-curveforequation(5.41).betakenwithintheinterval[−2,0].Figure5.29alsoapprovesthisreality,whichisplottedatthe21st-orderofHAMapproximation.1.´10-110.05-120.048.´10-126.´100.03ResΒ-124.´100.02-122.´100.0100.00-0.98-0.96-0.94-0.92-0.90-0.88-0.98-0.96-0.94-0.92-0.90-0.88hh√(a)Resversush(b)βversushFig.5.30.Residualerrorandratioforequation(5.41)atthe11st-orderoftheHAMapproximation.AsshowninFigures5.30(a–b),thediscreteresidualcomputedfrom(5.23)andtheratiocomputedfrom(5.26)(withp=2)atthe11th-orderofapproximationclearlyindicatethattheoptimalconvergencecontrolparam-eterliesintheintervalh∈[−0.95,−0.90].Actually,theoptimalvaluesateachorderofhomotopytruncationaretabulatedinTable5.17.Optimumsfrombothapproachesareobservedtoasymptotetothesamelimitingvalueh=−0.93astheapproximationorderincreases.Moreover,theratioβissosmallthattheconvergenceofthehomotopyapproximationsgivenbychoosingtheoptimumvaluesofhfromTable5.17isdoubtless.TheconvergenceoftheHAMapproximationstotheexactsolutionofthedampedDuffingequationcanbefurthervisualizedfromFigure5.31,whichdemonstratestheratiosforh=−0.95.InagreementwithTable5.17, 232October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5224M.TurkyilmazogluTable5.17.Theoptimumvaluesofhandtheratioβevaluatedfromtheminimumsofresidualandratioforequation(5.41).M26101420ha−0.9472−0.9315−0.9251−0.9296−0.9282hb−0.8771−0.9056−0.9243−0.9161−0.9253β0.014250.014460.001880.010560.00590aFromequation(5.23)bFromequation(5.26)averysmalllimitingvalueoftheratioβshowninFigure5.31revealstheconvergenceoftheHAM.1.00.80.6Β0.40.20.005101520kFig.5.31.AListplotoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.41).Figures5.32(a–b)exhibittheexactsolutionandthe21st-orderHAMapproximationsgivenbyh=−0.92,togetherwiththeirabsoluteerror.Theconvergencetakeplacesataconsiderablerateevenforthe5th-orderHAMapproximation,asrevealedinFigure5.32(a).TheuniformvalidityoftheHAMforthepresentproblemovertheentiresemi-infinitedomainisalsoconfirmedbytheabsoluteerrordisplayedinFigure5.32(b).5.5.4.6.Thomas–FermiequationLetusconsidernowoneofthemostimportantnonlinearordinarydiffer-entialequationthatoccursinmathematicalphysics,namelytheThomas–Fermiequation[27,37]:ru300u=,u(0)=1,u(∞)=0,(5.42)x 233October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod2251.00.81.2´10-8-80.61.´10-9È8.´10u0.4u--9Èue6.´100.2-94.´100.0-92.´10-0.20012345601020304050tt(a)HAMandnumericalsolutions(b)AbsoluteerrorFig.5.32.(a)SolutionsofdampedDuffingequation(5.41).Solidline:thenumeri-calsolution;Dashedline:the5th-orderHAMapproximation;Dottedline:theinitialapproximation.(b)Absoluteerrorofthe21st-orderHAMapproximation.whichhasasingularityatx=0sinceu00(0)→∞.Sofar,theaccuratevalueofu0(0)is−1.588071022611375313,givenby[38]usingHankel–Pad´emethod.AlthoughtheHAMpossessesseveralfreedomsinit,choosinglessusefulauxiliaryparametersmaynotalwaysyieldsatisfactoryseriesapproxima-tionstotheexactsolution.Thomas–Fermiequationisonesuchstrongnon-lineardifferentialequationthatrequiresmuchcarewhilebenefitingfromtheadvantagesoftheHAM.Otherwise,theaccuracyoftheapproximationsmightbefarfromthedesired,asmentionedbyLiao[3].Beingalertedaboutthisfact,inordertosolvetheThomas–Fermiequation(5.42),wefirstcon-vertitintomoreconventionalformfollowingZhaoetal.[37].Employingthetransformations√u(x)=g2(t),t=1+λx,(5.43)equation(5.42)turnsouttobeλ3gg000+3λ3g0g00−6(t−1)g2g0−4g3=0,(5.44)subjecttotheboundarycondition0g(1)=1,g(1)=g(∞)=0.(5.45)Toapproximatethesolutionof(5.42)andhence(5.44),theoptimalinitialguess,theauxiliarylinearoperatorandtheauxiliaryfunctionweresuppliedin[37],whichwemakeuseofthemhereas4+3+2g0(t)=−+,t3t4t5 234October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5226M.Turkyilmazoglu∂3∂2∂L=t3−5t2−100t−180,∂t3∂t2∂tt4H(t)=,λ3whereλ=5/16and=−86/17.Theseconstantsandtheauxiliarylinearoperatorwereextractedbyminimizingthesquaredresidualofthegoverningequationatsomeloworderofhomotopyapproximations,asmentionedbyZhaoetal.[37]indetails.-1.58-16.20-1.59-16.25LLH0-1.60H1-16.30u’g’’-1.61-16.35-1.62-16.40-2.0-1.5-1.0-0.50.0-2.0-1.5-1.0-0.50.0hh(a)u0(0)versush(b)g00(1)versushFig.5.33.Constanth-curvesforu0(0)in(5.42)andg00(1)in(5.44).Theh-curvesofu0(0)andg00(1)atthe21st-orderoftheHAMapprox-imationareshowninFigures5.33(a–b).Theintervalofconvergenceisanticipatedtobeabouth∈(−2,0)fromtheFigures5.33(a–b).Abetterestimationoftheconvergenceintervalisobtainedbysolvingtheequation|g00(1)/g00(1)|<1,whichgivesh∈[−1.8738,0).21200.000011.08.´10-60.8-60.66.´10ResΒ-64.´100.4-62.´100.200.0-1.7-1.6-1.5-1.4-1.3-1.7-1.6-1.5-1.4-1.3hh√(a)Resversush(b)βversushFig.5.34.Residualerrorandtheratioforequation(5.44)atthe21st-orderoftheHAMapproximation. 235October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod227Inordertogaintheoptimumvalueoftheconvergencecontrolparame-terh,Figures5.34(a–b)arethensketchedatthe21st-orderoftheHAMapproximation.TheresidualerrorinFigure5.34(a)iscalculatedusingthediscreteformula(5.23)andtheratioinFigure5.34(b)using(5.26)(withp=2),respectively.DuetothestrongnonlinearityoftheThomas–Fermiequation,moreandmoretermsareneededtobeevaluated.Itisfoundfromthediscreteerrorthattheoptimumhis−1.5618,betterthanthevaluegivenin[37]withh=−1.382.Thisisindeedthecase,sincearela-tiveerrorof5.9×10−7wasobtainedbyh=−1.382,seeTable1in[37],whileouroptimumvalueresultsinabetterrelativeerror1.3×10−7.ItismostintriguingtoanticipatefromFigure5.34(b)thattheratioisalmostconstantforsomevaluesofhbetween−2and0,theconstantbeingclosetoonebutfortunatelyless,indicatingthatthehomotopyseriessolution(8.27)convergeswithinasufficientlybroadrangeofh.Furthermore,theoptimumvaluesofhandthevaluesu0(0)aresummarizedinTable5.18atdifferentordersofhomotopyapproximations.FromTable5.18,thegoodperformanceoftheHAMapproachisapparentandtheaccuracyisgreatlyincreasedastheorderofhomotopyapproximationisgettinghigh.ThePad´e–approximantcanincreasetheaccuracyfurther,asmentionedbyZhaoetal.[37].Table5.18.Valuesofu0(0)andoptimumhforequation(5.44).M25101521u0(0)−1.61316215−1.58806931−1.58807014−1.58807132−1.58807123ha−1.38432974−1.17961548−1.35694215−1.46696045−1.56175956aEquation(5.23)TheconvergenceoftheHAMapproximationswhenh=−1.55forthecurrentphysicalproblemisalsoverifiedonTable5.19andinFigure5.35.Despitethefactthattheratioapproachesunity,itstillremainslessthan1,asshowninFigure5.34(b).Figure5.36welldemonstratesthattheHAMapproximationsoftheThomas–Fermiequation(5.42)convergesuniformlytotheexactsolutionoveralargeenoughdomain.5.5.4.7.GelfandequationTheGelfandequationu00+λeu=0,u(0)=u(1)=0,(5.46) 236October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.5228M.TurkyilmazogluTable5.19.Theratioβevaluatedwithh=−1.55atseveralorder(M)oftheHAMapproximations.M51020303540β0.697420.830230.928730.963560.973520.980991.00.80.6Β0.40.20.0010203040kFig.5.35.AlistplotoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.44).1.00.80.6u0.40.20.005101520xFig.5.36.SolutionsofThomas–Fermiequation(5.42):straightcurvefromtheexactsolutionanddashedcurvefromthe21st-order(M=21)homotopysolution.presentsanexponentialtypenonlinearity,whereλisaconstant.TheGelfandequation(5.46)representsthesteadystateofdiffusionandtrans-ferofheatconductionofathermalreactionprocessincombustion[39].LiandLiao[40]solvedthisequationbyfirstconvertingitintoanotherkindofdifferentialequationinwhichtheexponentialnonlinearityisnolongerpresent.However,thankstothetheoremprovidedin[41],thereisnoneedtotransformtheGelfandequation(5.46),whichcanbetreatedasitisby 237October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod229theHAM.Havingthisinmind,thesubsequentauxiliaryvariablesd2u0(t)=0,L=,H(t)=1dt2areusedtoapproximatetheexactsolutionof(5.46).-91.´100.50-108.´100.48-106.´100.46ResΒM=12-104.´100.44M=16-102.´100.42M=2000.40-1.40-1.38-1.36-1.34-1.32-1.30-1.34-1.32-1.30-1.28-1.26-1.24hh(a)ResversushatM=20(b)βversushforvaryingMFig.5.37.ResidualerrorandtheratioatMth-orderofHAMapproximationsforequa-tion(5.46).Takingλ=2asanexample,atthe20th-orderofapproximationthehomotopy(8.2)generatesanoptimumconvergencecontrolparameterh=−1.3434fromtheresidual(5.24)(seeFigure5.37(a))andh=−1.2916fromtheratio(5.25)withp=1(seeFigure5.37(b)).NotethatasMincreasesbothapproachesattainthenearlysameoptimumvaluesoftheconvergencecontrolparameterh.Itshouldberemindedthatequation(5.21)isnotanalyticallyintegrableatall,owingtothestrongnonlinearityof(5.46),henceanumericalintegrationisnecessarilycarriedoutwith500integrationpointsforthecomputationoftheresidual.Ontheotherhand,equation(5.25)isanalyticallyintegrabletoyieldtheexactvaluesoftheratioβ.Therefore,theadvantageofthepresentapproachdeservesaspecialmentionagainhere.Table5.20displaysthevaluesofu0(0),theabsoluteresidualerrorsResandtheratioβatseveralorderofapproximationsgivenbyh=−1.3.ItisfoundthattheHAMapproximationsconvergetoitsnumericalvalueu0(0)=1.248217518fairlyrapidly.Theintervalofconvergencecomputedfromtheconstanth-curvesus-ingbothu0(0)andu0(1)isrevealedinFigures5.38(a–b).Theyclearlyindicatethatwithintheinterval[−2,0)theconvergenceofthehomotopyapproximationispossible.Makinguseoftheratios|u0(0)/u0(0)|<12019and|u0(1)/u0(1)|<1,wefindtheintervalofconvergenceas[−2.0279,0),2019thatisconsistentwithFigures5.38(a–b).Thelimitingbehaviorofthera-tio(togetherwithh=−1.2)approximatelyas0.45revealedinFigure5.39 238October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5230M.TurkyilmazogluTable5.20.Valuesofu0(0),Resandβforequation(5.46).M25101520u0(0)1.1916666671.2480249651.2482109561.2482174661.248217517Resa8.997×10−29.876×10−59.807×10−67.324×10−88.634×10−10β0.9654761870.0555533960.5506837720.3741992200.415464040aEquation(5.21)1.300.0-0.21.25-0.4-0.6LLM=20,16,12H01.20H1u’u’-0.8-1.01.15M=20,16,12-1.2-1.41.10-2.0-1.5-1.0-0.50.0-2.0-1.5-1.0-0.50.0hh(a)u0(0)versush(b)u0(1)versushFig.5.38.Constanth-curvesforequation(5.46).issufficientinaccordancewiththeTheorem5.1in§5.3fortheconvergenceoftheHAMapproximationsoftheGelfandequation(5.46).Afurthersup-porttotheuniformconvergenceoftheHAMapproximationstotheexactsolutionof(5.46)comesfromFigures5.40(a–b).1.00.80.6Β0.40.20.0051015kFig.5.39.AlistplotoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.46). 239October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod2310.4-75.´100.3-74.´10Èu3.´10-7u0.2-Èue-72.´100.1-71.´100.000.00.20.40.60.81.00.00.20.40.60.81.0tt(a)HAMandnumericalsolutions(b)AbsoluteerrorFig.5.40.(a)SolutionsofGelfandequation(5.46).Solidline:theexactsolution;Dashedline:the20th-orderHAMapproximation;Dot-dashedline:the3rd-orderHAMapproximation;Dottedline:the1st-orderHAMapproximation.(b)Absoluteerror.5.5.4.8.UniformbeamactedbyaxialloadAstronglynonlineareigenvalueproblemarisingfromtheuniformbeamactedbyaxialloadisexpressedbythesystem(see,page324in[2])0000u+λsinu=0,u(0)=u(π)=0,(5.47)whereλisaneigenvaluecorrespondingtoacompressiveforce.Thesystem(5.47)issuppliedwithanextranormalizingconditionu(0)=γ.Here,weconsiderthecaseofγ=1.Thenonlineareigenvalueproblem(5.47)wastreatedbyLiao[2]andsowecloselyfollowthatsourceinourHAMapproach.LikeLiao[2],wechoosethefollowinginitialguess,theauxiliarylinearoperatorandtheauxiliaryfunctiond2u0(t)=cost,L=+1,H(t)=1,dt2andthenonlinearoperatorisdefinedby00sin[u(t;p)]N[u(t;p),λ(p)]=u(t;p)+λ(p).pItisknownfromtheworkofLiao[2]thatthecorrespondingpositiveeigenvalueisλ=1.137069.Makinguseoftheabovehomotopyapproach,thecurvesofu00(0)andλwithrespecttotheconvergencecontrolparameterharedepictedinFigure5.41(a–b)atdifferentordersofapproximationM.TheconvergenceregionsshowninFigure5.41(a–b)areresolvedexactlybymeansoftheratios(5.17)and(5.18)thatresultin[−1.9704,0),[−1.9787,0),[−1.9827,0), 240October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5232M.Turkyilmazoglu-0.951.1372-0.961.1370-0.971.1368LH0Λu’’M=26,18,10-0.98M=26,18,101.1366-0.991.1364-1.001.1362-2.0-1.5-1.0-0.50.0-2.0-1.5-1.0-0.50.0hh(a)u00(0)versush(b)λversushFig.5.41.Constanth-curvesforequation(5.47).fromu00(0)and[−2.0035,0),[−2.0099,0),[−2.0145,0),fromλ,atthe10th-order,18th-orderand26th-orderofapproximations,respectively.-111.4´100.6-111.2´100.5-111.´100.4-12Res8.´10Β0.3-126.´100.2-124.´10-120.12.´100.0-1.2-1.1-1.0-0.9-0.8-1.2-1.1-1.0-0.9-0.8hh√(a)Resversush(b)β(forλ)versushFig.5.42.Residualerrorandratioforequation(5.47)atthe24th-orderoftheHAMapproximation.Next,thecurveofsquaredresidualofthegoverningequation(5.47)andalsothecurveofratioforλversustheconvergencecontrolparameterhareshowninFigures5.42(a–b).Bothofthemclearlyindicatethattheoptimalconvergencecontrolparameterhatthe24th-orderofapproximationiscloseto−1:similartotheworkin[2],h=−1.0496fromthediscretesquaredresidual(5.23)andh=−1.0478fromthetheratioβassociatedwithλ,respectively.Notethatanalyticintegrationforthesquaredresidualisim-possibleatanyordersincethehomotopyseriestermsareimbeddedintothetrigonometricfunctionof(5.47).Ontheotherhand,theobviousadvan-tagehereistosearchfortheoptimumfromtheconsecutivehomotopyterms 241October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod233oftheeigenvalueλ,whichdoesnotrequirecumbersomeintegrations,butonlyaminimizationprocess.Toillustratethefurtherandmostimportantfeatureoftheratioapproach,atthe24th-orderoftheHAMapproxima-tion,only0.187secondsCPUtimeisneededtoevaluateminimumβofλ,whereas49.485secondsisconsumedforthecalculationofminimumofthediscreteresidualRes.Table5.21.Valuesofλandoptimumconvergencecontrolparameterhatdifferentordersofapproximationforequation(5.47).M35102026λ1.12500000001.13541666671.13706441391.13706872321.13706872330ha−1.058125369−1.078444218−1.129091502−0.998348379−1.0353924539aEquation(5.23)BesidestheaccuracyoftheHAMapproximationsasdemonstratedinTable5.21,thebehaviorofratioβrelatedtoλgivenbydifferentconver-gencecontrolparametersisrevealedeventuallyinFgures5.43(a–c).Theratioseemstobeoscillatoryinnaturefortheconvergencecontrolparame-tersinthevicinityoftheoptimalvalue,smoothforthesmallerones.Thus,morehomotopytermsareneededtobettervisualizetheratioforthehvaluescloseto−1,butunnecessaryforsmallenoughh.5.5.5.AnonlinearfractionaldifferentialequationFractionalcalculusisanareaofmathematicsthatisgrowingoutofthetraditionaldefinitionsofintegralandderivativeoperators,seeforinstancethebooks[42,43].Ithasbeenoftenusedinstudiesofviscoelasticmaterials,aswellasinmanyfieldsofscienceandengineeringincludingfluidflow,foodscience,rheology,diffusivetransport,electricalnetworks,electromagnetictheory,controltheoryandprobability,asmentionedin[44–47].Here,letusconsideronesuchnonlinearfractionaldifferentialequationDαu+u2=1,u(0)=0,t∈[0,1],(5.48)where0<α<1meanstheorderoffractionalderivativeDα.Withoutlossofgenerality,weconcentrateonα=1/2here.Notethatinthelimitαtendsto1,system(5.48)evolvesintothetraditionalphysicalproblempresentedin§5.5.4.2.Tosolvethefractionaldifferentialinitial-valueproblem(5.48),weusetheRiemann–LiouvillefractionalintegraloperatorJαoforderαandthe 242October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5234M.Turkyilmazoglu2.02.01.51.5Β1.0Β1.00.50.50.00.00510152005101520kk(a)h=−1(b)h=−0.82.01.5Β1.00.50.005101520k(c)h=−0.5Fig.5.43.ListplotsoftheratioβforλtorevealtheconvergenceoftheHAMsolutionsforequation(5.47).fractionalderivativeDαintheCaputossense,respectively,definedbyZtα1α−1Jf(t)=(t−τ)f(τ)dτ,(5.49)Γ(α)0Ztα1−α0Df(t)=(t−τ)f(τ)dτ,(5.50)Γ(1−α)0whereΓisthewell-knownGammafunction.TheHAMoutlinedin§5.2isnotdifficulttobeadaptedtogainap-proximateanalyticsolutionsof(5.48)aftertakingintoaccount(5.49)and(5.50)bymeansofthefollowingauxiliarylinearoperator,initialguessandauxiliaryfunctionααL=D,u0(t)=ct,H(t)=1,wherec=21/25isdeterminedbyoptimizingthesquaredresidualatthesixth-orderoftheHAMapproximation.Theconstanth-curvesfortheunknownu(1)atthe8th-order,14th-orderand20th-orderofapproximationsaredepictedinFigure5.44.Notethatthevalueofu(1)convergestoaconstantcloseto0.7fortheconvergence 243October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod2350.900.85LH10.80uM=20,14,80.750.70-1.2-1.0-0.8-0.6-0.4-0.20.0hFig.5.44.Constanth-curvesforequation(5.48).controlparametersh∈[−1.1,0.1].Thisconvergenceregionofhindicatedapproximatelybythegraphcanbeapprovedbysolvingtheinequalityin(5.17)foru(1),thatgivesexactlyh∈[−1.1946,0)atthe14th-orderoftheHAMapproximation.-70.403.´100.350.30-72.5´100.25ResΒ0.20-72.´100.150.101.5´10-70.05-0.75-0.74-0.73-0.72-0.71-0.70-1.0-0.9-0.8-0.7-0.6-0.5hh√(a)Resversush(b)βversushFig.5.45.Residualerrorandratioforequation(5.48)attheorderofhomotopyM=10.Makinguseoftheexactresidual(5.20)andtheratio(5.25)(withp=2),thebetterintervaloftheconvergencecontrolparameterandfurthertheoptimalvalueofitaredemonstratedinFigures5.45(a–b)atthe10th-orderofapproximation.Moreover,theoptimumvaluesoftheconvergencecontrolparameterhgivenbytheminimumofresidualsquareandtheratioatdifferentorderoftheHAMapproximationsareasshowninTable5.22,togetherwiththecorrespondingapproximationsofu0(1),theratioβandtheCPUtimesinseconds.ItisfoundthattheHAMapproximationsconvergequiterapidlybymeansofnearlythesameoptimumconvergencecontrolparameterhevaluatedfrombothofthesquaredresidualandtheratioapproaches.Notethat,theratioapproachismuchmoreefficientinterms 244October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5236M.Turkyilmazogluofthecomputationaltime,similartothecaseforthepreviousexamples.Itcanbeconjecturedherethatformorecomplicatedfunctions,thesquaredresidual(5.20)mightconsumemuchmoreCPUtimeowingto(5.49),evenitwouldbenecessarytoresorttothenumericalintegration,whereastheratioβanditsminimumcanalwaysbeobtainedfromequation(5.25).Table5.22.Theoptimumvaluesofhevaluatedfromtheminimumofresidualandtheratioforequation(5.48),togetherwiththecorrespondingCPUtimes(inparenthesis),u(1)andtheratioβ.M46810ha−0.70132(31.77)−0.71463(70.82)−0.72196(130.56)−0.72656(204.09)u(1)0.69889766320.69874895820.69873982240.6987392647hb−0.71209(8.66)−0.72522(15.95)−0.73065(27.86)−0.73360(48.76)β0.044450.051030.054420.05651aEquation(5.20)bEquation(5.25)WeeventuallydemonstratethelistplotofβinFigure5.46incaseoftheconvergencecontrolparameterh=−0.73.Theconvergenceofthehomotopyseriesisagainconfirmedforthenonlinearfractionaldifferentialequation(5.48),sinceβremainslessthanunity.1.00.80.6Β0.40.20.005101520kFig.5.46.AlistplotoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.48). 245October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod2375.5.6.Couplednonlineardifferentialequations5.5.6.1.VonKarmanviscousflowWeconsiderthesteadylaminarflowofaviscous,incompressible,electricallyconductingandrotatingunboundedfluidinthevicinityoftheequatorofarotatingsphere,governedbythesetofhighlynonlineardifferentialequationsandboundaryconditions[20]F00−HF0−F2−G2+1−mF=0,000G−HG−m(G−1)=0,F+H0=0,(5.51)F(0)=0,G(0)=λ,H(0)=s,F(∞)=0,G(∞)=1,wherem,sandλarephysicalparameters.Formoreinformationabouttheflowconfiguration,pleaserefertoTurkyilmazoglu[20].1.0G’(0)0.80.6F’(0)0.4-H(¥)0.20.0-1.2-1.0-0.8-0.6-0.4-0.20.0hFig.5.47.Constanth-curvesforthesystem(5.51).Forsimplicity,weusethesameauxiliarylinearoperatorLandthesameauxiliaryconvergencecontrolparameterhtosolvethecouplednonlinearODEs.WechoosethefollowinginitialguessesF(η)=ηe−η,G(η)=1+(λ−1)e−η,H(η)=−1+s+e−η(1+η),000andtheauxiliarylinearoperatord2L=−1.dη2Inthecaseofs=1/2,λ=0andm=1,atthe21st-orderoftheHAMapproximation,theconvergenceintervalforhisshowninFigure5.47bymeansofplottingtheunknownvaluesofF0(0),G0(0)and−H(∞)against 246October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5238M.Turkyilmazogluh.Dissimilartotheaboveexamples,althoughtheconvergencedomainvariesinaccordancewiththeconsideredquantity,thecommonintervalofconvergenceisseentobeconfinedtoapproximately[−1,−0.1].TheintervalsshowninFigure5.47arealsoverifiedviatheratiosin(5.17)suchthattheratiosF0(0)G0(0)H(∞)21,21,21F200(0)G020(0)H20(∞)give,respectively,themoreaccurateintervals[−1.0061,0),[−1.0078,0),[−0.9968,0).Thisexampleillustratesthat,forthesystemofequationslike(5.51),betterinformationcanbeaccessedthroughtheratioβgivenin(5.17),ratherthantheclassicalconstanth-curvesinFigure5.47.Eventhough,itmaybediscussedfromFigures5.47thattheintervalofconvergenceseemstocontainthevalueh=−1,buttheratioapproachtellsusthatthisisnottrue,asshownbelow.1.´10-81.08.´10-90.8-90.66.´10ResΒ-94.´100.4-92.´100.200.0-0.80-0.75-0.70-0.65-0.60-0.80-0.75-0.70-0.65-0.60hh√(a)Resversush(b)βversushforF1.01.00.80.80.60.6ΒΒ0.40.40.20.20.00.0-0.80-0.75-0.70-0.65-0.60-0.80-0.75-0.70-0.65-0.60hh(c)βversushforG(d)βversushforH√Fig.5.48.ResidualerrorResandratiosβforF,GandHatthe21st-orderoftheHAMapproximation. 247October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod239Theoptimalvalueoftheconvergencecontrolparameterhisfoundtobeabouth=−0.75atthe21st-orderofapproximationusingthediscretesquaredresidualZ∞Res(h)=g2(η)+g2(η)+g2(η)dη,1230whereg=F00−HF0−F2−G2+1−mF,g=G00−HG0−m(G−1),g=F+H0.123ThisoptimumvalueisbettervisualizedfromtheresidualinFigure5.48(a)andfurtherfromtheratio(5.25)(withp=2)inFigures5.48(b–d).Ta-ble5.23presentsthevaluesofF0(0),G0(0)andH(∞)usingthisoptimumconvergencecontrolparameteratdifferentorderoftheHAMapproxima-tions,M.TheHAMapproximationsforM≥20arethesame,andthus,ascomparedtothepreviousexamples,theconvergenceseemstotakeplaceataconsiderablyfastratetowardstheexactsolutionforthishighlynonlinearsystem.Table5.23.TheMth-orderapproximationofF0(0),G0(0)andH(∞)of(5.51)givenbyh=−3/4.M1020304050F0(0)0.570442440.570439120.570439120.570439120.57043912G0(0)0.809690650.809691930.809691930.809691930.80969193−H(∞)0.265399750.265403390.265403370.265403370.26540337Asalsohighlightedin[20],Figures5.49(a–c)demonstratetheratios(5.15)inTheorem5.1forF,GandH,whicharecomputedfromthehomotopyseries(8.27)bymeansofh=−0.75.Eventhoughsharposcil-lationsoccurforsmallnumberofterms,asthenumberoftermsincreases,theoscillationssettledownandtheratiostendtoalimitingvalueofap-proximately0.53.Theconvergenceforthecurrentphysicalsystem(5.51)isthenconfirmed,sincetheratiosremainlessthanunity.Finally,wepointtothefactthatthehomotopyperturbationmethod(HPM)proposedin1998isonlyaspecialcaseofthehomotopyanalysismethod(HAM)introducedfirstbyLiaoin1992[1].AspointedoutbyLiao[2],thisisobviousbysimplysettingh=−1inthehomotopyequation(8.2).Thismeansthatthehomotopyperturbationmethod(HPM)isonlyaspecialcaseoftheHAMwhenh=−1,andnotasflexibleastheHAMsothatdivergencebecomesinevitablefortheHPMtomostofstrongly 248October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5240M.Turkyilmazoglu3.02.02.51.52.0Β1.5Β1.01.00.50.50.00.00102030405001020304050kk(a)βversushforF(b)βversushforG3.02.52.0Β1.51.00.50.001020304050k(c)βversushforHFig.5.49.ListplotsoftheratiostorevealtheconvergenceoftheHAMsolutionsforequation(5.51)whenh=−3/4.nonlinearproblems.However,bymeansofchoosingpropervalueoftheconvergence-controlparameterh,theHAMprovidesuswithasimplewaytoguaranteetheconvergenceofapproximationseries.Thus,theconver-gencecontrolparameterplaysaveryimportantrole,whichdifferstheHAMfromallotheranalytictechniques.Forexample,asshowninFigures5.50(a–c),approximationsgivenbythehomotopyperturbationmethod(HPM),correspondingtotheHAMinthespecialcaseofh=−1,donotconvergetotheexactsolution,becausethelimitoftheratioobviouslyexceedsunity,reaching1.04.Therefore,whilemakinguseoftheso-calledhomotopyper-turbationmethod(HPM),theconvergenceoftheproducedapproximationseriesmustbemathematicallyjustified,whichwasunfortunatelynotad-heredinmostoftherelevantstudiesintheliterature. 249October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod2411.01.00.80.80.60.6ΒΒ0.40.40.20.20.00.0010203040010203040kk(a)βversushforF(b)βversushforG1.00.80.6Β0.40.20.0010203040k(c)βversushforHFig.5.50.ListplotsoftheratiostorevealthedivergenceoftheHPMsolutionsforequation(5.51).5.5.6.2.AsystemofequationsmodelingthesmokershabitinSpainConsidernowthesystemofnonlineardifferentialequations[48]n0(t)=µ(1−n)−βn(s+c),0s(t)=βn(s+c)+ρe+αc−(γ+λ+µ)s,(5.52)c0(t)=γs−(α+δ+µ)c,e0(t)=λs+δc−(ρ+µ)e,whichgovernthedynamicsofthesmokersinSpain.Thenecessaryexpla-nationsonthevariablesandparametersaregivenin[48],togetherwiththeinitialconditions.However,unlike[48],weusehered1L=+dt4astheauxiliarylinearoperator.Theso-calledh-curvesofn0(0),s0(0),c0(0)ande0(0)atthe25th-orderofapproximationareasshowninFigure5.51.TheysuggestthattheHAMapproximationsshouldbeconvergentwhenhisapproximatelyintheinterval[−1.8,−0.2].Theoreticalanalysisofthe 250October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5242M.Turkyilmazogluratiosβforthefourvariablesinequation(5.52)alsoresultsinthesimilarintervaloftheconvergence-controlparameter.0.0200.015e’(0)0.0100.0050.000n’(0)-0.005s’(0)andc’(0)-0.010-2.0-1.5-1.0-0.50.0hFig.5.51.Constanth-curvesfor(5.52)atthe25th-orderoftheHAMapproximation.TheresidualforthepresentsystemisdefinedbytheintegralZ∞Res(h)=[g2(t)+g2(t)+g2(t)+g2(t)]dt,12340where0g1=n−µ(1−n)+βn(s+c),0g2=s−βn(s+c)−ρe−αc+(γ+λ+µ)s,g=c0−γs+(α+δ+µ)c,g=e0−λs−δc+(ρ+µ)e.34However,unlikethepreviousexample,theratioβisdefinedhereasfollows1knk+1(t)kksk+1(t)kkck+1(t)kkek+1(t)kβ=+++,4knk(t)kksk(t)kkck(t)kkek(t)kwherethenormisassumedasL2.TheirresidualerrorResandratioβatthe25th-orderoftheHAMapproximationareshowninFigures5.52(a–b).Bothofthemsuggesttheoptimumvalueh≈−1.2.Moreover,thecorre-spondingoptimumvaluesofhandtheCPUtimesofthetwoapproachesatdifferentordersoftheHAMapproximationaregiveninTable5.24.Al-thoughtheCPUtimesofthetwoapproachesareclose,westillobservethebetterperformanceoftheratioapproach.TheconvergenceoftheHAMapproximationincaseofh=−1.18forthecurrentsmokinghabitproblemisalsoverifiedinTable5.25andFigure5.53.Despitethefactthattheratiokeepsoscillating,thelimitislikelytobeabout0.07.AsshowninFigure5.54,the25th-orderHAMapproximationincaseofh=−1.18agreeswellwiththeexactsolution.Thepresent 251October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod243-141.01.´10-158.´100.8-156.´100.6ResΒ-154.´100.4-152.´100.200.0-1.30-1.25-1.20-1.15-1.10-1.30-1.25-1.20-1.15-1.10hh√(a)Resversush(b)βversush√Fig.5.52.ResidualerrorResandratioforequation(5.52)atthe25th-orderofap-proximation.Table5.24.Theoptimumvaluesh,ratiosβandCPUtimesforequation(5.52)atdifferentordersofapproximationM.MhaCPUtimehbβCPUtime2−1.50362.67−1.42560.08331.905−1.24748.36−1.23080.07655.8910−1.180326.50−1.02520.067017.9120−1.176172.66−1.15110.061773.6824−1.175994.89−1.20940.057088.89aEquation(5.23)bEquation(5.26)HAMapproximationsaremuchbetterthanthosegivenin[48].NotethatGuerreroetal.[48]employedtheauxiliaryoperatorL=d.ItshouldbedtemphasizedthatthehomotopyPad´eapproximationhadtobeemployedin[48]soastogetconvergentresultsinalargetimeinterval,whicharealreadywellpredictedfromthepresentHAMapproximations.ThisillustratestheimportanceofchoosingabetterauxiliarylinearoperatorintheframeoftheHAM.Table5.25.Theratioβevaluatedwiththe50homotopytermsincaseofh=−1.18.M51020304050β0.124580.572510.071760.079330.075280.07185 252October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5244M.Turkyilmazoglu1.00.80.6Β0.40.20.001020304050kFig.5.53.AlistplotoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.52).0.5n0.40.3e0.2sc0.1020406080100tFig.5.54.Theexact(unbroken)andtheHAM(broken)solutionsforequation(5.52).5.5.7.Partialdifferentialequations5.5.7.1.Burger’sequationThefamousBurger’sequationut+uux=uxx(5.53)describesvariouskindofphenomena,suchastheturbulenceandtheshockwavetravelinginaviscousfluid[49].Equation(5.53)undertheinitialconditionu(x,0)=2x,(5.54)admitsanexactsolution2xu(x,t)=.(5.55)1+2t 253October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod245Togainthesolution(5.55)bymeansoftheHAM,wechoosethefol-lowinginitialguess,auxiliarylinearoperatorandauxiliaryfunction∂u0(x,t)=2x,L=,H(x,t)=1.∂tAsaresult,thehomotopy(8.2)turnsouttobe(1−p1−hput(x,t,p)+u(x,t,p)ux(x,t,p)−uxx(x,t,p)=0,(5.56)u(x,0,p)=2x.Thecorrespondinghomotopyseriessolutionof(5.53)readsu(x,t)=2x+4htx+4ht(1+h+2ht)x+4ht(1+h+2ht)2x+4ht(1+h+2ht)3x+4ht(1+h+2ht)4x+···,(5.57)whichisconvergent,if|un+1|lim=|1+h(1+2t)|<1,n→∞|un|wherethenormisinthesenseofabsolutevalue.Thus,accordingtoThe-orem5.1in§5.3,thisholdsexactlywhen12+h− 254October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5246M.TurkyilmazogluTheabovenonlinearPDEcanbesolvedintheframeoftheHAMusingthefollowinginitialguess,auxiliarylinearoperatorandauxiliaryfunction∂2u0(x,t)=(x+y+z−cosx−cosy−cosz)(1−t),L=,H(x,t)=1.∂t2Then,thehomotopy(8.2)forh=−1generatesthesubsequentclosed-formhomotopyseries∞Xt2nt2n+1u(x,t)=(x+y+z−cosx−cosy−cosz)−,(5.60)(2n)!(2n+1)!n=0whichleads,usingtheratioin(5.15),to|un+1|lim=0.n→∞|un|Therefore,accordingtoTheorem5.1in§5.3,thehomotopyseries(8.2)convergestotheexactsolution(5.58)forallt.5.5.7.3.AlinearpartialdifferentialequationofhighorderLetusconsiderthefollowinginitialvalueproblemgovernedbythelinearpartialdifferentialequation−xut+ux−2uxxt=0,u(x,0)=e,(5.61)whoseexactsolutionreadsu(x,t)=e−x−t.(5.62)Thezeroth-orderdeformationequation(8.2)thenbecomes(1−p)ut(x,t,p)−ph(ut(x,t,p)+ux(x,t,p)−2uxxt(x,t,p))=0,u(x,0,p)=e−x,(5.63)wherethefollowinginitialguess,auxiliarylinearoperatorandauxiliaryfunction−x∂u0(x,t)=e,L=,H(x,t)=1∂tareused.Whenh=−1,itappearsthatthehomotopyseriessolutionisconver-gentonlyfort≤0,thatisoutofphysicalinterest,sincetdenotestimeinequation(5.61).Ontheotherhand,whenh=1,thesuccessiveratioin(5.15)yieldsun+1(x,t)|t|=,un(x,t)n+1 255October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod247whoselimitasntendstoinfinityissimplyzero.Asmentionedbefore,thehomotopyperturbationmethod(HPM)isaspecialcaseofthehomotopyanalysismethod(HAM)whenh=−1.Thiswellexplainswhytheso-calledHPMisunabletoderiveconvergenthomotopyseries,andfurtherexplainswhythehomotopyseriessolution(8.27)withtheconvergencecontrolpa-rameterh=1representstherealphysicalsolutionovert≥0,asdiscussedbyLiangetal.[10]indetails.5.5.7.4.Korteweg-deVriesBurgers(KdVB)equationsTheKdVBequationhastheformut+uux−νuxx+µuxxx=0,(5.64)where,νandµareconstants.Thisnonlinearpartialdifferentialequation,whichwasoriginallyderivedby[51],isacornerstoneforsomesignificantphysicalproblemsinmathematicalphysics.Unliketheaboveexamples,weareinterestedinthenon-separablesolutionsof(5.64).Hence,weconsider(5.64)inaspecialcaseof=−6,ν=0andµ=1,togetherwiththeinitial2conditionu(x,0)=−2sechx.Substitutingthefollowinginitialguess,auxiliarylinearoperatorandauxiliaryfunction2∂u0(x,t)=−2sechx,L=,H(x,t)=1∂tinthezeroth-orderdeformationequation(8.2),thehomotopyapproxima-tionsatanyordercanbeeasilysucceeded.Afewofthemaregivenby2u1(x,t)=16htsechxtanhx4u2(x,t)=−8htsechx(−8ht+4htcosh(2x)−(1+h)sinh(2x))...TakingintoaccountthenormdefinedbyZ1Z∞||u(x,t)||=u2(ζ,τ)dζdτ,(5.65)00theratiogivenin(5.15)guarantiestheconvergenceofthehomotopyseries(8.27)withinthetimedomain[0,1]andspacedomain[0,∞),providedthatβ<1.Forinstance,thesuccessiveratiosu2u4u6u,u,u135 256October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5248M.Turkyilmazogluyieldtheconvergencecontrolparametersresidingintheintervals,respec-tively[−0.2545,0],[−0.0786,0],[−0.0472,0].Thisclearlyshowsthatasufficientlysmallconvergencecontrolparametershouldbechosentoensuretheconvergencewithinthementioneddomainofdefinition.Inadditiontothis,theaboveratiosindicatethath=−1isnotinsidethevalidregionofconvergencecontrolparameter,thus,theso-calledhomotopyperturbationmethodmaynotsucceedfortheKdVBproblem(5.64).Infact,inordertounderstandwhathappenswhenhissetto−1,letusconsiderthenormZ∞ku(x,t)k=u2(ζ,t)dζ,(5.66)0tocomplywiththeCorollary2.Then,thesuccessiveratiosu1u6u31u51u71u,u,u,u,u05305070resultinthetimeintervals[0,0.2795],[0,0.3493],[0,0.3834],[0,0.3870],[0,0.3886],pointingtotherealitythatthevalidityregionoftimedomainoftheho-motopyseriessolutionforh=−1iscloseto[0,0.4),whichexplainstheabovefailureofthehomotopyperturbationmethod(HPM)overtheinter-valt∈[0,1].5.5.7.5.CouplednonlinearsolitarywaveequationsConsidertwocouplednonlinearequationsut−uux−vx+uxx/2=0,(5.67)vt−uvx−vux−vxx/2=0,subjecttotheinitialconditions2u(x,0)=1+tanhx,v(x,0)=sechx,(5.68)forthelongsolitarywavesindeepwater[52–54].Forthesakeofsimplicity,weusethesameauxiliarylinearoperatorL=∂andthesameconvergencecontrolparameterhforthetwopartial∂tdifferentialequationsin(5.67),andtheinitialguesses2u0(x,t)=1+tanhx,v0(x,t)=sechx. 257October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod249Takingintoconsiderationthenormfromequation(5.66),thesuccessiveratiosconcerninguu2u6u10u15u,u,u,u15914producethevalidregions[0,2.2361],[0,1.6413],[0,1.6116],[0,1.5977],andthesuccessiveratiosconcerningvv2v6v10v15v,v,v,v15914producethevalidregions[0,1.1832],[0,1.3974],[0,1.4616],[0,1.4962],fortimet,incaseoftheconvergencecontrolparameterh=−1.Thecorresponding10th-orderHAMapproximationreadsu(x,t)=1+tanhx+t9sech10x(1−ttanhx)1 +t7sech8x−3−5t2+t3+4t2tanhx31 +t5sech6x59+12t2+7t4−3t15+15t2+7t4tanhx45134246+tsechx(−945−945t−378t−85t945 246+t945+630t+189t+34ttanhx)1 +tsech2x(52835+2t2945+189t2+18t4+t614175 −t14175+4725t2+630t4+45t6+2t8tanhx),12246810v(x,t)=sechx(14175+28350t+9450t+1260t+90t+4t14175+tsechx(−155925t9sech9x −10(2835+2t2945+189t2+18t4+t6)sinhx +14175t7sech7x9+22t2−10ttanhx −945t5sech5x(105+225t2+209t4−40t3+5t2tanhx)+15t3sech3x(4725+8820t2+6615t4+2816t6 −630t9+12t2+7t4tanhx)+3tsechx(−14175−23625t2−13230t4−3825t6−682t8 +20t945+945t2+378t4+85t6tanhx))). 258October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5250M.Turkyilmazoglu5.5.7.6.Anonlinearage-structuredpopulationmodelThefollowingnonlinearpartialdifferentialequationZ∞−xeut(x,t)+ux(x,t)=−1+u(x,t)dxu(x,t),u(x,0)=(5.69)02modelstheage-structuredpopulations[55,56],wherexandtdenoteageandtime,andu(x,t)istheage-specificdensityofindividualsofagexattimet,respectively.Themaximumagethatanindividualofthepopulationmayreceiveisassumedtogrowindefinitelysothat0≤x<∞andweconfinethetimetot∈[0,1].IntheframeoftheHAM,weusethefollowingauxiliarylinearoperator,initialguessandauxiliarylinearoperator∂−xL=,u0(x,t)=e/2,H(x,t)=1.∂t10864utH0,0L2uttH0,0L0-2.0-1.5-1.0-0.50.0hFig.5.55.Constanth-curvesforequation(5.69).Similarly,theconstanth-curvesofut(0,0)andutt(0,0)atthe20th-orderofapproximationareasshowninFigure5.55,whichsuggeststhattheconvergencecontrolparameterhforconvergentseriesshouldbewithintheinterval[−2,0].Thebetterestimationoftheintervalisrecoveredfromtheratio(5.17),whichgives[−2,0]forut(0,0)and[−1.9474,−0.0526]forutt(0,0),respectively.However,ascommentedbefore,theobtainedin-tervalsfromtheconstanth-curvesarelocallygoodapproximates.Betterconvergencecontrolparametercanbegainedbymeansoftheratio(5.15)togetherwiththenormgivenin(5.65).Hence,theratiosu2u6u10u20u30u,u,u,u,u1591929resultinthethesubsequentintervalsoftheconvergencecontrolparameterh:[−1.4474,0],[−1.3849,0],[−1.3655,0],[−1.3498,0],[−1.3443,0].As 259October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod251comparedtotheintervalsgivenbytheconstanth-curves,theaboveinter-valsarebettersincetheyarecomputedfromnormsovertheentireregions,unlikethelocalonesobtainedbymeansoftheh-curvesinFigure5.55.-151.3´10-150.0381.2´10-151.1´100.036ResΒ-151.´100.034-169.´100.032-0.790-0.788-0.786-0.784-0.782-0.780-0.800-0.795-0.790-0.785-0.780hh√(a)Resversush(b)βversushFig.5.56.Residualerrorandratioforequation(5.69)attheorderofhomotopyM=20.Figures5.56(a–b)demonstratetheresidualandtheratioatthe20th-orderoftheHAMapproximation,usingtheexactintegrationsbasedonthenorm(5.65).Moreover,Table5.26presentstheoptimalvaluesofh,theratiosβandtheusedCPUtimes.Similartotheordinarydifferentialequationsmentionedbefore,theratioapproachishighlysuccessfulforthisnonlinearpartialdifferentialequationinobtainingtheoptimalvaluesoftheconvergencecontrolparametersh,whichareveryclosetothoseobtainedbymeansofthesquaredresidualapproachthathoweverunfortunatelyneedsmorecomputationaltimes.Table5.26.Theoptimumvaluesh,ratiosβandCPUtimesforequation(5.69)atdifferentordersofapproximationM.√MhaResCPUtimehbβCPUtime2−0.72594.6028×10−30.98−0.72370.00490.335−0.75863.0671×10−54.34−0.76490.01980.9810−0.77438.9322×10−98.64−0.77910.02723.1220−0.78468.4875×10−1637.56−0.78750.032217.36aEquation(5.20)bEquation(5.25)TheconvergenceoftheHAMapproximationincaseofh=−0.79totheexactsolutionofthenonlinearage-structuredpopulationmodel(5.69)isfurtherconfirmedbyFigure5.57,whichclearlydemonstratestheratioless 260October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5252M.Turkyilmazogluthanunity.Therefore,thisexampleonceagainvalidatestheratioapproachevenforstronglynonlinearpartialdifferentialequations.1.00.80.6Β0.40.20.0051015202530kFig.5.57.AlistplotoftheratioβtorevealtheconvergenceoftheHAMsolutionsforequation(5.69).5.6.ConcludingremarksInthischapter,thehomotopyanalysismethod(HAM)hasbeeninvesti-gatedwithanaimtostudytheconvergenceconditionsofthehomotopyseriessolutionsofthenonlinearalgebraicequationsaswellasthenon-lineardifferential-difference,integro-differential,fractionaldifferentialandordinary/partialdifferentialequations,andalsosystemofdifferentialequa-tions.Inthecontextofthetheoremsandtheircorollaries,weillustratethatiftheconvergencecontrolparameter,theauxiliarylinearoperator,theinitialguessandsoonareproperlychosenintheframeoftheHAM,thehomotopyanalysisresultsmayindeedconvergetotheexactsolutionofconsiderednonlinearproblems.TheconceptoftheconvergencecontrolparameteremployedintheHAMtechniquehasbeenthoroughlyreviewed.Anoveldefinitionfortheoptimalconvergencecontrolparameterisgiveninthischapter,whichisdistinctfromthewell-knownoptimumapproachbasedonminimizingthesquaredresidual.Thenewoptimumapproachfortheconvergencecontrolparame-terisbasedontheminimizationoftheratioofthehomotopyseriesterms.Itoftengivesoptimumconvergencecontrolparameterclosetothosegainedbythesquaredresidualapproachatafiniteorderofapproximations.Whentheorderofapproximationtendstoinfinity,themajorityoftheconsideredexamplesclearlyimplythecoincidenceoftheoptimalconvergencecon-trolparametersgivenbythetwoapproaches.Differenttypesofnonlinear 261October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.5AConvergenceConditionoftheHomotopyAnalysisMethod253equationshavebeenusedinthischaptertoverifythetheoremsandthecorollariesprovedinthischapter.Bymeansoftheratioapproach,notonlytheconvergenceofthehomo-topyseriesisguaranteed,butalsotheintervaloftheconvergencecontrolparameterensuringtheconvergenceoftheHAMresultsisanalyticallyde-termined.Inparticular,theintervalofhforconvergenceestimatedbythetraditionalconstanth-curvesisoftenworkedoutbymeansofthera-tiomethod.Beingabletobetterdeterminetheboundsoftheintervaloftheconvergencecontrolparameterh,theratioapproachwellexplainswhytheso-calledhomotopyperturbationmethod(HPM),whichisactuallyaspecialcaseoftheHAM,mightfailoronlycorrespondtosomerestrictedintervalsofthespaceandtimevariablesofconsideredphysicalproblems.Theratioapproachseemspromising,sinceittakesconsiderablylesscomputationaltimethanthetraditionalways.Ourexamplesillustratethatthisisindeedthecase,ifbothofthetraditionalsquaredresidualandtheratioapproachesareusedtoestimatetheintervaloftheconvergencecontrolparameters.Forsomestronglynonlinearproblems,liketheGelfandproblem,theeigenvalueproblemsinvolvingtrigonometricfunctionsandsoon,exactevaluationofthesquaredresidualisnotpossibleinmostcases,whereastheratioconcerningthehomotopytermscanbecomputeduptoanyorderofapproximation.Particularly,forfractionaldifferentialequa-tions,theadvantageoftheratioapproachcannotbedisregarded.TheminimizationoftheratioapproachtogainoptimalhstillrequireslessCPUtimesthantheminimizationofthesquaredresidualapproach.Thisisalsogenerallytrueifdiscretesquaredresidualversustheexactratioisconcerned.Furthermore,forproblemsinvolvinganunknownparame-ter,suchastheundampedDuffingoscillator,theeigenvalueproblems,theparameterizeddifferentialequationsandsoon,theratioapproachhastheadvantagethatboththeoptimalvalueofconvergencecontrolparameterandtheconvergenceanalysiscanbepursuedonlyfromtheratioscon-cerningtheparameter,leavingasidetheneedofusingresidualsforothervariables,andthusavoidingheavyintegrations.Evenifthereisnosuchkindofunknownparameterinthedifferentialequation,apseudo-likepa-rametercanalwaysbeintroducedusinganunknownboundaryconditiontogetherwithasuitablescaling,sothattheadvantageofnewmethodcanalwaysbebenefited.Ourexamplesclearlyshowthattheproposedratioapproachgeneratesalmostthesameoptimalconvergencecontrolparametersasthosegivenbythesquaredresidualapproach,aslongastheorderofHAMapproximation 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267October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6Chapter6HomotopyAnalysisMethodforSomeBoundaryLayerFlowsofNanofluidsTasawarHayatDepartmentofMathematics,Quaid-I-AzamUniversity45320,Islamabad44000,Pakistanpensyt@yahoo.comMerajMustafaResearchCentreforModelingandSimulation(RCMS),NationalUniversityofSciencesandTechnology(NUST),Islamabad44000,PakistanThischapterdescribestheapplicationofhomotopyanalysismethod(HAM)tothenonlinearboundaryvalueproblemsarisinginnanofluiddynamics.Theseriessolutionsforsomefundamentalproblemsinfluidmechanics(includingaxisymmetricflowpastastretchingsheet,un-steadyboundarylayerflowdevelopedduetotheimpulsivemotionofanextensiblesurface,squeezingflowbetweenparalleldisksandbound-arylayerflowofsecondgradefluid)involvingnanofluidsarecomputedanddiscussed.Contents6.1.Background.....................................2606.2.Unsteadyboundary-layerflowofnanofluidpastanimpulsivelystretchingsheet.........................................2636.3.Axisymmetricflowofnanofluidoveraradiallystretchingsheetwithconvectiveboundaryconditions................................2716.4.Squeezingflowofnanofluidbetweenparalleldisks................2756.5.Boundarylayerflowofnon-Newtoniannanofluidoverastretchingsheet...280References.........................................288259 268October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6260T.HayatandM.Mustafa6.1.BackgroundFluidflowproblemscharacterizingeitherviscousornon-Newtonianfluidsareusuallymodeledintermsofhighlynonlinearboundaryvalueproblems.ObviouslyonerequiresanefficientandeasytouseanalyticaltechniquelikeHAMtosolvesuchproblemsanddeepenourunderstandingtothesecompli-catednonlinearphenomena.TheresearchershavethereforewidelyappliedHAMtosolvevariousnonlinearproblemsarisinginfluidmechanics.Forinstance,anapproximateanalyticsolutionofthetwo-dimensionallaminarviscousflowoverasemi-infiniteplatehasbeenobtainedbyLiao[1].Theconvergenceofthesolutionswasensuredthroughtheproperselectionoftheauxiliaryparameter.TheobtainedsolutionwasinanexcellentagreementwiththeHowarth’snumericalsolution.LiaoandCampo[2]usedHAMtofindtheseriessolutionoftemperaturedistributioninaBlasiusflow.Thedimensionlessheattransferrateattheplatewasalsocalculated.Theanalyticresultswerefoundincompleteagreementwiththeobtainednu-mericalsolutionforreasonablylargevaluesofthePrandtlnumber.Liao[3]computedseriessolutionsformagnetohydrodynamic(MHD)flowofpower-lawfluidpastastretchingsheet.Theanalyticsolutionswereobtainedforrealpowerlawindexintermsofrecursiveformulasofconstantcoefficientsandthesewereindecentagreementwiththenumericalsolutionforallthevaluesofmagneticparameter.Healsoobtainedasimpleanalyticfor-mulaforcalculatingskinfrictioncoefficientonamovingsheettofindwideapplicationsinmetallurgicalprocesses.HomotopysolutionsforunsteadyboundarylayerflowduetoimpulsivelystretchingplatehavebeenprovidedbyLiao[4].Differentfromthepreviousperturbationsolutions,theHAMsolutionswereconvergentforallvaluesofthedimensionlesstime.Thevali-dationtothefindingswasgiventhroughtheobtainednumericalsolutioninseveraldifferentcases.SajidandHayat[5]obtainedthelocalsimilarityso-lutionsforthetwo-dimensionalflowofMaxwellfluidaboveacontinuouslymovingflatplateinaquiescentambientfluid(theso-calledSakiadisflow)byHAM.Theconvergencewasguaranteedforafixedrangeoftheauxiliaryparameterbydisplayingthesocalled~-curves.Theconvergentnumericalresultsforvariousparametricvalueswerealsogiven.InanotherpaperSajidandHayat[6]furtherinterpretedthatsecondorderorsecondgradefluidsareinadequateindescribingtheshearthinningorthickeningbehaviorofvariouspolymericliquids.Forthispurposetheyconsideredaflowoffourthgradefluiddownaverticalcylinder.Themodelednonlinearproblemwassolvedforthehomotopyanalyticsolution.Theyalsoprovedthatobtained 269October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids261solutionincaseofhomotopyperturbationmethod(HPM)isaspecialcaseofHAManditoffersdivergentsolutionsforstrongnonlinearities.Seriesso-lutionforunsteadyaxisymmetricflowoverastretchingsheetwasprovidedbySajidetal.[7].Avalidrangeoftheauxiliaryparameterwasidentifiedthrough~-curves.Hayatetal.[8]investigatedtheviscoelasticeffectsintheBlasiusflowbyHAM.DifferentfromLiaoandCampo[2]theviscousdissipationeffectswerealsoconsidered.Similarinitialguessesandthelin-earoperatorsforthevelocityandtemperaturefunctionswereselectedasinLiao[1],andLiaoandCampo[2].Localsimilaritysolutionswereob-tainedandconvergencewasguaranteedforvariousvaluesofviscoelasticparameter.HomotopysolutionsforFalkner–SkanflowhavebeenreportedbyAbbasbandyandHayat[9].TheyhaveobtainedthenumericalvaluesofdimensionlessvelocitygradientontheplatebyCrocco’stransformation.BasedontheideagiveninLiao[1],inadditiontotheauxiliaryparameter~anotherparameterγwasintroducedintheinitialguess.Propervaluesof~andγwereidentifiedsothattheseriessolutionwouldconvergeforallthevaluesofembeddedparameters.Homotopy-Pademethodwasusedtocomputethenumericalvaluesofskinfrictioncoefficientandthesevalueswereshowntobeinaverygoodagreementwiththoseobtainednumeri-callyandbyCrocco’stransformation.Inanotherpaper,AbbasbandyandHayat[10]consideredtheunsteadyflowofspecialthirdgradefluidoveraporousunsteadystretchingsheet.Thevelocitydistributionoutsidetheboundarylayerwasassumedtobetimedependent.Thedevelopeddiffer-entialsystemwassolvedforthehomotopysolutions.Foracceleratingtheconvergence,homotopy-Pademethodwasadoptedandnumericalvaluesofvelocitygradientonthesheetwerecomputedforvariousparametricvalues.ThesevalueswerecomparedwiththenumericalsolutionobtainedthroughthecomputationalsoftwareMathematicaandwereshowntobeindecentagreement.Xuetal.[11]exploredthemultiplesolutionsoftheNavier–Stokesequationsrepresentingtheviscousflowinaporouschannelwithmovingwallsbyoptimalhomotopyanalysisapproach.Theypredictedthemultiplesolutionsoftheboundarylayerequationsbyadjustingtheaux-iliaryparameter.Rashidietal.[12]providedthehomotopyanalyticsolu-tionsforheattransferintheflowofmicropolarfluidembeddedinaporousmedium.Thethermalradiationeffectwasalsoconsidered.TheydefinedtheaveragedresidualerrorsofthecorrespondingdifferentialequationsandusingtheideagivenbyLiao[13]theoptimalvaluesoftheauxiliaryparam-eterweredetermined.Theyconcludedthat,bymeansoftheconvergencecontrol(auxiliary)parameter,HAMcanbeappliedtobothstronglyand 270October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6262T.HayatandM.MustafaweaklynonlinearproblemswhichisafundamentalqualitativedifferencebetweenHAMandotheranalyticalmethods.Nanofluidisaliquidinwhichnanometer-sizedparticles(callednanopar-ticles)madeupofmetals,oxides,carbidesorcarbonnanotubesaresus-pendedinthebasefluidsuchasoils,water,ethyleneglycoletc.Choi[14]experimentallyfoundthatadditionofthesenanoparticlesinthebasefluidappreciablyenhancestheeffectivethermalconductivityofthefluid.Non-homogeneousequilibriummodelproposedbyBuongiorno[15]revealsthatsuchmassiveincreaseinthethermalconductivityoccursduetothepresenceoftwomaineffectsnamelytheBrowniandiffusionandthethermophoreticdiffusionofnanoparticles.ReviewstudiesonnanofluidshavebeenmadebyDaungthongsukandWongwises[16],WangandMujumdar[17,18],Eastmanetal.[19]andKakaandPramuanjaroenkij[20].KuznetsovandNield[21]havereportedthepioneeringworkontheboundarylayerflowbehaviorofnanofluidsutilizingthisBuongiorno’smodel.Theyformulatedthemomentum,transportandenergyequationsvalidforanynanofluid.Theequationswerefirstlysimplifiedunderusualboundarylayerassump-tionsandthensolvednumericallyforboundarylayerflowpastaverticalplate.BasedontheideadevelopedbyKuznetsovandNield[21]vari-ousnanofluidboundarylayerflowproblemswereconsideredbythediffer-entresearchersandtheirsolutionswereobtainedbytraditionalnumeri-calmethodsincludingexplicit/implicitfinitedifferenceschemes,shootingmethodwithRunge-Kuttaintegrationtechniques,finiteelementmethodetc.Despiteofthecomplexnatureoftheseequations,theanalyticsolutionbecameanexcessivechallengetotheresearchersfromdifferentquarters.Mustafaetal.[22]wereprobablythefirsttocomputehomotopyanalyticsolutionsforstagnation-pointflowofanincompressiblenanofluidtowardsastretchingsheet.Theresultswerecomparedwiththepreviousnumeri-calstudiesandwereshowntobeincompleteagreement.Homotopybasedanalyticsolutionformixedconvectionflowofnanofluidpastaverticalcylin-derhasbeenprovidedbyDinarvandetal.[23].SeriessolutionsforflowofnanofluidpastanexponentiallystretchingsheetwerecomputedbyNadeemetal.[24].Mustafaetal.[25]providedanalyticsolutionsforsqueezingflowofnanofluidbetweenparalleldisks.Theresultingcouplednonlinearbound-aryvaluesproblemwasformulatedandsolutionscorrespondingtotheop-timalvaluesoftheauxiliaryparameterwerepresented.Ellahietal.[26]developedthehomotopyanalyticsolutionsforflowofthirdgradenanofluidwithtemperaturedependentviscosity.Non-orthogonalstagnation-pointflowofsecondgradenanofluidtowardsastretchingsurfacehasbeenexam- 271October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids263inedbyNadeemetal.[27].TheHAMsolutionforflowofviscousnanofluidwithporousmediumandslipconditionhasbeenconstructedbyZhengetal.[28].TheunsteadyfilmflowofnanofluidinducedbyastretchingsurfaceisanalyzedbyXuetal.[29].HAMisimplementedforthedevel-opmentandanalysisofseriessolution.InthesubsequentsectionswewilldiscusssomeinterestingboundarylayerflowsofnanofluidsinwhichHAMisappliedtoyieldexplicitseriessolutionsofavarietyofnonlinearproblems.6.2.Unsteadyboundary-layerflowofnanofluidpastanimpulsivelystretchingsheetMustafaetal.[30]consideredtheunsteadyincompressibleflowofnanofluidoverastretchingsheet(situatedaty=0withavelocityuw(x)=ax,temperatureTwandnanoparticlesconcentrationCw.Thex-andy-axesaretakenalongandperpendiculartothesheetrespectivelyandtheflowisconfinedtoy≥0.WedenoteT∞andC∞theambienttemperatureandnanoparticlesconcentrationrespectively.Theequationsgoverningtheunsteadyflowofanincompressiblenanofluidcanbesimplifiedunderusualboundarylayerassumptionsas(seeKuznetsovandNield[21])∂u∂v+=0,(6.1)∂x∂y∂u∂u∂u∂2u+u+v=ν,(6.2)∂t∂x∂y∂y2"#∂T∂T∂T∂2T∂C∂TD∂T2∗T+u+v=αm+τDB+,(6.3)∂t∂x∂y∂y2∂y∂yT∞∂y∂C∂C∂C∂2CD∂2TT+u+v=DB+.(6.4)∂t∂x∂y∂y2T∞∂y2Theconsideredboundaryconditionsaret<0:v=0,u=0,T=T∞,C=C∞foranyx,yu=ax,v=0,T=Tw,C=Cwaty=0,(6.5)t≥0:u,v→0,T→T∞,C→C∞asy→∞.Hereuandvarethevelocitycomponentsalongx-andy-directionsre-spectively,νthekinematicviscosity,αmthethermaldiffusivity,DBtheBrownianmotioncoefficient,DTthethermophoreticdiffusioncoefficient 272October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6264T.HayatandM.Mustafaandτ∗=(ρc)/(ρc)theratioofeffectiveheatcapacityofthenanoparti-pfclematerialtoheatcapacityofthefluid.FollowingLiao[4]weseekthesimilaritysolutionofEqs.(7.1)–(7.5)ofthefollowingform(√qψ=xaνξf(ξ,η),η=ay,ξ=1−exp(−τ),τ=at,νξ(6.6)θ(ξ,η)=T−T∞,φ(ξ,η)=C−C∞,Tw−T∞Cw−C∞whereψisthestreamfunction,τthedimensionlesstimeandξandηthesimilarityvariables.SubstitutionofEq.(7.6)intoEqs.(7.2)–(7.5)yieldsthefollowingdifferentialequations2!∂3f∂2f∂fη∂2f∂2f+ξf−+(1−ξ)−ξ=0,(6.7)∂η3∂η2∂η2∂η2∂η∂ξ21∂θ∂θη∂θ∂θ0002+ξf+(1−ξ)−ξ+Nbθφ+Ntθ=0,(6.8)Pr∂η2∂η2∂η∂ξ∂2φ∂φη∂φ∂φNt∂2θ+Scξf+Sc(1−ξ)−ξ+=0,(6.9)∂η2∂η2∂η∂ξNb∂η2subjecttotheboundaryconditions∂ff(ξ,0)=0,=1,θ(ξ,0)=1,φ(ξ,0)=1,∂ηη=0∂f→0,θ(ξ,+∞)→0,φ(ξ,+∞)→0,(6.10)∂ηη=+∞wherePr=ν/αisthePrandtlnumber,Sc=ν/DBtheSchmidtnumber,(ρc)pDB(Cw−C∞)(ρc)pDT(Tw−T∞)Nb=,Nt=(ρc)fν(ρc)fT∞νaretheBrownianmotionparameterandthethermophoresisparameter,respectively.TheskinfrictioncoefficientCx,thelocalNusseltnumberNuandthefxlocalSherwoodnumberSharegivenbyxτwxqwxjwCf=2,Nux=,Sh=,(6.11)ρuwk(Tw−T∞)DT(Tw−T∞)wherethewallshearstress(τw),wallheatflux(qw)andthemassfluxfromthesheet(jw)aredefinedby∂u∂T∂Cτw=µ,qw=−k,jw=−DB.(6.12)∂y∂y∂yy=0y=0y=0 273October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids265Intermsofdimensionlessquantities(7.6)oneobtains∂2f(ξ,η)ξ1/2Re1/2Cx=,xf∂η2η=01/21/2∂θ(ξ,η)Nur=Nux/ξRex=−,∂ηη=01/21/2∂φ(ξ,η)Shr=Sh/ξRex=−,∂ηη=0whereNurandShrdenotethereducedNusseltandSherwoodnumberscharacterizingtherateofheatandnanoparticlestransferatthesheet.Wenowconsiderthefollowingspecialcases:(i)Initialunsteadyflow.Thissolutioncorrespondstoξ=0(τ=0),wheref(0,η)=fu(η),θ(0,η)=θu(η)andφ(0,η)=φu(η).ThusEqs.(7.7)–(7.10)reduceto000100fu+ηfu=0,(6.13)2100100002θu+ηθu+Nbθuφu+Ntθu=0,(6.14)Pr20010Nt00φu+Scηφu+θu=0,(6.15)2Nbsubjecttotheboundaryconditions00fu(0)=0,fu(0)=1,fu(∞)=1,θu(0)=φu(0)=1,θu(∞)=φu(∞)=0,(6.16)whereprimedenotesdifferentiationwithrespecttoη.Eq.(7.15)subjecttotheboundaryconditions(7.18)admitstheexactsolution(seeLiao[4])2η2−ηfu(η)=η1−erf+√1−exp,(6.17)2Π4whereerfistheerrorfunctiondefinedasZs2−z2erf(s)=√edz.(6.18)Π0Thusskinfrictioncoefficientintheinitialunsteadyflowbecomes1/21/2x1ξRexCf=−√.(6.19)Π(ii)Finalsteady-stateflow.Forthissolutionξ=1asτ→∞,wheref(1,η)=fs(η),θ(1,η)=θs(η)andφ(1,η)=φs(η).Eqs.(7.7)–(7.9)now 274October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6266T.HayatandM.Mustafabecome000000f+ff−f2=0,(6.20)ssss10000002θs+fsθs+Nbθsφs+Ntθs=0,(6.21)Pr000Nt00φs+Scfsφs+θs=0,(6.22)Nbsubjecttothesameboundaryconditions(7.18)withindexs.HereEq.(7.22)withtherelevantboundaryconditionshastheexactsolutionfs=1−exp(−η).Basedontheruleofsolutionexpressionandthein-volvedboundaryconditions,theinitialguessesf0,θ0andφ0off(η),θ(η)andφ(η)arechosenasf0(ξ,η)=1−exp(−η),θ0(ξ,η)=φ0(ξ,η)=exp(−η),(6.23)andtheauxiliarylinearoperatorsareselectedas∂3f∂f∂2θ∂2φLf(f)=∂η3−∂η,Lθ(θ)=∂η2−θ,Lφ(φ)=∂η2−φ.(6.24)Ifq∈[0,1]isanembeddingparameterand~denotesthenon-zeroauxiliaryparametersthenthegeneralizedhomotopicequationsareconstructedasfollows:(1−q)Lf[fˆ(ξ,η;q)−f0(η)]hi=q~Nffˆ(ξ,η;q),(6.25)(1−q)Lθ[Θ(ξ,η;q)−θ0(η)]hi=q~Nθfˆ(ξ,η;q),Θ(ξ,η;q),Φ(ξ,η;q),(6.26)(1−q)Lφ[Φ(ξ,η;q)−φ0(η)]hi=q~Nφfˆ(ξ,η;q),Θ(ξ,η;q),Φ(ξ,η;q),(6.27)subjecttotheboundaryconditionsfˆ(ξ,η;q)=0,∂fˆ(ξ,η;q)=1,∂fˆ(ξ,η;q)=0,(6.28)η=0∂η∂ηη=0η→∞Θ(ξ,η;q)|=1,Θ(ξ,η;q)|=0,(6.29)η=0η→∞Φ(ξ,η;q)|=1,Φ(ξ,η;q)|=0,(6.30)η=0η→∞ 275October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids267inwhichthenon-linearoperatorsNf,NθandNφarehiNffˆ(ξ,η;q)∂3fˆ(ξ,η;q)=∂η3!2ˆ2∂f(ξ,η;q)∂fˆ(ξ,η;q)+ξfˆ(ξ,η;q)−∂η2∂η!η∂2fˆ(ξ,η;q)∂2fˆ(ξ,η;q)+(1−ξ)−ξ,(6.31)2∂η2∂η∂ξhiNθfˆ(ξ,η;q),Θ(ξ,η;q),Φ(ξ,η;q)1∂2Θ(ξ,η;q)=Pr∂η2∂Θ(ξ,η;q)∂Θ(ξ,η;q)∂Φ(ξ,η;q)+ξfˆ(η,p)+Nb∂η∂η∂η2∂Θ(ξ,η;q)η∂Θ(ξ,η;q)+Nt+(1−ξ)−ξΘ,(6.32)∂η2∂ηhiNφfˆ(ξ,η;q),Θ(ξ,η;q),Φ(ξ,η;q)∂2Φ(ξ,η;q)=∂η2∂Φ(ξ,η;q)Nt∂2Θ(ξ,η;q)+Scfˆ(ξ,η;q)+∂ηNb∂η2η∂Φ(ξ,η;q)+Sc(1−ξ)−ξΦ.(6.33)2∂ηByTaylor’sseriesX∞fˆ(ξ,η;q)=f(ξ,η)+f(ξ,η)qm,(6.34)0mm=1X∞Θ(ξ,η;q)=θ(ξ,η)+θ(ξ,η)qm,(6.35)0mm=1X∞Φ(ξ,η;q)=φ(ξ,η)+φ(ξ,η)qm,(6.36)0mm=1 276October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6268T.HayatandM.Mustafawheremˆ1∂f(ξ,η;q)fm(ξ,η)=,m!∂ηmp=01∂mΘ(ξ,η;q)θm(ξ,η)=,m!∂ηmq=01∂mΦ(ξ,η;q)φm(ξ,η)=,m!∂ηmq=0thefinalsolutionsareretrievedatq=1.Thefunctionsfm,θmandφmareobtainedfromthedeformationofEqs.(7.27)–(7.32).Explicitlymth-orderdeformationproblemscorrespondingtoEqs.(7.27)–(7.32)areL[f(ξ,η)−χf(ξ,η)]=~Rf(ξ,η),(6.37)fmmm−1fmθLθ[θm(ξ,η)−χmθm−1(ξ,η)]=~θRm(ξ,η),(6.38)φLφ[φm(ξ,η)−χmφm−1(ξ,η)]=~φRm(ξ,η),(6.39)subjecttotheboundaryconditions(∂fm(ξ,η)∂fm(ξ,η)fm(ξ,0)=0,∂θ=0,∂θ=0,η=0η→+∞(6.40)θm(ξ,0)=0,θm(ξ,∞)=0,φm(ξ,0)=0,φm(ξ,∞)=0,where∂3fmX−1∂2f∂f∂ffm−1km−1−kkRm(ξ,η)=∂η3+ξfm−1−k∂η2−∂η∂ηk=0η∂2f∂2fm−1m−1+(1−ξ)−ξ,(6.41)2∂η2∂η∂ξmX−1θ∂θk∂θm−1−k∂φk∂θm−1−k∂θkRm(ξ,η)=ξfm−1−k+Nb+Nt∂η∂η∂η∂η∂ηk=01∂2θη∂θ∂θm−1m−1m−1++(1−ξ)−ξ,(6.42)Pr∂η22∂η∂ξ∂2θmX−1∂φNt∂2θφm−1km−1Rm(ξ,η)=∂η2+Scfm−1−k∂η+Nb∂η2k=0η∂φm−1∂φm−1+Sc(1−ξ)−ξ,(6.43)2∂η∂ξand(0,m≤1,χm=1,m>1. 277October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids269AspointedoutbyLiao[1]thatconvergenceoftheseriessolutionsgiveninEqs.(6.34)–(6.36)largelydependsonthechoiceofauxiliaryparameter~.Inordertoobtainanappropriatevalueof~,the~-curvesforthefunctionsf,θandφat15th-orderofapproximationsareplottedinFig.6.1(a)and6.1(b).Herethepermissiblerangeof~liesintheflatportionof~-curves.Thusitisclearthatintervalofconvergenceforf,θandφis[−0.7,−0.35]whenξ=1/2.Ourcomputationsshowthatseriessolutionsareuniformlyconvergentforallvaluesofparameterswhen~=−0.4.Theinitialunsteadyandsteady-stateproblemsfortemperatureandconcentrationaresolvednumericallythroughcommandNDSolveofthesoftwareMathematica.The10th-orderhomotopysolutioniscomparedwiththenumericalsolutionfortheinitialunsteadyflow(ξ=0).ItiswitnessedthatboththesolutionsareinexcellentagreementfordifferentvaluesofNbandNt(seeFig.6.2).InadditionthenumericalvaluesofreducedNusseltandSherwoodnumbersintheinitialunsteadyflowhavebeencomparedwiththoseobtainedvianumericalsolution(seeTable6.1).The25th-orderHAMsolutionsseemtobeinaverygoodagreementwiththenumericalsolutionsforallthevaluesofNbandNt.Nb=Nt=110,Ξ=12,Pr=Sc=1Nb=Nt=110,Ξ=0,Pr=Sc=1-0.4(a)-0.2(b),0LHΞ-0.6,0LΦ'HΞ-0.4,-0.8Φ',,0LΞ-1.0,0LΞ'HΘ'HΘ-0.6,-1.2f''HΞ,0L,,0L-1.4Φ'HΞ,0L,0LHΞf''HΞ,0LHΞ''''-0.8f-1.6Θ'HΞ,0LfΦ'HΞ,0L-1.8Θ'HΞ,0L-1.0-1.0-0.8-0.6-0.4-0.20.0-1.0-0.8-0.6-0.4-0.20.0ÑÑ(a)(b)Fig.6.1.~-curvesforthefunctionsf,θandφwhen(a)ξ=1/2and(b)ξ=0.Fig.6.3(a)plotsthetemperatureprofilesfordifferentvaluesofdimen-sionlesstimeτ.Wenoticethatprofilescontinuouslydevelopfromrestasτincreasesuntilthesteady-statesituationisachieved.Thisobservationleadstotheconclusionthatthermalboundarylayerthickensaswemovefrominitialunsteadyflow(ξ=0)tothefinalsteady-stateflow(ξ=1).Furtheritisrevealedthatθ(0,+∞)oftheinitialsolutiontendstozeromuchfasterthanθ(1,+∞)ofthesteady-statesolution.Nanoparticlescon-centrationprofilesφfordifferent(dimensionless)timesarepresentedin 278October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6270T.HayatandM.MustafaPr=Sc=1.0Nb=0.2,Pr=Sc=1.01.01.0(a)(b)0.80.8Nb,Nt=0.1,0.5,1,1.5L0.6L0.6Nt=0.01,0.1,0.2,0.3ΗΗH0,0,Θ0.4ΦH0.40.20.20.00.001234560123456ΗΗ(a)(b)Fig.6.2.ComparisonofHAMandnumericalsolutionsfortheinitialunsteadyflowatξ=0.Lines:10th-orderHAMsolutionsat~=−0.4;Points:Numericalsolutions.Nb=Nt=0.1,Pr=Sc=1Nb=Nt=0.1,Pr=Sc=11.0HaL1.0HbL0.80.8Τ=0.01,0.1,0.25,0.5,1,10L0.6Τ=0.01,0.1,0.25,0.5,1,10L0.6,Η,ΗΞHΞΘΦH0.40.40.20.20.00.001234501234567ΗΞΗΞ(a)(b)Fig.6.3.Temperatureandconcentrationprofilesforvariousvaluesofunsteadinesspa-rameterτ.Fig.6.3(b).Thebehaviorofdimensionlesstimeτonφisquitesimilartothataccountedfortemperatureθ.CombinedbehaviorofNbandNtonthereducedNusseltnumberissketchedinFig.6.4(a).ThereisadecreaseinthemagnitudeofNurwhenNbandNtareincreased.Interestingly,thisreductionisuniforminbothstrongerandweakerBrownianmotionandthermophoresiseffects.ThereducedSherwoodnumberShrcorrespondingtothedatagiveninFig.6.4(a)hasbeenplottedin6.4(b).WenoticethatforaweakerBrownianmotion,thereducedSherwoodnumberdrasticallydecreaseswithanincreaseinNb.HoweverthereducedSherwoodnumberisnegligiblyaffectedforsufficientlystrongerBrownianmotioneffect.Ontheotherhand,thereisaslightincreaseinthemasstransferatthesheetwithanincreaseinBrownianmotionforaweakerthermophoreticeffect.How-everwhenthestrengthofthermophoreticeffectisincreasedi.e.,Ntchanges 279October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids271Ξ=0.5,Pr=Sc=1Ξ=0.5,Pr=Sc=10.55HaL0.50.50Nb=0.1,0.3,0.5,0.7,0.9,1.10.450.0,0LLΞ0.40,0LLHbL'HHΞ-0.5H-Θ0.35H-Φ'Nur0.30Shr-1.00.25-1.50.20Nb=0.1,0.2,0.3,0.4,0.5,0.60.51.01.52.00.51.01.52.0NtNt(a)(b)Fig.6.4.DimensionlessheatandmasstransferratesversusNtfordifferentvaluesofNb.from0.1to2,thereisamassiveincreaseinthenanoparticlesconcentra-tionatthesheetwithanincreaseinNb.ThisisnotsurprisingsincetheincreasingvaluesofNtgiverisetotheeffectivemovementofnanoparticlesfromthesheettothefluid.Table6.1.Comparisonof25th-orderHAMsolutionswiththenumericalsolutionsforthesteady-stateflow(ξ=1).NurShrNbNtHAMNumericalHAMNumerical0.10.10.53727440.53727440.22445100.22445540.50.43514300.43514270.52756730.52756841.00.32828220.32828140.56349620.56349681.50.24286160.24286040.57415300.57415350.50.50.38615840.38615340.34718420.34718691.00.33433310.33431090.18889240.18889421.50.29132130.29129900.08420040.08424422.00.25564060.25558930.01632270.01640966.3.AxisymmetricflowofnanofluidoveraradiallystretchingsheetwithconvectiveboundaryconditionsTheflowofnanofluid(obeyingtheBuongiornomodel[15])pastaradiallystretchingsheethasbeeninvestigatedbyMustafaetal.[31].Thecylin-dricalcoordinatessystem(r,θ,z)isadopted.TheazimuthalcomponentvofvelocityV=[u,v,w]vanishesidenticallyduetotherotationalsym-metryoftheflow(∂/∂θ≡0).Thesheetatz=0isstretchedalongthe 280October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6272T.HayatandM.Mustafaradialdirectionwiththevelocityuw=cr.HereTwdenotestheconvec-tivetemperatureofthesheetwhileCwisthenanoparticlesconcentrationatthesheet.TheambienttemperatureandconcentrationaredenotedbyT∞andC∞respectively.UsingthemathematicalmodelformulatedbyKuznetsovandNield[21]thetransportequationsincylindricalcoordinatesareasunder∂uu∂w++=0,(6.44)∂rr∂z∂u∂u1∂pˆ∂2u∂2u1∂uuu+w=−+ν++−,(6.45)∂r∂zρf∂r∂r2∂z2r∂rr2∂w∂w1∂pˆ∂2w∂2w1∂wu+w=−+ν++,(6.46)∂r∂zρf∂z∂r2∂z2r∂r∂T∂T∂2T1∂T∂2Tu+w=α++∂r∂z∂r2r∂r∂z2∂C∂T∂C∂T+τDB+∂r∂r∂z∂z""22##DT∂T∂T+τ+,(6.47)T∞∂r∂z∂C∂C∂2C1∂C∂2Cu+w=DB++∂r∂z∂r2r∂r∂z2D∂2T1∂T∂2TT+++.(6.48)T∞∂r2r∂r∂z2Theboundaryconditionsfortheproblemunderconsiderationare∂Tu=uw(r)=cr,−k=hs(Tw−T),C=Cwaty=0,∂zu→0,T→T∞,C→C∞asy→∞,(6.49)inwhichuandwarethevelocitycomponentsalongrandzdirectionsrespectively,ˆpisthepressure,νisthekinematicviscosity,αisthethermaldiffusivity,DBistheBrownianmotioncoefficient,DTisthethermophoreticdiffusioncoefficient,kisthethermalconductivityandτ=(ρc)p/(ρc)fistheratioofeffectiveheatcapacityofthenanoparticlematerialtoheat 281October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids273capacityofthefluid.Implementingthefollowingtransformationsrc0√η=z,u=crf(η),w=−2cνf(η),νT−T∞C−C∞θ(η)=,φ(η)=,(6.50)Tw−T∞Cw−C∞intoEqs.(6.45)and(6.46)andeliminatingthepressuregradientfromtheresultingequationswefinallyobtain(seeAriel[32])000000f+2ff−f2=0,(6.51)andEqs.(6.47)and(6.48)inviewofEq.(6.50)become10000002θ+2fθ+Nbθφ+Ntθ=0,(6.52)Pr000Nb00φ+2Lefφ+θ=0,(6.53)Ntsubjecttotheboundaryconditions00f(0)=0,f(0)=1,θ(0)=−Bi(1−θ(0)),φ(0)=1,0f(∞)→0,θ(∞)→0,φ(∞)→0.(6.54)wherePr=ν/αisthePrandtlnumber,Le=ν/DBistheLewisnumber,NbistheBrownianmotionparameter,NtisthethermophoresisparameterpandBi=hsνistheBiotnumber.Itisworthmentioningherethatkctheboundaryconditions(6.54)reducetothethecaseofconstantwalltemperature(θ(0)=1)whenBi→∞.FurtherNb=Nt=0correspondstothesituationinwhichBrownianmotionandthermophoresiseffectsareabsent.TheskinfrictioncoefficientCf,thelocalNusseltnumberNuandthelocalSherwoodnumberSharegivenbyµ(∂u+∂w)r(∂T)r(∂C)∂z∂rz=0∂zz=0∂zz=0Cf=,Nu=−,Sh=−,ρu2wk(Tw−T∞)DB(Cw−C∞)Using(6.50)intheaboveformulagives0000Re1/2C=f(0),Nu/Re1/2=−θ(0)=Nur,Sh/Re1/2=−φ(0)=Shr,rfrrwhereRer=uwr/νdenotesthelocalReynoldsnumber.TheanalyticsolutionsofEqs.(6.51)–(6.53)subjecttotheboundaryconditions(6.54)havebeenobtainedbyhomotopyanalysismethod(HAM).Followinginitialguessesandlinearoperatorsforthefunctionsf,θandφareselectedBif0(η)=1−exp(−η),θ0(η)=exp(−η),φ0=exp(−η),(6.55)1+Bi 282October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6274T.HayatandM.Mustafa00000000Lf(f)=f−f,Lθ(θ)=θ−θ,Lφ(φ)=φ−φ.(6.56)Wedenotethenon-zeroauxiliaryparametersforthefunctionsf,θandφby~f,~θand~φrespectively.InviewoftheEqs.(6.55)and(6.56),thehigherorderdeformationequationsareformulatedwhicharesolvedbysymboliccomputationalsoftwareMathematica.Theadmissiblevaluesoftheauxiliaryparameterscanbedeterminedfromthelinesegmentparallelto~-axisintheFig.6.5.Itcanbeseenthatthepermissiblerangeofvaluesof~f,~θand~φare−1≤~f≤−0.4,−1.4≤~θ≤−0.5and−1.2≤~φ≤−0.7respectively.Theseriessolutionsarefoundtobeconvergentinthewholespatialdomainwhen~f=~θ=~φ=−1(seeTable6.2).FurtherthegoverningdifferentialsystemhasalsobeensolvednumericallythroughthesoftwareMathematica.Thissoftwareusestheshootingmethodasadefaultmethodforthenumericalsolutionofnonlinearboundaryvalueproblems.Fig.6.6isdisplayedjusttoseethecomparisonofanalyticandnumericalsolutions.Itisclearthatdataobtainedbyboththesolutionsisvirtuallysimilardemonstratingthevalidationofcurrentfindings.ThesimultaneousbehaviorofNbandNtonthereducedNusseltnumberissketchedinFig.6.7(a).ThereisadecreaseinthemagnitudeofNurwhenNbandNtareincreased.Interestingly,thisreductionisuniforminbothstrongerandweakerBrownianmotionandthermophoresiseffects.Fig.6.7(b)plotsthereducedsherwoodnumberShragainstNtfordifferentvaluesofNb.Itisseenthatforaweakerthermophoreticeffect,thechangeintheBrownianmotionparameterslightlyaffectsthereducedSherwoodnumber.HoweverthenanoparticlesconcentrationatthesheeteffectivelyincreaseswhenNbisincreasedfrom0.1to0.5.MoreoverthereducedSherwoodnumberisfoundtodecreaseuponincreasingthestrengthofthermophoreticeffect.ThisisnotsurprisingsincetheincreasingvaluesofNtgiverisetotheeffectivemovementofnanoparticlesfromthesheettothefluid.Table6.3showsthenumericalvaluesofNurandShrfordifferentvaluesofPr,LeandBi.AsseenearlierinthegraphicalresultsthatthermalboundarylayerthicknessdecreasesandprofilesgetclosertotheboundarywhenPrisincreased.ThereducedNusseltandSherwoodnumbers(whichareproportionaltotheinitialslopes)arethereforeincreasingfunctionsofPrandLerespectively.FurtheritisnoticedthatanincreaseintheBiotnumberBicorrespondstothehigherconvectiveheatingatthesheetwhichresultsintheincreaseoftherateofheattransferatthesheet.MoreoveritisobservedthatlargervaluesofBihavealittleimpactonthereducedNusseltnumberNur. 283October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids275Nb=Nt=0.1,Bi=0.5,Pr=Le=1.5-1.1735-0.1(a)(b)-0.2-1.1736-0.3-1.17370L'HΦΦ'H0LH0L'',-0.4f0L'HΘ'H0L-1.1738Θ-0.5-1.1739-0.6-1.1740-0.7-1.2-1.0-0.8-0.6-0.4-0.2-1.5-1.0-0.50.0ÑΘ,ÑΦÑf(a)(b)Fig.6.5.~-curvesforthefunctionsf,θandφ.Nt=0.1,Bi=0.5,Pr=Le=1.0Nt=0.1,Bi=0.5,Pr=Le=1.0(a)1.0(b)0.60.8LL0.6HΗ0.4Nb=0.1,1.0,2.0HΗNb=0.1,0.2,2ΘΦ0.40.20.20.00.002468100246810ΗΗ(a)(b)Fig.6.6.Comparisonof10th-orderHAMsolutionswiththenumericalsolution:Lines:HAMsolution;Points:Numericalsolution.6.4.SqueezingflowofnanofluidbetweenparalleldisksThestudyofsqueezingflowsbetweenparalleldiskshasparticularrelevanceinmanyindustrialapplicationswhichincludepolymerprocessing,compres-sion,injectionmodeling,transientloadingofmechanicalcomponentsandthesqueezedfilmsinpowertransmission.Theapplicationofmagneticfieldinsuchflowsallowsustopreventtheunpredictabledeviationoflubricationviscositywithtemperatureincertainextremeoperatingconditions.Theseminalworkonthetopicunderlubricationapproximationhasbeenstud-iedbyStefan[33].Mustafaetal.[25]recentlyconsideredthesqueezingflowofnanofluidbetweenparalleldisksinwhichoneoftheseisimper-meableandtheotherisporous.Thedisksareseparatedbyadistanceh(t)=H(1−at)1/2.AmagneticfieldofstrengthB(t)=B(1−at)−1/2is0appliedperpendiculartothedisks.HereTwandCwdenotethetempera-tureandnanoparticlesconcentrationatthelowerdiskwhilethetempera- 284October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6276T.HayatandM.MustafaBi=0.5,Pr=Le=1.5Bi=0.5,Pr=Le=1.50.28HaLHbLNb=0.1,0.2,0.3,0.4,0.5,0.60.50.260.00LL0LL'H'H0.24H-ΘH-Φ-0.5Nur0.22Shr-1.0Nb=0.1,0.15,0.2,0.3,0.4,0.50.20-1.50.51.01.52.00.51.01.52.0NtNt(a)(b)Fig.6.7.DimensionlessheatandmasstransferratesfordifferentvaluesofNbandNt.Table6.2.ConvergencerateofHAMsolutionswhenNb=Nt=0.1,Pr=Le=1and~f=~θ=~φ=−0.8.Parenthesisshowsthenumericalsolutions.1/21/21/2OrderofRerCfNu/RerSh/Rerapproximations1−1.1666670.31456790.73333335−1.1738970.30784420.654052510−1.1737560.30782300.650142115−1.1737120.30782250.650153020−1.1737210.30782250.650154125−1.1737210.30782250.650145135−1.1737210.30782250.6501451(−1.17372)(0.307825)(0.650145)tureandconcentrationattheupperdiskareThandChrespectively.Theupperdiskatz=h(t)movestowardsorawayfromthestationarylowerdiskwiththevelocitydh/dt.Theequationsgoverningtheconservationofmass,momentum,energyandnanoparticlesconcentrationincylindricalcoordinatesaresimilartothoseconsideredintheprevioussectionwithanadditionofmagneticfieldtermi.e.,−σ/ρB(t)2uinther-componentofthemomentumequation.Theboundaryconditionsforthisproblemare(seeDomairryandAziz[34]andJoneidietal.[35].)dhu=0,w=,T=Th,C=Chatz=h(t),dtw0u=0,w=−√,T=Tw,C=Cwatz=0.(6.57)1−at 285October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids277Table6.3.ValuesofreducedNusseltandSherwoodnumbers00Nur=−θ(0)andShr=−φ(0)fordifferentvaluesofPr,LeandBiwhenNb=Nt=0.1.PrLeBiNurShr0.410.50.2300780.7108030.720.2819480.67191720.3539340.605337100.4226500.5153910.40.3105570.1936140.70.3089000.44636720.3059051.160300100.3030213.2228901.00.10.0889800.7935450.40.2669010.6769281.00.4434520.5615462.00.5678590.480446Usingthefollowingtransformationsar0aHzu=f(η),w=−√f(η),η=√,2(1−at)1−atH1−atB0T−ThC−ChB(t)=√,θ=,φ=,(6.58)1−atTw−ThCw−Chintotherelevantequationsandtheneliminatingthepressuregradientfromtheresultingequationswefinallyobtain000000000000002f−S(ηf+3f−2ff)−Mf=0.(6.59)Nowdimensionlessenergyandequationofmasstransferbecome0000000θ+PrS(2fθ−ηθ)+PrNbθφ+PrNtθ2=0,(6.60)0000Nt00φ+LeS(2fφ−ηφ)+θ=0,(6.61)Nbwiththeboundaryconditions0f(0)=A,f(0)=0,θ(0)=φ(0)=1,10f(1)=,f(1)=0,θ(1)=φ(1)=0,(6.62)2whereSisthesqueezenumber,Aisthesuction/blowingparameter,MistheHartmannumber,NbistheBrownianmotionparameter,Ntisthe 286October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6278T.HayatandM.Mustafathermophoreticparameter,PristhePrandtlnumberandLeistheLewisnumberwhicharedefinedasrwaH2σB2H2νν00A=,S=,M=,Pr=,Le=,aH2νναDe(ρc)pDB(Cw−Ch)(ρc)pDT(Tw−Th)Nb=,Nt=.(ρc)fν(ρc)fTmνThecontinuityequationisidenticallysatisfied.ItisworthmentioningherethatA>0indicatesthesuctionoffluidfromthelowerdiskwhileA<0representsinjectionflow.ForNb=Nt=0,theproblemreducestothecaseofordinaryfluid(inwhichBrownianmotionandthermophoreticeffectsarenegligible).Thephysicalquantitiesofinterestaretheskinfrictioncoeffi-cientCfr,reducedNusseltnumberNurandreducedSherwoodnumberShrwhicharedefinedbyτrz|z=h(t)HqwHjwCfr=2,Nu=,Sh=,−aHk(Tw−Th)DB(Cw−Ch)ρ2(1−at)1/2where∂u∂wτrz=µ+,∂z∂rz=h(t)∂Tqw=−k,∂zz=h(t)∂Cjw=−DB.∂zz=h(t)Intermsofvariables(6.58)wehave2H001/202RerCfr=f(1),Nur=(1−at)Nu=−θ(1),r1/21/20raH(1−at)Shr=(1−at)Sh=−φ(1),Rer=.2νTheruleofsolutionexpressionandtheinvolvedboundaryconditionsdirectustoselecttheinitialguesses312f0(η)=(−1+2A)η+(3−6A)η+A,θ0(η)=φ0(η)=1−η,(6.63)2andtheauxiliarylinearoperatorsd4dd2d2Lf≡dη4−dη,Lθ≡dη2,Lφ≡dη2,(6.64) 287October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids279respectively.Aftersolvingthehigherorderdeformationproblemsweseektheoptimalvaluesoftheconvergencecontrolparameters.Inthisparticularcase~1denotestheauxiliaryparameterforthefunctionfwhile~2istheauxiliaryparameterforthefunctionsθandφ.Todeterminetheoptimalvaluesoftheseparameters(thosewillensurerapidconvergence)wedefinetheaveragedresidualserrorsforthefunctionsf,θandφas(seeLiao[13]fordetails)21XKXmEm,1(~1)=Nffj(i∆x),(6.65)Ki=0j=021XKXmXmEm,2(~2)=Nθθj(i∆x),φj(i∆x),(6.66)Ki=0j=0j=021XKXmXmEm,2(~2)=Nφθj(i∆x),φj(i∆x),(6.67)Ki=0j=0j=0where∆x=1/KandK=20.HereNf,NθandNφdenotethenon-linearoperatorscorrespondingtotheequations(6.59)–(6.61).Theaboveaveragedresidualerrorscanbeplottedversustherespectiveauxiliarypa-rameterstodeterminetheconvergenceregionofthesolutions.Foragivenorderofapproximationsm,theoptimalvaluesof~1and~2canbedeter-minedbyminimizingtheaveragedresidualerrorgiveninEqs.(6.65)–(6.67)usingthecommandMinimizeofthesoftwareMathematica8.0.InTa-bles6.4and6.5,theoptimalvaluesof~1and~2forthefunctionsf,θandφcorrespondingtovariousvaluesoftheparametersaregiven.HerethecorrespondingaveragedresidualsarerepresentedasE∗,E∗andE∗m,1m,2m,3respectively.ForafurthercheckattheaccuracyofourcomputationswecomparedHAMsolutionswiththenumericalsolutionsobtainedthroughthecommandNDSolveofthesoftwareMathematica8.0.TheresultsareinanexcellentagreementwhichcanbeseenfromTables6.6and6.7.More-overthevelocityprofilesanddimensionlessheatandmasstransferrateshavebeenportrayedintheFigs.6.8and6.9. 288October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6280T.HayatandM.MustafaTable6.4.Optimalvaluesof~1fordifferentphysicalparameters.A=2A=−2MSOptimalvalueofE∗OptimalvalueE∗m,1m,1of~1of~101−0.8683.06×10−14−0.8898.32×10−122−0.8552.23×10−15−0.8331.00×10−123−0.8561.21×10−15−0.7729.18×10−145−0.7911.04×10−11−0.7201.64×10−1111/10−0.9048.97×10−24−0.8686.16×10−241/2−0.9163.38×10−22−0.9061.05×10−201−0.8621.68×10−14−0.8744.92×10−122−0.7531.75×10−6−0.9061.05×10−20Table6.5.Optimalvaluesof~2fordifferentphysicalparametersincaseofM=S=1,A=2and~1=−0.862.NbNtOptimalvalueE∗OptimalvalueE∗m,2m,3of~2forθof~2forφ1/101/10−0.9081.47×10−11−0.9362.51×10−101/2−0.9151.62×10−13−0.9411.18×10−121−0.9319.10×10−13−0.8944.11×10−133/2−0.9214.51×10−12−0.8862.86×10−131/2−0.9612.93×10−10−0.9132.87×10−101−1.0211.21×10−8−0.9532.97×10−83/2−0.9552.37×10−6−1.0271.18×10−52−0.8201.00×10−4−0.9164.12×10−46.5.Boundarylayerflowofnon-NewtoniannanofluidoverastretchingsheetManyindustrialfluidssuchasclaycoatingandsuspensions,cosmeticprod-ucts,oilsandgrease,drillingmuds,coalwaterorcoal-oilslurriesandcertainpaintsexhibitanonlinearrelationshipbetweenstressandshearrateandthesearetreatedasnon-Newtonian.Thediversecharacteristicsofsuchfluidsleadtothefactthatthesecannotbedescribedbyusingasinglecon-stitutiverelationshipbetweenstressandshearrate.Generally,theconsti-tutiveequationsofnon-Newtonianfluidsyieldthemathematicalproblemswhichareofthehigherorders,morenonlinearandcomplicatedthanthecorrespondingproblemforaNavier–Stokesfluid.Therheologicalmodelofsecondgradehasbeenwidelyaddressedbytheresearchersinviewofitssimplicity.Thismodelcanpredicttheeffectsofnormalstressesintheflow. 289October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids281Table6.6.Resultsofskinfrictioncoefficientf00(1)fordifferentvaluesofMandS.f00(1)MSHAMnumerical017.533165797.5331657928.263872318.2638723039.097325729.09732573511.349289011.349289011/108.975523948.975523941/28.34924788.3492457817.721946017.7219460126.940773266.94077334Table6.7.ResultsofreducedNusseltnumberNurandreducedSherwoodnumberShrfordifferentvaluesofNbandNtincaseofA=2,M=S=Pr=Le=1and~1=−0.862.NurShrNbNtHAMnumericalHAMnumerical1/101/100.526285400.526285390.866046660.866046661/20.634332530.634332530.530128140.5301281410.786363850.786363840.486039190.486039193/20.955699550.955699540.469861570.469861571/21.176821191.176821190.401807180.4018071811.485812071.485811940.126193340.126193303/21.823052761.823053540.390830800.3908398822.179159912.179227951.167777231.16800856Mustafaetal.[36]consideredthesteadyflowandheattransferofsecondgradefluidwhenthenanoparticlesareintroducedintothebasefluid.ThefluidiselectricallyconductinginthepresenceofanappliedmagneticfieldofstrengthB0.Thex-andy-axesaretakenalongandperpendiculartothesheetrespectively.Thevelocityofthestretchingsheetisuw=cx(wherec>0isapositiveconstant).Tfdenotestheconvectivesurfacetem-peraturewhileT∞istheambienttemperaturesuchthatTf>T∞.ThenanoparticlesconcentrationatthesheetisdenotedbyCw.HoweverC∞istheambientconcentration.Thustheboundarylayerequationsgovern-ingtheconservationofmass,momentum,energyandnanoparticlesvolume 290October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6282T.HayatandM.MustafaM=1S=144HaLHbL3A=-23A=-222LLHΗ1S=0.1,2HΗ1f'f'M=0,500-1-1A=2A=2-2-20.00.20.40.60.81.00.00.20.40.60.81.0ΗΗ(a)(b)Fig.6.8.Velocityfieldf0fordifferentvaluesofSandM.A=2,Pr=M=S=Le=1A=2,Pr=M=S=Le=11.8HaLHbL2.51.6Nb=0.1,0.15,0.2,0.3,0.4,0.5Nb=0.1,0.5,1.0,1.5,21LL2.01LL1.4'H'H1.2H=-Θ1.5H=-Φ1.0NurShr1.00.80.60.50.51.01.52.00.10.20.30.40.5NtNt(a)(b)Fig.6.9.DimensionlessheatandmasstransferratesfordifferentvaluesofNbandNt.fractionare(seeCortell[37]formoredetails)∂u∂v+=0,(6.68)∂x∂y∂u∂u∂2uα∂u∂2u∂3u∂u∂2v∂3u1u+v=ν++u++v∂x∂y∂y2ρ∂x∂y2∂x∂y2∂y∂y2∂y3σ2−B0u,(6.69)ρ"#∂T∂T∂2T∂C∂TD∂T2Tu+v=ξ+τDB+,(6.70)∂x∂y∂y2∂y∂yT∞∂y∂C∂C∂2CD∂2TTu+v=DB+.(6.71)∂x∂y∂y2T∞∂y2 291October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids283Therelevantboundaryconditionsare∂Tu=uw(x)=cx,−k=hs(Tf−T),C=Cwaty=0,∂y∂uu→0,→0,T→T∞,C→C∞asy→∞.(6.72)∂yHereα1isthematerialfluidparameter,kisthethermalconductivityandhsistheheattransfercoefficientandξisthethermaldiffusivity.Therestofthequantitiesarealreadydefinedintheprevioussections.Invokingthefollowingdimensionlessvariablesrc0√η=y,u=cxf(η),v=−cνf(η),νT−T∞C−C∞θ(η)=,φ(η)=,(6.73)Tf−T∞Cw−C∞equation(6.68)isidenticallysatisfiedandEqs.(6.69)–(6.71)takethefol-lowingforms:000000000000022iv2f+ff−f+α2ff−f−ff−Mf=0,(6.74)10000002θ+fθ+Nbθφ+Ntθ=0,(6.75)Pr000Nt00φ+Lefφ+θ=0,(6.76)Nbsubjecttotheboundaryconditions00f(0)=0,f(0)=1,θ(0)=−Bi(1−θ(0)),φ(0)=1,0f(∞)→0,θ(∞)→0,φ(∞)→0,(6.77)wherercα1ννhsνα=,Pr=,Le=,Bi=,µξDBkc(ρc)pDB(Cw−C∞)(ρc)pDT(Tf−T∞)Nb=,Nt=.(6.78)(ρc)fν(ρc)fT∞νHereαisthedimensionlesssecondgradefluidparameter,PristhePrandtlnumber,LeistheLewisnumber,NbistheBrownianmotionparameter,Ntisthethermophoresisparameter,andBiistheBiotnumber,respectively.Itisworthmentioningherethatwhenα=0thegoverningEqs.(6.74)–(6.77)forviscousfluidcase(originallyconsideredbyMakindeandAziz[38]arededuced.FurtherNt=Nb=0correspondstothecasewhenBrownian 292October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6284T.HayatandM.Mustafamotionandthermophoreticeffectsareabsent.Moreovertheanalysisforconstantwalltemperature(θ(0)=1)canberecoveredbysettingBi→∞.HerethephysicalquantitiesofinterestaretheskinfrictioncoefficientCf,thelocalNusseltnumberNuandthelocalSherwoodnumberShdefinedbyτwxqwxjwCf=,Nu=−,Sh=−,(6.79)ρu2wk(Tw−T∞)DB(Cw−C∞)whereτwisthewallskinfriction,qwisthesurfaceheatfluxandjwisthewallmassfluxgivenby∂u∂2u∂2u∂v∂uτw=µ+α1u+v−2,∂y∂x∂y∂y2∂y∂yy=0∂T∂Cqw=k,jw=DB.∂y∂yy=0y=0UsingEq.(6.73)inEq.(6.79)oneobtains0001/21/2RexCf=(1+3α)f(0),Nu/Rex=−θ(0)=Nur,0Sh/Re1/2=−φ(0)=Shr.xItisworthmentioningherethattheexactsolutionofEq.(6.74)satisfyingtheboundaryconditionsgiveninEq.(6.77)isgivenby(seeCortell[37])r1−exp(−Kη)1+M2f(η)=,K=.(6.80)K1+αTheanalyticsolutionsofEqs.(6.74)–(6.76)subjecttotheboundarycon-ditions(6.77)havebeencomputedbyhomotopyanalysismethod(HAM).Theappropriateinitialguessessatisfyingtherelevantboundaryconditionsforf,θandφarechosenasBif0(η)=1−exp(−η),θ0(η)=exp(−η),φ0(η)=exp(−η).(6.81)1+BiFurtherwehaveselectedthefollowingauxiliarylinearoperators00000000Lf(f)=f−f,Lθ(θ)=θ−θ,Lφ(φ)=φ−φ.(6.82)ThehigherorderdeformationproblemscorrespondingtoEqs.(6.74)–(6.76)havebeensolvedusingcomputationalsoftwareMathematica.Wehavede-notedtheso-callednon-zeroauxiliaryparametersforthefunctionsf,θandφby~f,~θand~φrespectively.Theappropriatevaluesoftheseparam-etershavebeendeterminedbyplottingthe~-curvesinFig.6.10.Itisfoundthatadmissiblevaluesoftheseparametersare−0.9≤~f≤−0.3, 293October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids285Α=0.5,M=1.0Nb=Nt=0.1,Α=Bi=0.5,M=1.0,Pr=Le=1.5-1.10-0.1(a)(b)-1.12-0.2-0.3-1.140L'HΦΦ'H0LH0L'',-0.4f0L-1.16'HΘΘ'H0L-0.5-1.18-0.6-1.20-0.7-1.2-1.0-0.8-0.6-0.4-0.20.0-1.0-0.8-0.6-0.4-0.20.0ÑΘ,ÑΦÑf(a)(b)Fig.6.10.~-curvesforthefunctionsf,θandφat20th-orderofapproximations.−1.1≤~θ≤−0.3and−1≤~φ≤−0.4respectively.Aftersubstitut-ingtheexactsolutiongiveninEq.(6.80)intodimensionlessequationsfortemperatureandconcentrationgivenin(6.75)and(6.76)respectively,theresultingdifferentialequationshavebeensolvednumericallybyusingMathematica.ThenumericalsolutionsagreewellwiththehomotopysolutionsascanbeseenfromFig.6.11.Α=0.5,Nt=0.1,Bi=0.5,M=Pr=Le=1.0Α=0.5,Nt=0.1,Bi=0.5,M=Pr=Le=1.01.00.7HaLHbL0.60.80.5LL0.6HΗ0.4Nb=0.1,1,2HΗNb=0.1,0.2,2ΘΦ0.30.40.20.20.10.00.00246802468ΗΗ(a)(b)Fig.6.11.Comparisonofnumericalandseriessolutions.Points:Numericalsolutions,Lines:10th-orderHAMsolutionsat~f=−0.7,~θ=~φ=−1.2.Figure6.2displaysthecomparisonofnumericalandhomotopysolutionsfordifferentvaluesofparameters.Averygoodagreementisfoundbetweenthesolutions.Figure6.12plotsthetemperatureandconcentrationprofilesagainstthesimilarityvariableηfordifferentvaluesofparameters.Itisclearfromthisfigurethatanincreaseinthesecondgradefluidparame-terorequivalentlythenormalstressdifferencesshiftstheprofilestowardstheboundarycausingareductioninthethermalboundarylayerthickness.Thetemperatureprofilesmonotonicallydecreasewithanincreaseinηand 294October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6286T.HayatandM.MustafaNb=Nt=0.1,Bi=0.5,Pr=Le=1.0Nb=Nt=0.1,Bi=0.5,Pr=Le=M=1.01.0(a)(a)Α=0.00.280.8Α=0.5Α=2.00LL'HL0.6Α=100.27HΗH-ΘΦ0.4Nur0.260.2Α=0,0.5,1.5,3,40.250.002468100.00.51.01.52.0ΗM(a)(b)Fig.6.12.Temperatureandnanoparticlesconcentrationfordifferentvaluesofα.Nb=Nt=0.1,Bi=0.5,Pr=Le=1.0Nb=Nt=0.1,Bi=0.5,Pr=Le=1.00.60(a)(b)0.550.280LL0LL0.50'H'H0.27H-ΘH-Φ0.45NurShr0.260.40Α=0,0.5,1.5,3,40.35Α=0,0.5,1.5,3,40.250.00.51.01.52.00.00.51.01.52.0MM(a)(b)Fig.6.13.DimensionlessheatandmasstransferratesfordifferentvaluesofαandM.asymptoticallyreachthezerovalueasη→∞representingthecharacter-isticofboundarylayerflow.Figure6.13isdisplayedtoseetheinfluenceofsecondgrade(viscoelastic)parameteronthereducedNusseltandSher-woodnumbers.Wehaveseenearlierinthegraphicalresultsthatthermalandconcentrationboundarylayersreducewhentheviscoelasticeffectsin-tensify.Thisreductioniscompensatedwiththeincreaseintherateofheatandmasstransferattheboundingsurface.Table6.8providesthenumer-icalvaluesofreducedNusseltnumberforsomevaluesofNbandNt.InaccordancewithMakindeandAziz[38]aslightdecreaseinthemagnitudeofNurwithanincreaseinNtisobservedforaweakerBrownianmotion.SuchreductioninNurbecomesprominentforlargerBrownianmotionpa-rameterNb.ThevaluesofreducedSherwoodnumberShrcorrespondingtothedatagiveninTable6.1havebeenprovidedinTable6.9.ThereisadecreaseinthemagnitudeofShrwithanincreaseinNt.ThisdecreaseismorepronouncedforastrongerBrownianmotion.Furtherthereduced 295October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.6HAMforSomeBoundaryLayerFlowsofNanofluids287Table6.8.Comparisonofnumericalresults(usingshootingmethod)ofreducedNusseltnumberNur=−θ0(0)fordifferentvaluesofNbandNtwiththe25th-orderHAMapproximations(inparenthesis,us-ing~f=−0.7,~θ=~φ=−1.2)incaseofα=Bi=0.5andPr=Le=M=1.0.NurNtNb=0.2Nb=0.4Nb=0.60.10.247493(0.24749)0.234266(0.23427)0.227170(0.22717)0.20.245638(0.24564)0.232295(0.23230)0.218641(0.21864)0.30.243761(0.24376)0.230303(0.23030)0.216546(0.21655)0.40.241863(0.24186)0.228292(0.22829)0.214434(0.21443)0.50.239944(0.23994)0.226260(0.22626)0.212305(0.21231)Table6.9.Comparisonofnumericalresults(usingshootingmethod)ofreducedNusseltnumberShr=−φ0(0)fordifferentvaluesofNbandNtwiththe25th-orderHAMapproximations(inparenthesis,us-ing~f=−0.7,~θ=~φ=−1.2)incaseofα=Bi=0.5andPr=Le=M=1.0.ShrNtNb=0.2Nb=0.4Nb=0.60.10.464939(0.46494)0.510999(0.51010)0.526370(0.52637)0.20.381010(0.38101)0.472329(0.47233)0.502799(0.50280)0.30.298766(0.29877)0.434532(0.43453)0.479824(0.47982)0.40.218223(0.21822)0.397613(0.39761)0.457448(0.45745)0.50.139396(0.13940)0.361579(0.36158)0.435673(0.43567)SherwoodnumberincreaseswithanincreaseinNb.FurthertheresultsarecomparedwiththosereportedbyMakindeandAziz[38].AsalreadynoticedinthegraphicalresultsthatthermalandconcentrationboundarylayersthinwhenPrandLeareincreased.ThereducedNusseltandSher-woodnumbers(whichareproportionaltotheinitialslopes)arethereforeincreasingfunctionsofPrandLerespectively.ItisalsoobservedthatmagnitudesofNurandShrareincreasingfunctionsofBi. 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299October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7Chapter7HomotopyAnalysisMethodforFractionalSwift–HohenbergEquationS.Das∗andK.VishalDepartmentofAppliedMathematics,IndianInstituteofTechnology(BHU),Varanasi-221005,India∗subirdas08@hotmail.comInthischapter,homotopyanalysismethodisusedtoobtainapproxi-mateanalyticsolutionofthetime-fractionalSwift–Hohenbergequationwithagiveninitialcondition.ThefractionalderivativesareconsideredintheCaputosense.Effectsofparametersontheconvergenceoftheapproximateseriessolutionbyminimizingtheaveragedresidualerrorarecalculatednumericallyandpresentedthroughgraphsandtablesfordifferentparticularcases.Contents7.1.Introduction.....................................2927.2.Basicideasoffractionalcalculus..........................2957.3.Basicideasofthehomotopyanalysismethod..................2967.4.Solutiongivenbythehomotopyanalysismethod................2987.5.Numericalresultsanddiscussion.........................3017.6.Conclusion......................................304References.........................................306291 300October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.7292S.DasandK.Vishal7.1.IntroductionTheSwift–Hohenberg(S–H)equationwasderivedbyJackSwiftandPierreHohenberg[1]intheyear1977as2∂u∂2=µu−1+u+γu2−u3,(7.1)∂t∂x2whereµandγareparameters,whichisamodelpattern-formingequationusedasamodelforafluidwhichisthermallyconvecting.Thisevolutionequationisaparabolicequationinvolvingafourthorderspatialderivative.ItisaneffectivemodelequationforavarietyofsystemsinPhysicsandMechanics.ThemathematicalmodelfortheRayleigh–BenardconvectioninvolvestheNavier–Stokesequationscoupledwiththetransportequationfortemperature.TheSwift–Hohenbergequationhasimportantroleindif-ferentbranchesofphysics,rangingfromhydrodynamicstononlinearoptics,suchasTaylor–Couetteflow[2]andinthestudyoflasers[3].Theequa-tionalsoplaysanimportantroleinthestudyofpatternformation[4].Large-timebehaviourofsolutionsoftheSwift–Hohenbergequationonaone-dimensionaldomainhasbeenstudiedbyPeletierandRottschafer[5].Inthelastfewdecades,fractionaldifferentialequations(FDEs)havebeenfocusedduetotheirfrequentappearancesinvariousapplicationsinfluidmechanics,viscoelasticity,biology,physics,electricalnetwork,controlthe-oryofdynamicalsystems,chemicalphysics,opticsandsignalprocessing,whicharesuccessfullymodelledbylinearandnon-linearfractionalorderdifferentialequations[6–8].Duetoitsimportantapplicationsinengineer-ingandphysics,theauthorsaremotivatedtosolvethefollowingSwift–Hohenbergequationwithfractionalordertimederivativeα(0<α≤1):∂αu(x,t)∂4u(x,t)∂2u(x,t)2++2+(1−µ)u(x,t)−2u(x,t)∂tα∂x4∂x2+u3(x,t)=0,(x,t)∈Ω,(7.2)subjecttotheinitialcondition1πxu(x,0)=sin,(7.3)10lαwhereµistherealbifurcationparameterand∂istheCaputoderivative∂tαoforderα.In[5]itisshownthatifµ≤0,thenthesteadycaseofsaidequationhasonlythetrivialsolutionandhence,foreveryx∈(0,l),u(x,t)→0ast→∞,Sowehavetakenthecaseµ>0.In2010,Akyildizetal.[9]have 301October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7HAMforFractionalSwift–HohenbergEquation293solvedtheSwift–Hohenbergequationforthestandardmotion.In2011,theS–HequationwithfractionalordertimederivativeisstudiedbyVishaletal.[10].Recently,theS–HequationinthepresenceofdispersionwithfractionalordertimederivativeisstudiedbyVishaletal.[11].Previously,modelingwasmainlyrestrictedtolinearsystemsforwhichanalyticaltreatmentistractable.Butduetotheadventofpowerfulcom-putersandwithimprovedcomputationaltechniques,nowadaysitispos-sibletotackleevennonlinearproblemstosomeextent.Nonlinearityisaphenomenonthatisexhibitedbymostofthesystemsinnatureandhasgainedincreasingpopularityduringlastfewdecades.Mostofthenonlin-earproblemsdonothaveapreciseanalyticalsolution;especiallyitishardtoobtainitforthefractionalordernonlinearequations.Thesetypesofequationsshouldbesolvedbyvariousapproximateanalyticalmethodsorbyusingdifferentnumericalmethods.Owingtothedifficultyofobtainingsolutionsinclosedformforfractionalordersystems,numericalsimulationsarequiteoftenusedtoinvestigatethebehaviorofthesystems.Obviously,theinteger-ordermodelcanbeviewedasaspecialcasefromthemoregeneralfractionalordermodel.Namely,theinteger-ordermodelcanbere-trievedbyputtingallfractionalordersofthederivativesequaltounity.Inotherwords,theultimatebehaviorofthefractionalordersystemresponsemustconvergetotheresponseoftheinteger-orderversionofthemodel.Recently,thefractionaldifferentialequations(FDEs)havegainedmuchattentionduetothefactthatfractionalordersystemresponseultimatelyconvergestotheintegerordersystemresponse(Podlubny[6],MillerandRoss[12],OldhamandSpanier[13]).Analysisoffractionalpartialdiffer-entialequations,whichareobtainedfromtheclassicalequationsinMathe-maticalPhysicsbyreplacingthefirstordertimederivativebyafractionalderivativeoforderαsatisfying0<α≤1,hasbeenafieldofgrowinginterestasevidentfromliteraturesurvey.Thisshowsthatfractionalcalcu-lusistheextensionofclassicalmathematicswherederivativesaretakenasrational,irrationalandcomplexorders.Inthelasttwodecades,fractionaldifferentialequationshavebeenwidelyusedbytheresearchersnotonlyinscienceandengineeringbutalsoineconomicsandfinance.Itisalsoapowerfultoolinmodelingmultiscaleproblems,characterizedbywidetimeorlengthscale.Theattributeoffractionalordersystemsforwhichtheyhavegainedpopularityintheinvestigationofdynamicalsystemsisthattheyallowgreaterflexibilityinthemodel.Anintegerorderdifferen-tialoperatorisalocaloperator.Whereasthefractionalorderdifferentialoperatorisnon-localinthesensethatittakesintoaccountthefactthat 302October28,201311:14WorldScientificReviewVolume-9inx6inAdvances/Chap.7294S.DasandK.Vishalthefuturestatenotonlydependsuponthepresentstatebutalsouponallofthehistoryofitspreviousstates.AnimportantcharacteristicoftheseevolutionequationsisthattheygeneratethefractionalBrownianmotion(FBM)whichisageneralizationofBrownianmotion(BM).Forphysicalsystems,oneshouldhavetokeepinmindtwothingsforapplicationoffractionalorderinthesystemformakingadecisivestepforthepenetrationofmathematicsoffractionalcalculusintoabodyofnat-uralsciences.Firstlytoanalyze,theimportanceandphysicalinfluenceofthememoryeffectsontimeorspaceorboth.Secondly,togiveproperinterpretationisofgeneralmeaningofnonintegeroperators.However,fractionalcalculushasscarcelybeenappliedtoecologicalproblemsduetotheirnonlinearnatureanddifficultiesfacedwhileconfrontingtheproblemswithfractionalderivativeswhichareaddressedbyDasetal.[14–16].Re-searchersareworkinghardtoovercomeit.Themainadvantageofthefractionalcalculusisthatfractionalderivativesprovideanexcellentinstru-mentforthedescriptionofmemoryandhereditarypropertiesofvariousmaterialsandprocesses.HomotopyAnalysisMethod(HAM)proposedbyS.J.Liao[17]isbasedonhomotopy,afundamentalconceptintopologyanddifferentialgeome-try.Itisananalyticalapproachtogettheseriessolutionsoflinearandnonlineardifferentialequations.Thedifferencewiththeotherperturba-tionmethodsisthatthismethodisindependentofsmall/largephysicalparameters.Anotherimportantadvantageofthismethodascomparedtotheotherexistingperturbationandnon-perturbationmethodliesintheflexibilitytochooseproperbasefunctiontogetbetterapproximatesolu-tionoftheproblems.Thismethodofferscertainadvantagesoverroutinenumericalmethods.Numericalmethodsusediscretizationwhichgivesrisetoroundingofferrorscausinglossofaccuracy,andrequireslargecom-putermemoryandtime.Thiscomputationalmethodisbettersinceitdoesnotinvolvediscretizationofthevariablesandhenceisfreefromround-ingofferrorsanddoesnotrequirelargecomputermemoryortime.Thismethodhasbeensuccessfullyappliedbymanyresearchersforsolvinglinearandnon-linearpartialdifferentialequations[18–25].Dasetal.[26]havesuccessfullyappliedthemethodtoinvestigatetheinfluencesofauxiliaryparametertofindtheregionofconvergencethroughh-curveanalysisforsolvingthefractionaldiffusionequation.In2010,Liao[27]hasgiventheoptimalhomotopyanalysismethodwhichisgeneralizationofHAM,whereonecanensuretheconvergenceofseriessolutionbymeansofoptimalvalueofconvergence-controlparameter.Recently,VishalandDas[28]havesuc- 303October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7HAMforFractionalSwift–HohenbergEquation295cessfullyusedoptimalhomotopyanalysismethodtoobtainapproximateanalyticalsolutionoftimefractionalnonlineardiffusionequationinthepresenceofanexternalforceandanabsorbantterm.Inanotherarticle,Dasetal.[29]havesolvedapproximatesolutionoftelegraphequationwithfractionalordertimederivativeforbothfractionalBrownianmotionsandstandardmotion.Themethodisveryeffectiveandhasbeenproventobeaveryefficienttoolwhenappliedtosolvevariouslinearandnonlinearfractionalorderdifferentialequationshavingphysicalrelevanceinphysicalmodelsaswellasbiologicalmodels([30–39]).Thebeautyofthischaptercanbeattributedtothesuccessfulapplica-tionofmoreaccurate,flexibleandverypowerfulanalyticalmethodHAMforthesolutionofthenonlinearfractionalorderS–Hequationandthecon-vergenceoftheseriessolutionthroughminimizationofaveragedresidualerrorwithproperchoicesofoptimalvaluesofauxiliaryhomotopyparameterconfirmingthevalidityandpotentialofthemethod.Thesalientfeaturesofthechapterarethenumericalandgraphicalpresentationsoftheeffectsofbifurcationparameterandthelengthofthedomainonthesolutionwithtimeandalsotheperiodicnatureofthesolutionsfordifferentvaluesofparameters.7.2.BasicideasoffractionalcalculusInthissection,wegivesomedefinitionsandpropertiesofthefractionalcalculus[6]whichareusedinthischapter.Definition7.1.Arealfunctionf(t),t>0,issaidtobeinthespaceCµ,µ∈<,ifthereexistsarealnumberp>µ,suchthatf(t)=tpf(t),where1f(t)∈C(0,∞),anditissaidtobeinthespaceCn,ifandonlyiff(n)∈C,1µµn∈N.Definition7.2.TheRiemann–LiouvillefractionalintegraloperatorJαoftorderα≥0,ofafunctionf∈Cµ,µ≥−1,isdefinedasZtα1α−1Jtf(t)=(t−ξ)f(ξ)dξ,α>0,t>0,(7.4)Γ(α)0J0f(t)=f(t),twhereΓ(α)isthewell-knowngammafunction.SomeofthepropertiesoftheoperatorJαf(t),whichareneededhere,areasfollows:Forf∈C,tµµ≥−1,α,β≥0andγ≥−1, 304October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7296S.DasandK.Vishal(i)JαJβf(t)=Jα+βf(t),ttt(ii)JαJβf(t)=JβJαf(t),tttt(iii)Jαtγ=Γ(γ+1)tα+γ.tΓ(α+γ+1)Definition7.3.ThefractionalderivativeDαoff(t),intheCaputosensetisdefinedasZtα1n−α−1nDtf(t)=(t−ξ)f(ξ)dξ,(7.5)Γ(n−α)0forn−1<α 305October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7HAMforFractionalSwift–HohenbergEquation297Itisobviousthatwhentheembeddingparameterq=0andq=1,thezeroth-orderdeformationequation(7.8)hasthesolutionφ(x,t;0)=u0(x,t)andφ(x,t;1)=u(x,t)respectively.Thusasqincreasesfromzerotounity,thesolutionφ(x,t;q)variesfromtheinitialguessu0(x,t)totheexactsolutionu(x,t).Expandingφ(x,t;q)inTaylorserieswithrespecttoq,onehasX∞kφ(x,t;q)=u0(x,t)+uk(x,t)q,(7.9)k=1where1∂kφ(x,t;q)uk(x,t)=.k!∂qkq=0TheconvergenceoftheseriesgiveninEq.(7.9)dependsontheconvergence-controlparameter~.Ifitisconvergentatq=1,onehasX∞u(x,t)=φ(x,t;1)=u0(x,t)+uk(x,t),(7.10)k=1whichmustbeoneofthesolutionsoftheoriginalfractionalorderdifferentialequation,asprovenbyLiao[17].Nowdefiningthevectoru˜n(x,t)=(u0(x,t),u1(x,t),u2(x,t),...,un(x,t)),(7.11)themth-orderdeformationequationreadsL[um(x,t)−χmum−1(x,t)]=~H(x,t)Rm(˜um−1(x,t),(7.12)withtheinitialconditionum(x,0)=0,(7.13)whereRm(˜um−1(x,t))1∂m−1∂αφ(x,t;q)=−N[φ(x,t;q)](7.14)(m−1)!∂qm−1∂tαq=0and0,m≤1,χm=(7.15)1,m>1.Nowthesolutionofmth-orderdeformationequation(7.12)form≥1readsu(x,t)=χu(x,t)+~L−1[H(x,t)R(˜u(x,t))]+c,(7.16)mmm−1mm−1 306October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7298S.DasandK.Vishalwherecistheintegrationconstantwhichisdeterminedbytheinitialcon-ditiongiveninEq.(7.13).7.4.SolutiongivenbythehomotopyanalysismethodTosolveEq.(7.2)byhomotopyanalysismethod,wechoosetheinitialapproximation1πxu0(x,t)=sin,(7.17)10landthelinearauxiliaryoperator∂αφ(x,t;q)L[φ(x,t;q)]=,(7.18)∂tαwiththepropertyL[c]=0,(7.19)wherecisintegralconstant,φ(x,t;q)isanunknownfunction.Furthermore,intheviewofEq.(7.2),wehavedefinedthenonlinearoperatoras∂αφ(x,t;q)∂4φ(x,t;q)∂2φ(x,t;q)N[φ(x,t;q)]=++2∂tα∂x4∂x223+(1−µ)φ(x,t;q)−2φ(x,t;q)+φ(x,t;q).(7.20)Bymeansofthehomotopyanalysismethod,Liao[17]hasconstructedtheso-calledzeroth-orderdeformationequationas(1−q)L[φ(x,t;q)−u0(x,t)]=q~N[φ(x,t;q)],(7.21)whereq∈[0,1]istheembeddingparameter,~6=0istheconvergence-controlparameter,u0(x,t)istheinitialguessofu(x,t).Itisobviousthatwhentheembeddingparameterq=0andq=1,Eq.(7.21)becomesφ(x,t;0)=u0(x,t),φ(x,t;1)=u(x,t),respectively.Nextweexpandφ(x,t;q)asinEq.(7.9).IftheauxiliarylinearoperatorL,theinitialguessu0(x,t)andtheconvergence-controlparameter~areproperlychosensothattheseriesdescribedinEq.(7.9)convergesatq=1asdiscussedinprevioussection,thenu(x,t)definedasinEq.(7.10)willbeoneofthesolutionsofourconsideredproblemgiveninEq.(7.2). 307October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7HAMforFractionalSwift–HohenbergEquation299Differentiatingthezero-orderdeformationEq.(7.21)mtimeswithre-specttoqandthendividingitbym!andfinallysettingq=0,onehasthesocalledmth-orderdeformationequationasL[um(x,t)−χmum−1(x,t)]=~Rm[um−1(x,t)],(7.22)withtheinitialconditionum(x,0)=0,(7.23)where∂αu(x,t)∂4u(x,t)m−1m−1Rm[˜um−1(x,t)]=∂tα+∂x4∂2u(x,t)m−1+22+(1−µ)um−1(x,t)∂xmX−1XimX−1+um−1−iujui−j−2uium−1−i,(7.24)i=0j=0i=0andχmisdefinedinEq.(7.15).Then,wehaveu(x,t)=χu(x,t)+~JαR[u(x,t)]+c,(7.25)mmm−1tmm−1whereZtα1α−1Jt[f(t)]=(t−ξ)f(ξ)dξ,Γ(α)0andtheintegrationconstantcisdeterminedbytheinitialconditionEq.(7.23).NowfromEq.(7.25),thevaluesum(x,t)form=1,2,3,...canbeobtainedandtheseriessolutionsarethusgained.FinallytheapproximatesolutionisobtainedbytruncatingtheseriesasXmu˜m(x,t)=ui(x,t).(7.26)i=0ItisclearfromEq.(7.26)that˜um(x,t)containsoneconvergence-controlparameter~,whichdeterminestheconvergenceregionandrateofthehomotopy-seriessolution.AsgivenbyLiao[27],atthemth-orderofapproximation,onecandefinetheexactsquareresidualasZZ"m#!2X∆m=Nui(x,t)dxdt(7.27)Ωi=0 308October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7300S.DasandK.VishalHowever,itwasillustratedbyLiao[27]thattheexactresidualerror∆mdefinedbyEq.(7.27)needstoomuchCPUtimetocalculateeveniftheorderofapproximationisnotveryhigh.Thus,toovercomethisdifficultyi.e.,todecreasetheCPUtime,weuseheretheso-calledaveragedresidualerrordefinedby"#!21XMxXMtXmEm=Nui(j∆x,k∆t),(7.28)(Mx+1)(Mt+1)j=1k=1i=0whereMx=Mt=14and∆x=∆t=0.5fortheconsideredproblem.Theoptimalvalueof~canbeobtainedbymeansofminimizingtheso-calledaveragedresidualerrorEq.(7.28),correspondingtothenonlinearalgebraicequationdEm=0.(7.29)d~AnasymptoticexpansionoraTaylorseriesexpansioncanoftenbeaccel-eratedquitedramatically(oreventurnedfromdivergenttoconvergent)bybeingre-arrangedintoaratiooftwosuchexpansions.APad´eapproximationPmkmk=0akxPn(x)=Pnk(7.30)k=0bkx(normalizedbyb0=1)generalizestheTaylorseriesexpansionwiththesametotalnumberofcoefficientsmX+nT(x)=cxk.(7.31)m+nkk=0FromatruncatedTaylorseriesexpansiongivenbyEq.(7.31),onedeter-minesthecorrespondingPad´ecoefficientsbyrequiringthatifEq.(7.30)isexpandedasTaylorseriesandtheresultshallmatchallthetermsgiveninEq.(7.31).OneofthemainapplicationsofPad´eapproximationsistoextractasmuchinformationaswecanformapowerseriesexpansionthatisknownonlytoafewterms.ConversionfromTaylortoPad´eformusuallyacceleratesconvergence,andoftenallowsgoodaccuracyevenoutsidetheradiusofconvergenceofapowerseries(whichincaseofdivergentasymp-toticexpansionsmaybezero).Theso-calledhomotopy-Pad´etechnique[22]wasproposedbycombiningtheabove-mentionedtraditionalPad´etechniquewiththehomotopyanalysismethod.Hence,inordertocalculatethe[m,n] 309October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7HAMforFractionalSwift–HohenbergEquation301homotopy-Pad´eapproximantofφ(x,t;q),firstthetraditionalPad´etech-niqueisusedwithexpansionaboutthehomotopyembeddingparameterqasPmkk=0Am,k(x,t)qφ(x,t;q)=Pn(7.32)Bm,k(x,t)qkk=0wherethecoefficientsAm,kandBm,karedeterminedbythefirstseveralapproximationsu0(x,t),u1(x,t),u2(x,t),.........um+n(x,t).(7.33)Then,settingq=1inEq.(7.32)andusingEq.(7.10),wehavetheso-called[m,n]homotopy-Pad´eapproximantPmk=0Am,k(x,t)Pm(x,t)u(x,t)≈Pn=.(7.34)k=0Bm,k(x,t)Qm(x,t)The[1,1]and[2,2]homotopy-Pad´eapproximantsintermsofthebasisfunc-tionsukreaduu+u2−uu01120u≈u1−u2andP2(x,t)u≈,(7.35)Q2(x,t)whereP(x,t)=uu2+uu2+u3−uuu−u2u−uuu−2uuu20212201313023123+uu2+uuu+u2u−uuu,0301414024Q(x,t)=u2+u2−uu+uu−uu−uu.22313142324Likewise,the[i,j]homotopy-Pad´eapproximantcanbeobtainedfromthetermsinthe(i+j)th-orderseriesapproximation.7.5.NumericalresultsanddiscussionInthissection,thenumericalresultsofprobabilitydensityfunctionu(x,t)forthenon-lineartimefractionalSwift–Hohenbergequationhavebeenob-tainedfortheone-dimensionaldomain.Theoptimalvaluesof~andmini-mumaveragedresidualerrorareprovidedthroughTables7.1to7.6andaredisplayedthroughFig.7.1.Theeffectsoftheparameterµandthelengthofthedomainlonsolutionprofileofu(x,t)arepresentedgraphicallythrough 310October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7302S.DasandK.VishalTable7.1.Optimalvalueof~incaseofl=10,µ=2andα=1.OrderofApprox.~Em10.1892702.79031×10−330.0825672.36788×10−350.0534142.26236×10−3Table7.2.Optimalvalueof~incaseofl=8,µ=2andα=1.OrderofApprox.~Em10.1746222.84695×10−330.0765132.40528×10−350.0495512.29546×10−3Table7.3.Optimalvalueof~incaseofl=10,µ=2andα=0.75.OrderofApprox.~Em10.3001542.23036×10−330.1961661.83711×10−350.1193031.59495×10−3Table7.4.Optimalvalueof~incaseofl=8,µ=2andα=0.75.OrderofApprox.~Em10.2735152.26256×10−330.1794711.84955×10−350.1093711.59816×10−3Figs.7.2to7.9.ItisseenfromFigs.7.2to7.9,whicharedrawnforµ=2,α=1,0.5andl=10,8,6,2thatthebehavioursofsolutionsareperiodicinnaturefordifferenttime.ItisclearfromTables7.1and7.2thatoptimalvaluesof~are0.053414and0.049551forl=10andl=8respectivelyforµ=2,α=1inthecaseof5th-orderapproximation.Tables7.3and7.4exhibitthatoptimalvaluesof~are0.119303and0.109371forl=10andl=8respectivelyforµ=2,α=0.75.ItisalsoseenfromtheTable7.5andTable7.6thatoptimalvaluesof~are0.293335and0.272750forl=10andl=8 311October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7HAMforFractionalSwift–HohenbergEquation303Table7.5.Optimalvalueof~incaseofl=10,µ=2andα=0.5.OrderofApprox.~Em10.5657591.55442×10−330.4180521.09422×10−350.2933358.21554×10−4Table7.6.Optimalvalueof~incaseofl=8,µ=2andα=0.5.OrderofApprox.~Em10.4883071.56634×10−330.3640191.07741×10−350.2727507.79888×10−4Fig.7.1.PlotsofaverageresidualerrorEmvs.~forµ=2,α=0.5andl=10.respectivelywhenµ=2,α=0.5.ItisalsoobservedfromTables7.1,7.3and7.5,whicharecalculatedforµ=2,l=10thatasthevaluesofαdecreasetheoptimalvalueof~increasesandgoesawayfrom~=−1,thecaseofusualHAM.Similarly,Tables7.2,7.4and7.6,whicharecalculatedforµ=2,l=8showthatasthevaluesofαdecreasestheoptimalvalueof~increasesandgoesawayfrom~=−1.Weseethattheoptimalvalueoftheconvergencecontrolparameterdependsnotonlyonthevaluesofαandl,butalsoonthenumberoftermsemployedintheseriesapproximationofthesolution.Wehavefoundthatthevalueoftheresidualerrordecreaseswhileweincreasethenumberoftermsoftheseries.ThisshowsthatHAMisagoodalternativetoanumericalmethod.FurtherHAMprovidesanalyticalsolutionswhichcanbereadilyavailable 312October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7304S.DasandK.VishalFig.7.2.Plotsofprofileofu(x,t)vs.xatµ=2,α=1,l=10fordifferentvaluesoftat5thorderofapproximation.Fig.7.3.Plotsofprofileofu(x,t)vs.xatµ=2,α=0.5,l=10fordifferentvaluesoftat5thorderofapproximation.Fig.7.4.Plotsofprofileofu(x,t)vs.xatµ=2,α=1,l=8fordifferentvaluesoftat5thorderofapproximation.forotherpurposes.7.6.ConclusionInthischapterthreeimportantgoalsareachieved.Firstoneisthesuccess-fulapplicationofthehomotopyanalysismethodforfindingthesolution 313October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7HAMforFractionalSwift–HohenbergEquation305Fig.7.5.Plotsofprofileofu(x,t)vs.xatµ=2,α=0.5,l=8fordifferentvaluesoftat5thorderofapproximation.Fig.7.6.Plotsofprofileofu(x,t)vs.xatµ=2,α=1,l=6fordifferentvaluesoftat5thorderofapproximation.Fig.7.7.Plotsofprofileofu(x,t)vs.xatµ=2,α=0.5,l=6fordifferentvaluesoftat5thorderofapproximation.ofthenonlinearS–Hequationwithfractionalordertimederivative.Sec-ondly,theoptimalvalueoftheconvergence-controlparameterthatgivesrisetoaconvergentseriessolution.Thirdoneisthesuccessfulnumericalandgraphicalpresentationsoftheeffectsofbifurcationparameterontheprobabilitydensityfunction.Theeffectsofbifurcationparameterinspatialdomainarealsodescribedduringthestudy.TheSwift–Hohenbergequation 314October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.7306S.DasandK.VishalFig.7.8.Plotsofprofileofu(x,t)vs.xatµ=2,α=1,l=2fordifferentvaluesoftat5thorderofapproximation.Fig.7.9.Plotsofprofileofu(x,t)vs.xatµ=2,α=0.5,l=2fordifferentvaluesoftat5thorderofapproximation.describestheconvectiveheatcurrentinaRayleeigh–Benardcellandthenatureofhydrodynamicstability.Theequationfurtherallowsthestudyofconvectivepatternthatresultsasafunctionoftheforcingmecanism.AclearconclusionfromthenumericalresultsisthattheHAMprovideshighlyaccuratenumericalsolutionsforthenonlinearpartialdifferentialequationwithfractionaltimederivative.Ithasbeenobservedthattheproperchoiceoftheconvergencecontrolparametergreatlyimprovestheconvergencerateoftheapproximateseriessolution.References[1]J.B.SwiftandP.C.Hohenberg,Hydrodynamicsfluctuationsattheconvec-tiveInstability,Phys.Rev.A.15,319–328(1977).[2]P.C.HohenbergandJ.B.Swift,EffectsofadditivenoiseattheonsetofRayleigh-BenardConvection,Phys.Rev.A.46,4773–4785(1992).[3]L.Lega,J.V.MoloneyandA.C.Newell,Swift-HohenbergEquationforlasers,Phys.Rev.Lett.73,2978–2981(1994). 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317October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8Chapter8HAM-BasedPackageNOPHforPeriodicOscillationsofNonlinearDynamicSystemsYinpingLiuDepartmentofComputerScienceEastChinaNormalUniversity,Shanghai200241,Chinaypliu@cs.ecnu.edu.cnBasedonWu’seliminationmethodandthehomotopyanalysismethod(HAM),theMaplepackageNOPH(version1.0.2)isdevelopedforperiod-icallyoscillatingsystemsofcenterandlimitcycletypes,whichdeliversaccurateapproximationsoffrequency,meanofmotionandamplitudesofoscillationautomatically.SincetheHAMisvalidforhighlynonlin-earproblems,thepackagecanbeusedtofindaccurateapproximatesolutionsofnonlinearoscillationsystemswithstrongnonlinearity.Forsystemswithphysicalparameters,itcanprovidepossibleconstraintcon-ditionsonparameters.Inthischapter,webrieflydescribethediscussedproblem,anefficientalgorithm,aMaplepackageNOPHandsoon.Asimpleusersguideisgivenintheappendixofthischapter.TheMaplepackageNOPHisfreeavailable(Accessed20Jan2013,willbeupdatedinthefuture)athttp://numericaltank.sjtu.edu.cn/NOPH.htm.309 318October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8310Y.-P.LiuContents8.1.Introduction.....................................3108.2.Problemdescription................................3128.3.Algorithm......................................3138.3.1.Ruleofsolutionexpression.........................3148.3.2.Thezero-orderdeformationequation...................3148.3.3.Thehigher-orderdeformationequation..................3168.3.4.Solvingnonlinearalgebraicequationswithparameters..........3188.3.5.Thechoiceoftheconvergence-controlparameter~...........3208.3.6.Thehomotopy-Pad´etechnique.......................3218.4.ThepackageNOPH.................................3228.5.ApplicationsofNOPH...............................3248.5.1.Example1..................................3258.5.2.Example2..................................3278.5.3.Example3..................................3318.5.4.Example4..................................3358.5.5.Example5..................................3388.6.Discussionsandconclusions............................342AppendixA.TheprocedurescontainedintheNOPH................342AppendixB.Asimpleuser’sguide...........................343B.1.Example1......................................344B.1.1.Allphysicalparametersareunknown...................344B.1.2.Inthecaseofα=5,a=1/2........................345B.2.Example2......................................347B.2.1.Allphysicalparametersareunknown...................348B.2.2.Inthecaseof=2.............................349B.3.Example3......................................352B.3.1.Allphysicalparametersareunknown...................352B.3.2.Inthecaseofε=1/2............................354B.3.3.Inthecaseofε=1/2,µ=8.......................355References.........................................3578.1.IntroductionNonlinearoscillationisatypeofcommonnonlinearinitial-valueproblems,whichcanbeseenanywhereanytimeinscienceandengineering.Con-siderableinterestshavebeenfocusedonthestudyofanalyticsolutionsofnonlinearoscillations,forthesesolutionsmaygivemoreinsightsintointernalaspectsofnonlinearproblems.Foremostamongtheanalytictech-niquesareperturbationmethodsintermsofasmall/largephysicalpa-rameter[1–4].Thebasicpremiseofperturbationmethodsistotransferanonlinearproblemtoasetofinfinitenumberoflinearsub-problemsbymeansofasmall/largephysicalparameter.Unfortunately,manynonlinearequationsdonotcontainsuchtypesofsmall/largephysicalparameters. 319October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations311Besides,perturbationapproximationsaregenerallyonlyaccurateenoughforweaklynonlinearproblems,butoftenbreakdownwhenthephysicalpa-rametersbecomelarge.Inaddition,restrictedbysuchtypesofsmall/largephysicalparameters,theperturbationtechniquescannotprovidefreedomtochoosetheequationtypeofthelinearsub-problems.Thiscanregularlyresultinmajordifficultieswhensolvingnonlinearoscillationproblems.Toovercometherestrictionsofperturbationmethods,somenon-perturbationmethods,suchastheartificialsmallparametermethod,theδ-expansionmethodandAdomian’sdecompositionmethod,havebeendeveloped[5–7].However,itisapitythattheseperturbationandnon-perturbationmethodscannotprovideaconvenientwaytoadjustandcontroltheconvergenceofapproximationseries.HAM[8,9]isananalyticapproximationmethodforstronglynonlin-earproblems,whichhasbeensuccessfullyappliedtosolvemanynonlinearproblemsinscienceandengineering[8–23].Thisisespeciallytrueforsev-eralsolutionsthatwerenotidentifiedusinganalyticorevennumericaltechniques,aspresentedbyLiao[14].Comparedwithotheranalyticornumericaltechniques,HAMhasthefollowingadvantages.•HAMdoesnotneedanysmall/largephysicalparameters.Therefore,itsuseisvalidinagreaternumberofnonlinearproblems.•HAMintroducesanauxiliaryparameter,calledconvergence-controlparameter,whichprovidesasimplewaytocontrolandadjusttheconvergenceofapproximationsolutions[8,9,15,16].•HAMprovidesgreatfreedomtochoosetheproperequation-typeandbasefunctionstoapproximatesolutionofnonlinearproblemseffi-ciently.•UsingHAM,accurateapproximationsoftherequiredsolutioncanbeobtained,aswellaspossibledependentrelationsofthephysicalparam-etersfromthesystemunderconsideration.Theserelationsarehelpfulforbetterunderstandingstheinneraspectofnonlinearproblems.•MultiplesolutionsofnonlinearproblemscanbegainedbymeansofHAM.TheseadvantagesofHAMimplythatitispossibletodevelopsomegeneralsymboliccomputationpackagesforsometypesofnonlinearproblems.Fol-lowingaHAM-basedMathematicapackageBVPhforcoupled,highlynon- 320October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8312Y.-P.Liulinearboundary-value/eigenvalueproblemsa,thisMaplepackageNOPH1.0.2hasbeensuccessfullydevelopedforstronglyperiodicnonlinearoscillations,theoldversionofthepackagehasbeenreported[24],thecurrentversionofthepackagecanworkforsomesingleparametricorforcedoscillation.However,asthereareexternalandinternalatleasttwodifferentfrequen-ciesinthesetwotypesofoscillations.Therefore,wewillcontinuetode-velopnewalgorithmtosolveparametricorforcedoscillations,especiallywithvectorfrequency.Thesepackagesarefreelyavailableonthewebsitehttp:/numericaltank.sjtu.edu.cn/.8.2.ProblemdescriptionConsiderperiodicsolutionsofnonlinearoscillationsgovernedbyU¨i(t)=fi[U(t),U˙(t),U¨(t)],1≤i≤κ,(8.1)wherethedependentvariableU(t)hasκcomponentsU1(t),...,Uκ(t),theindependentvariabletdenotestime,thedotdenotesdifferentiationwithre-specttot,andfi[U(t),U˙(t),U¨(t)]iseitheranalgebraicorrationalfunctionofU(t),U˙(t)andU¨(t).NotethatEq.(8.1)israthergeneralwhichcanbeusedtodescribelotsofperiodicoscillationsofnonlinearproblemsinscienceandengineering.Inthischapter,thefocusismainlyonsingleconservativeoscillationsandself-excitedoscillations.Theformercorrespondstoperi-odicoscillationsofthecentertype,andthelattertoperiodicoscillationsofthelimitcycletype.Bythewayweshouldmentionthatself-excitedoscillatorscanonlyappearasrelaxationoscillationwhenthecoefficientofthemainnonlineartermislargeenough.Astheintermediateexpressions“swells”sofast,thecurrentversionofthepackageNOPHdoesnotworkforarelaxationoscillator.ItshouldbeemphasizedthatEq.(8.1)doesnotnecessarilycontainsmall/largephysicalparameters.LetΩ={ω1,...,ωκ}denotethesetoffrequenciesoftheperiodicoscil-lations.Forsimplicity,wejustconsiderthecasethatΩcontainssingleωinwhatfollows.Formultidimensionalsystems,ifωi(1≤i≤κ)aredifferent,wecouldreduceΩto{ω}bythepretreatments:ω2=p2ω,...,ωκ=pκω.Inthiscase,pi(i=2,...,κ)canbelookeduponaspositiveparameters,andtheywillbedeterminedbyusingthesecondmainstrategyin§8.3.4.Itshouldbepointedoutthataccordingtothepropertyofoscillations,thealgorithmcanonlydealwithintegerpi(2≤i≤κ),anditmaynotworkwellforrationalorirrationalratiospi(2≤i≤κ).aThepackageisfreeavailableonlinehttp://numericaltank.sjtu.edu.cn/BVPh.htm. 321October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations313DefinethemeanofmotionsZT1δi=Ui(t)dt,i=1,...,κ,(8.2)T0whereT=2π/ωistheperiodofoscillation.NotethatiffiinEq.(8.1)containsonlyoddnonlinearityaboutUi(t),themeanofmotionδiisequaltozero.Withoutlossofgenerality,letusconsidernonlinearperiodicoscillationswithinitialconditionsUi(0)=ai+δi,U˙i(0)=ωbi,i=1,...,κ−1,(8.3)Uκ(0)=aκ+δκ,U˙κ(0)=0,whereai,biandδiareunknownstobedeterminedlater.Physically,thefrequencyωcanberegardedasatime-scale.Hence,introducingthetrans-formationsτ=ωt,Ui(t)=δi+ui(τ),1≤i≤κ,Eqs.(8.1)and(8.3)becomeω2u¨(τ)=f[δ+u(τ),ωu˙(τ),ω2u¨(τ)],(8.4)iisubjecttoinitialconditionsuκ(0)=aκ,u˙κ(0)=0,ui(0)=ai,u˙i(0)=bi,1≤i≤κ−1,(8.5)inwhichuandδhaveκcomponentsu1(τ),...,uκ(τ)andδ1,...,δκ,re-spectively.InEqs.(8.4)and(8.5),apartfromtheunknownfunctionsui(τ)andi=1,...,κ,thereare3κunknownphysicalquantities,includ-ingthefrequencyω,themeanofmotionδi(1≤i≤κ),theamplitudesaj,bj(1≤j≤κ−1)andaκ.Alloftheseunknownscanbedeterminedbytheso-called“ruleofsolutionexpression”intheframeofHAM,asdescribedlater.8.3.AlgorithmAlthoughHAMhasbeenwidelyappliedwhilesolvingnonlinearproblemsinscienceandengineering,itisnotveryeasytoapplytoasystemofcouplednonlinearoscillations.Thereasonliesinthatonehastosolveκcouplednonlineardifferentialequationstogetherwithasetof3κhighlynonlinearalgebraicequationsrelatedtotheunknownphysicalquantitiesω,δi(1≤i≤κ),aj,bj(1≤j≤κ−1)andaκ.Inthissection,HAMiscombinedwithWu’seliminationmethodtodevelopanefficientalgorithm. 322October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8314Y.-P.Liu8.3.1.RuleofsolutionexpressionIngeneral,asystemofcouplednonlinearoscillatorsismorecomplicated.Here,thefocusisonlyonperiodicsolutions,whichcanbeexpressedintermsofthebasefunctions{cos(mτ),sin(mτ)|m=1,2,3,...}.(8.6)Thisistheso-called“ruleofsolutionexpression”.Tosearchforperiodicsolutions,initialguesseswerechosenuκ,0(τ)=aκ,0cos(τ),(8.7)ui,0(τ)=ai,0cos(τ)+bi,0sin(τ),i=1,...,κ−1,whereai,0andbi,0areunknownstobedeterminedlater,andanauxiliarylinearoperator∂2Φ(τ,q)L[Φ(τ,q)]=ω2+Φ(τ,q)(8.8)0∂τ2withpropertyL[C1sin(τ)+C2cos(τ)]=0,inwhichq∈[0,1]istheembedding-parameter,ω0istheinitialguessofthefrequencyω,Φ(τ,q)isafunctionofτandq,whileC1andC2areintegralconstants.8.3.2.Thezero-orderdeformationequationBasedonEq.(8.4),thefollowingnonlinearoperatorsaredefined∂2Φ(τ,q)N[Φ(τ,q),Ω(q),∆(q)]=Ω2(q)−f∆(q)+Φ(τ,q),i∂τ2i∂Φ(τ,q)∂2Φ(τ,q)2Ω(q),Ω(q),(8.9)∂τ∂τ2whereΩ(q)and∆(q)arefunctionsofq,Φ(τ,q)and∆(q)haveκcom-ponents,correspondingtothefunctionu(τ)andthemeanofmotionδ,respectively.HAMisbasedoncontinuousvariationsΦ(τ,q),Ω(q)and∆(q),astheembedding-parameterqvariesfrom0to1,Φ(τ,q)variesfromtheinitialguessu0(τ)totheexactsolutionu(τ).SodoesΩ(q)fromtheinitialguessω0totheexactfrequencyω,and∆(q)fromtheinitialguessδ0totheexactmeanofmotionδ.Then,withtheaidofhomotopyintopology,thefollowingequationisconstructed(calledthezeroth-orderdeformationequation)(1−q)L[Φ(τ,q)−u0(τ)]=q~Ni[Φ(τ,q),Ω(q),∆(q)],(8.10) 323October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations315subjecttoinitialconditions∂Φj(τ,q)Φi(0,q)=¯ai(q),=¯bj(q),∂ττ=0∂Φκ(τ,q)=0,1≤i≤κ,1≤j≤κ−1,(8.11)∂ττ=0where¯ai(q)and¯bj(q)arefunctionsofq,correspondingtoaiandbj,respec-tively.Whenq=0,itfollowsfrom(8.7)and(8.10)thatΦ(τ,0)=u0(τ),Ω(0)=ω0,∆(0)=δ0.(8.12)Whenq=1,since~6=0,thezeroth-orderdeformationequation(8.10)isequivalenttoNi[Φ(τ,1),Ω(1),∆(1)]=0,1≤i≤κ,(8.13)whichisexactlythesameastheoriginalequation(8.4),providedΦ(τ,1)=u(τ),Ω(1)=ω,∆(1)=δ.(8.14)Thus,accordingto(8.12)and(8.14),astheembedding-parameterqin-creasesfrom0to1,Φ(τ,q)variescontinuouslyfromtheinitialguessu0(τ)totheexactsolutionu(τ).SodoesΩ(q)fromtheinitialguessω0totheexactfrequencyω,and∆(q)fromtheinitialguessδ0totheexactmeanofmotionδ.AccordingtoTaylor’stheoremandusing(8.12),Φi(τ,q),∆i(q),Ω(q),a¯i(q)and¯bi(q)areexpandedinthepowerseriesofqasfollowsX+∞Φ(τ,q)=u(τ)+u(τ)qn,ii,0i,nn=1X+∞X+∞nnΩ(q)=ω0+ωnq,∆i(q)=δi,0+δi,nq,n=1n=1X+∞X+∞a¯(q)=a+aqn,¯b(q)=b+bqn,(8.15)ii,0i,njj,0j,nn=1n=1where1≤i≤κ,1≤j≤κ−1,and1∂nΦ(τ,q)1∂n∆(q)u(τ)=i,δ=i,i,nn!∂qni,nn!∂qnq=0q=01∂nΩ(q)1∂na¯(q)1∂n¯b(q)ω=,a=i,b=j.nn!∂qni,nn!∂qnj,nn!∂qnq=0q=0q=0 324October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8316Y.-P.LiuAssumingthattheparameter~isproperlychosensothattheaboveseriesareconvergentatq=1,anddueto(8.12)thereisX+∞X+∞ui(τ)=ui,0(τ)+ui,n(τ),ω=ω0+ωn,n=1n=1X+∞X+∞X+∞δi=δi,0+δi,n,ai=ai,0+ai,n,bj=bj,0+bj,n,(8.16)n=1n=1n=1where1≤i≤κand1≤j≤κ−1.AttheMth-orderapproximations,thereisXMMX−1MX−1u˜i(τ)≈ui,n(τ),ω˜≈ωn,δ˜i≈δi,n,n=0n=0n=0MX−1MX−1a˜i≈ai,n,˜bj≈bj,n,1≤i≤κ,1≤j≤κ−1.(8.17)n=0n=08.3.3.Thehigher-orderdeformationequationDifferentiatingthezero-orderdeformationequation(8.10)ntimeswithrespecttoq,thendividingthembyn!,andfinallysettingq=0,thenth-orderdeformationequationcanbeobtainedL[ui,n(τ)−χnui,n−1(τ)]=~Ri,n,1≤i≤κ,(8.18)subjecttoinitialconditionsuκ,n(0)=aκ,n,u˙κ,n(0)=0,ui,n(0)=ai,n,u˙i,n(0)=bi,n,1≤i≤κ−1,(8.19)where1∂n−1N[Φ(τ,q),Ω(q),∆(q)]R=i,1≤i≤κ,(8.20)i,n(n−1)!∂qn−1q=0and(0,n≤1,χn=(8.21)1,n>1.Theaboventh-orderdeformationequation(8.18)isasetofκlin-eardifferentialequationsaboutui,n(τ)and1≤i≤κ.However,apartfromκunknownfunctionsui,n(τ),thereare3κunknownparameters 325October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations317ωn−1,ai,n−1,δi,n−1(1≤i≤κ)andbj,n−1(1≤j≤κ−1).Accordingtothepropertiesoftrigonometricfunctions,Ri,ncanbeexpressedbyφX(i,n)ψX(i,n)i,ni,ni,nRi,n=A0+Ajcos(jτ)+Bjsin(jτ),1≤i≤κ,j=1j=1wheretheintegersφ(i,n)andψ(i,n)dependonEq.(8.4)andtheordern.Tosearchforaperiodicsolutionandavoidtheso-calledsecularterms,i,ni,ni,nA0,A1andB1,where1≤i≤κ,mustvanish,i.e.,i,ni,ni,nA0=0,A1=0,B1=0,1≤i≤κ.(8.22)Equations(8.22)provideuswithasetof3κalgebraicequationstode-terminetheabovementioned3κunknowns,i.e.,ωn−1,ai,n−1,bj,n−1andδi,n−1,where1≤i≤κand1≤j≤κ−1.Then,itiseasytoobtainthesolutionsforthenth-orderdeformationequationφX(i,n)i,ni,ni,nAjui,n(τ)=C1sin(τ)+C2cos(τ)+2cos(jτ)1−jj=2ψX(i,n)i,nBj+sin(jτ),1−j2j=2i,ni,nwhere1≤i≤κ,the2κintegralconstantsC1andC2aredeterminedbythe2κinitialconditionsgivenin(8.19).ItistobestressedthatEq.(8.4)ismoregeneral,whichincludesin-gleconservativeoscillationsandself-excitedoscillations,etc.Forsin-gleconservativeoscillationthebasefunctions(8.6)canbesimplifiedas{cos(mτ)|m=1,2,3,...},sinceinthiscase,allcoefficientsofsinefunc-tionsbecomezero.ThusEqs.(8.22)arereducedto1,n1,nA0=0,A1=0,whichareusedtodetermineωn−1andδ1,n−1.Notethatdifferenttypesofoscillationshavedifferentcharacteristics,forexample,theamplitudeforasingleconservativeoscillationisfixed,oritisevenknownsometimes,buttheamplitudeforaself-excitedoscillationsystemisunknownandneedstobedetermined.Theproblemcanberemovedwhentheconservativeoscillationamplitudeisunknown,asthismeansitcanbeconsideredtobeaparameterandtreatedassuch. 326October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8318Y.-P.Liu8.3.4.SolvingnonlinearalgebraicequationswithparametersItshouldbepointedoutthat,Eqs.(8.22)arelinearwhenn>1,butare3κcouplednonlinearalgebraicequationswhenn=1.Forasinglenonlinearoscillation,itiseasytosolveEqs.(8.22)forω0,δ1,0anda1,0.However,forcouplednonlinearoscillation,itisdifficulttosolvethecorrespondingnon-linearalgebraicEqs.(8.22)forω0,δi,0,ai,0andbj,0,where1≤i≤κand1≤j≤κ−1,whichmaybecomeabottleneckforthewholealgorithm.Toovercomethisdifficultyandtoimprovetheefficiencyofthisprogram,var-ioustechniqueswereapplied,apartfromthedivide-and-conquertechniqueandfreezetechnique.Here,thethreemainstrategiesarelistedbelow:•Firstly,Wu’seliminationmethod[25]isapowerfultoolforsolvingnon-linearalgebraicequations.Wangetal.[26,27]completelyimplementedWu’seliminationmethodanddevelopedaMaplepackageCharSets,inwhichtheCsolvesolvercanbedirectlyusedtosolvethenonlinearal-gebraicequations.InthisMaplecode,Eqs.(8.22)aresolvedbymeansofthepackageCharSets,whosegreatestadvantageisthatitcanavoidmissingsolutions.Besides,sincetheordersofvariablescanbespecifiedwhensolvingnonlinearalgebraicequationsbyusingthepackage,theor-dersofvariablescanbeoptimizedsoastoimprovethecomputationalefficiency.Inaddition,theCharSetspackageisveryefficient,duetosomeexcellenttechniqueshavingbeenemployedinit,suchasdecompos-ingexpression,extractingsubexpressionaswellassortingandspecifyingtheordersofequationstobesolved,andsoon.•Secondly,inEqs.(8.22),usuallythedegreesofparametersinvolvedinthegoverningequationsarelowerthanthoseofunknowns,sotreatingtheseparametersasunknownsmightgreatlyweakenthedifficultyoftheproblem.Doingthisitispossibletoobtaincompactspecialsolutionswithparameterconstraints.•Thirdly,evenwiththeabovetwostrategies,forsomecouplednonlinearoscillatingsystems,theobtainedequations(8.22)arestillverydifficulttodealwith.Inthiscase,someextraconditionstosimplifyEqs.(8.22)mightbeimposed.Withthisstrategy,thecomplicatedalgebraicequa-tions(8.22)couldbegreatlysimplified,andthensomespecialsolutionsmaybeobtained.Here,anexampleisgiventoillustratehowtoapplythesetechniquestosimplifyandsolveanonlinearalgebraicequations.Considerathree 327October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations319coupledself-excitedoscillatorU¨−ε(1−U2)U˙+U=εµ(W−U),V¨−ε(1−V2)V˙+V=εµ(W−V),(8.23)W¨−ε(1−W2)W˙+p2W=εµ(U+V−2W),withinitialconditions(U(0)=a1,U˙(0)=b1,V(0)=a2,V˙(0)=b2,(8.24)W(0)=a3,W˙(0)=0,whereε,µandpareunknownparameters,anda1,a2,a3,b1andb2areunknownstobedetermined.Asthissystemonlycontainsoddnonlinearity,δi=0andi=1,2,3.Directcalculations,followingtheprocedureofHAM,yieldεωb(b2+a2−4)−4[εµa+a(ω2−1−εµ)]=0,01,01,01,03,01,00322εω0(a1,0+b1,0a1,0)−4(b1,0+εω0a1,0+εµb1,0−ω0b1,0)=0,εωb(b2+a2−4)−4[εµa+a(ω2−1−εµ)]=0,02,02,02,03,02,00(8.25)εω(a3+b2a)−4(b+εωa+εµb−bω2)=0,02,02,02,02,002,02,02,0022εµa1,0+ω0a3,0−pa3,0−2εµa3,0+εµa2,0=0,34a3,0ω0−4µb2,0−ω0a3,0−4µb1,0=0.ItisdifficulttosolveEqs.(8.25)fortheunknownsω0,a1,0,b1,0,a2,0,b2,0anda3,0,becausethehighestdegreesofa1,0,b1,0,a2,0,b2,0anda3,0areall3,andthehighestdegreeofω0is2.Duetotheknownfactthatthesolutionexpressionofacubicequationiscomplicated,itisnaturalthatthesolutionexpressionsofEqs.(8.25)wouldbehuge.Itbecomesevenworsewhentheintermediateexpressions“swell”sofastthatthememorywilloverflowfinally.Therefore,treatingtheparametersε,µandpasunknownsmayreducesuchtypesofdifficulties.Ifsolvingthecomplicatedequations(8.25)withsomeorallofε,µ,p,ω0,a1,0,b1,0,a2,0,b2,0anda3,0asunknowns,unfortunately,atrivialsolutionmaybeobtainedwiththeconstraintε=0,oritmaytaketoolongtoobtaintheresult,oreveninsomecasesthememorycouldoverflow.Inordertoobtainnontrivialsolutions,someextra 328October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8320Y.-P.Liuconditionscouldbeimposed,suchas{b1,0=0},toreduceEqs.(8.25)asω02a1,0−a1,0−εµa1,0+εµa3,0=0,222εω0b2,0(b2,0−4+a2,0)−4a2,0(ω0−1−εµ)−4εµa3,0=0,εωa(4−b2−a2)+4b(1+εµ−ω2)=0,02,02,02,02,00(8.26)εµa+ω2a−p2a−2εµa+εµa=0,1,003,03,03,02,034a3,0ω0−4µb2,0−ω0a3,0=0,2a1,0−4=0.Clearly,theequationsin(8.26)aremuchsimplerthanthosein(8.25).Treatingtheparametersε,µandpasunknownsandsolvingEqs.(8.26)withtheMaplepackageCharSets,thefollowingsolutionsareobtained√{ω0=1+εµ,a1,0=∓2,a2,0=±2,b1,0=0,b2,0=0,a3,0=0},1−p2p{µ=,ω0=2−p2,a1,0=a2,0=∓2,b1,0=b2,0=0,a3,0=±2},2ε{p=∓1,ω0=1,a1,0=−2,a2,0=−2,b1,0=0,b2,0=0,a3,0=−2},{p=∓1,ω0=1,a1,0=2,a2,0=2,b1,0=0,b2,0=0,a3,0=2}.Aspappearsintheformofp2inEqs.(8.23),itiseasytoseethatifU(t),V(t)andW(t)aresolutionsofthesystems(8.23)and(8.24),then−U(t),−V(t)and−W(t)arealsosolutionsforthesystem.Therefore,therearethreeessentiallydifferentsolutionsin(8.27),givenby√{ω0=1+εµ,a1,0=2,a2,0=2,b1,0=0,b2,0=0,a3,0=0},1−p2p{µ=,ω0=2−p2,a1,0=a2,0=a3,0=2,b1,0=b2,0=0},2ε{p=1,ω0=1,a1,0=2,a2,0=2,b1,0=0,b2,0=0,a3,0=2}.Obviously,thereareparameterconstraintsforthelasttwosolutions.Itiseasytoseethatunderthedifferentconditionsimposed,theobtainedsolutionsforthesystems(8.23)and(8.24)wouldbedifferent.8.3.5.Thechoiceoftheconvergence-controlparameter~Followingtheabovesteps,afamilyofsolutionseriescanbeobtainedintheconvergence-controlparameter~.Howcanapropervalueof~bechosentoguaranteeafastenoughconvergenceofthesolutionseriesforanygivenphysicalparameters?Theoptimalvalueof~isdeterminedbyminimizing 329October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations321thesquaredresidualerrorofEq.(8.4),i.e.,XκZ2π2Λ(~)=i(~,τ)dτ,i=10wherei(~,τ)=Ni[u˜(τ),ω,˜δ˜],u˜(τ)={u˜1(τ),...,u˜κ(τ)},δ˜={δ˜1,...,δ˜κ}and1≤i≤κ.8.3.6.Thehomotopy-Pad´etechniquePad´eapproximantexpandsafunctionasaratiooftwopowerseries.WhenPMkPNkafunctionisgivenintheformofR(x)=(k=0akx)/(1+k=1bkx),P∞kthenR(x)issaidtobeaPad´eapproximanttotheseriesf(x)=k=0ckx,ifkkddR(0)=f(0),R(x)=f(x),k=1,2,...,M+N.dxkx=0dxkx=0(8.27)Theconditions(8.27)provideM+N+1equationsfortheunknownsa0,...,aMandb1,...,bN.ThePad´etechniquecanbecombinedwithHAMtogeneratetheso-calledhomotopy-Pad´emethod,asshownbyLiao[8].Thegeneralprocedureisasfollows.First,employthetraditionalPad´etechniquetotheseries(8.15)abouttheembedding-parameterq∈[0,1]toobtainthe[m,n]Pad´eapproximant! !XmXnkkTPm,n=Wk(τ)q1+Wm+k(τ)q,k=0k=1whereWk(τ),k=0,1,...,m+n,arefunctionsdeterminedbythefirstm+n+1componentsui,j(τ),j=0,1,...,m+n.Then,settingq=1,theso-called[m,n]homotopy-Pad´eapproximantwasobtained! !XmXnHPm,n=Wk(τ)1+Wm+k(τ).k=0k=1Moredetailscanbefoundin[8].Ithasbeenfoundthatthehomotopy-Pad´eapproximantusuallycon-vergesfasterthanthecorrespondingtraditional[m,n]Pad´eapproximant.Inmanycases,the[m,m]homotopy-Pad´eapproximantdoesnotdependupontheconvergence-controlparameter~.Ingeneral,thehomotopy-Pad´e 330October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8322Y.-P.Liutechniquecangreatlyenlargetheconvergenceregionofsolutionseries.Hence,itisemployedinthecode.8.4.ThepackageNOPHAlthoughthealgorithmdescribedin§8.3isrelativelysimpleinprinciple,thecalculationswouldbeverycomplicatediftheyhadtobeperformedbyhand.Inthissection,basedonthepackageCharSets,aMaplepackageNOPHispresented,whichcanbeusedtoautomaticallyderivehomotopyanalysissolutionsfornonlinearoscillationequationsintheformsof(8.1)and(8.3).InNOPH,themaininterfaceismain(eqns,inicon,[horder,solorder]),inwhicheqnsrepresentsanoscillationequationstobesolvedandiniconrepresentsthecorrespondinginitialconditions.Notethateqnsshouldbeputinalist,asshouldinicon.Theothertwoparametershorder(thedefaultvalueis8)andsolorder(thedefaultvalueis10)areoptional,whichareusedtocontroltheprocessflowoftheprogram.Inordertomeetdifferentneeds,thecomputationsintheprogramaredividedintotwoparts:thefirstpartisdependentontheconvergence-controlparameter~,andthesecondpartisfreeoftheconvergence-controlparameter~.Theyaredescribedasfollows:(1)Whenthereareunknownparametersineqnsorinicon,justperformthefirstpartcomputation,namely,calculatethefirsthordercompo-nentsofthesolutionseriestoserveasanapproximationoftherequiredsolution.Inthiscase,theoptionalparametersolordercanbeomitted,andtheprogramwilloutputthe[horder/2,horder/2]homotopy-Pad´eapproximateexpressionofthefrequencyωaswellasthemeanofmo-tionsδi,i=1,...,κ.(2)Ifthereisnounknownparameterineqnsorinicon,firstobtainthefirsthordercomponentsofthesolutionseriesbythefirstpartcomputation,andfurthercalculatethesquaredresidualerroroftheinputequationstodeterminetheoptimalvalueof~.Oncethevalueof~isdetermined,amoreaccurateapproximatesolution(i.e.,continuetocalculatefromhorder+1tosolorderorder)maybeobtainedbythesecondpart 331October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations323computation.Inthiscase,theprogramwilloutputdifferentorderapproximationsandthecorrespondingerrorsofthefrequencyω,aswellasthemeanofmotionsδi,i=1,...,κ,etc.Inaddition,comparisongraphsofthedifferentorderapproximationswillbeproducedfortherequiredsolutions.Asdescribedin§8.3,therearedifferentcharacteristicsfordifferenttypesofoscillations.Therefore,theoutputoftheprogramisflexibleanddiverse,notonlydependingonthenumberofunknownparameters,butalsode-pendinguponthetypeofoscillations.Theoutputoftheprogramcanbesummerizedasfollows.•Iftherearemorethanoneunknownparametersininputsystem,theprogramdelivershomotopyPad´eapproximantsofωandδ.Foroscil-latorwithjustoddnonlinearity,δwillbedegeneratedtozero.Thebelowissame.•Ifthereisjustoneunknownparameterininputsystem,theprogramoutputsnotonlyhomotopyPad´eapproximantsofωandδ,butalsocomparisongraphsfordifferentorderapproximantsofωandδ.•Ifthereisnounknownparameterininputsystem,forconservativeoscillator,theprogramoutputsapproximantsofωandδ,aswellascomparisongraphfordifferentorderapproximationsoftherequiredsolution;forself-excitedoscillatororcoupledoscillator,theprogramdeliversapproximantsofω,δ,andtheamplitudesai,bj,meanwhile,theprogramalsooutputslimitcyclesoftherequiredsolutions.Similarly,theprocessflowoftheprogrammainlydependsontypesofnon-linearoscillations.Therearethreedifferentcases.•Forconservativeoscillator,thesolvingprocessiscompletelyautomatic.•Forself-excitedoscillator,theobtainednonlinearalgebraicequationsforai,jandbi,jmayexistmultiplesolutions.Inthiscase,usershavetoselectonesolutionandgoonfurthercalculationdirection.•Forcouplednonlinearoscillator,apartfrommultiplesolutions,theob-tainednonlinearalgebraicequationsforai,jandbi,jmaybecometoocomplicatedtobedealtwith.Inthiscase,theprogramprovidesseveralinputinterfacessoastoperformmanualintervention. 332October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8324Y.-P.LiuApartfromalltheoutputresultslistedinthenextsection,ifneces-sary,userscouldalsogetcomponentsoftherequiredsolutionseries.Thecomponentsofobtainedsolutionseriesarestoredinasetthatisnamedas‘export’.Allthesesetsarelistedasfollows:•Knowu:thecomponentssetofUi(t)andi=1,...,κ.•Knowomega:thecomponentssetofthefrequencyω;andPadeomega:thehomotopy-Pad´eapproximationssetofω.•Knowdelta:thecomponentssetofthemeanofmotionδiandi=1,...,κ,inaccordance;andPadedelta:thehomotopy-Pad´eapproxi-mationssetofδiandi=1,...,κ.•Knowabc:thecomponentssetoftheamplitudesai,bi,i=1,...,κ−1andaκ.ItshouldbenotedthatKnowabcissetexclusivelyforself-excitedoscillatingsystems.Itisverysimpletolistthecontentsoftheabovesets.Forexample,todisplaythesetKnowu,onejustneedstoinputthesetnameKnowuafterthecommandprompt“>”,thatis,>Knowu;TheproceduresofthepackageareoutlinedinAppendixA.AsimpleuserguideisgiveninAppendixB,threedifferenttypesofexamplesaregiventoshowhowtousethepackageNOPH.TheMaplepackageNOPHisfreeavailableonlineviathewebsitehttp://numericaltank.sjtu.edu.cn/NOPH.htm.8.5.ApplicationsofNOPHInthissection,someexamplesaregiventoshowtheeffectivenessofthepackage.Morethan30oscillationequationshavebeensolvedusingthepackageNOPH.ItshouldbementionedthatalltheresultsshowninthissectionwereobtainedonalaptopwithIntelCorei5-2410MCPUastheprocessorandafrequencyof2.30GHz.Duetospacelimitations,onlyfivedifferenttypesofnonlinearoscillationsystemsareconsideredhere.Todemonstratetheconvergenceoftheobtainedhomotopyanalysisso-lutions,thesesolutionsarecomparedwithexactsolutionsornumericalso-lutionsobtainedusingaFehlbergfourth-fifthorderRunge–Kuttamethod(rkf45)withdegreefourinterpolant.Asrkf45doesnotworkforasystem 333October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations325withunknownparameters,eachunknownparameterwasassignedthevaluedeliveredbythepackageNOPH.8.5.1.Example1Considerthefreeoscillationofaconservativesystemwithoddnonlinearity[8]U¨+U+U3=0,subjecttoinitialconditionsU(0)=a,U˙(0)=0.√Forthissystem,introducingthetransformationV(t)=U(t),yieldsV¨+V+V3=0,(8.28)subjecttoinitialconditionsV(0)=b,V˙(0)=0,(8.29)√whereb=aisanunknownparameter.Thereisonlyoneparameterbinthesystem(8.28)and(8.29),thepackageautomaticallydelivers(byhorder=5)the[1,1]homotopy-Pad´eapproximation√(279b4+768b2+512)3b2+4ωP1,1=2(285b4+768b2+512),andthe[2,2]homotopy-Pad´eapproximationω=[574118847b14+5452491348b12+22121717760b10P2,2+49711251456b8+66831974400b6+53758656512b4.p+23957864448b2+4563402752][23b2+4(195618321b12+1584935424b10+5335944192b8+9556131840b642+9602662400b+5133828096b+1140850688)].Moreover,thepackageautomaticallydeliversafiguretocompareapprox-imationsofωatdifferentorders.AsshowninFig.8.1,theabove[1,1]and[2,2]homotopy-Pad´eapproximationsofωagreequitewellevenfor0≤b<+∞. 334October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8326Y.-P.Liu800600400omega[1,1]homotopy-padeapproximant200[2,2]homotopy-padeapproximant02004006008001000bFig.8.1.Thehomotopy-Pad´eapproximantsofωinEq.(8.28)asafunctionofb.Tofurthershowtheconvergenceofobtainedsolutionseries,thecaseofb=1isconsidered.Inthiscase,accordingto(13)and(14)givenin[28],anexactsolutionforthesystem(8.28)and(8.29)isintheform√√ofV(t)=cn(t2;1),thefrequencyω=2π/K(1/4),whereK(1/4)=R42π12−1(1−sinθ)2dθisthecompleteellipticintegralofthefirstkind.In04thiscase,allphysicalparametersareknown,whentakinghorder=5andsolorder=10,thepackageautomaticallydelivers~=−1.Atthesametime,thepackagealsooutputsapproximantsofω,andacomparisongraphfordifferentorderapproximationsofV(t)within10seconds.TheapproximatesolutionsarecomparedwiththeaboveexactoneinFig.8.2.8th-orderHAMapproximation10th-orderHAMapproximation1Exactsolution0.5V0-0.5-102468tFig.8.2.ThecomparisonofV(t)forEq.(8.28)whenb=1andtaking~=−1. 335October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations327AsshowninFig.8.2,the8th-orderand10th-orderHAMapproximationsofV(t)agreeverywellwiththeexactsolutionshownabove.Besides,theith-orderapproximationofωandthecorresponding[i,i]homotopy-Pad´eapproximantsaregiveninTables8.1and8.2,respectively.Thefrequencydeliveredbythepackageisingoodagreementwiththeexactone,thehomotopy-Pad´eapproximantsofωconvergefasterthanthecorrespondinghomotopyapproximants.Table8.1.Theith-orderapproximationsofωinEq.(8.28)whenb=1andtaking~=−1.iω11.322921.317831.3178Table8.2.The[i,i]homotopy-Pad´eapproximationsofωinEq.(8.28)whenb=1.iω11.317821.31788.5.2.Example2Forfreeoscillationofaconservativesystemwithquadraticnonlinearity,thepackageautomaticallydeliversapproximationsofboththefrequencyωandthemeanofmotionδi(i=1,...,κ).Forexample,considerafreeoscillationwithquadraticnonlinearity[8]U¨(t)+U(t)+εU2(t)=0,(8.30)subjecttoinitialconditionsU(0)=1/2+δandU˙(0)=0,whereεisaparameterandδisthemeanofmotion.Whenεisunknown,thepackageautomaticallydelivers(byhorder=7)within52secondsthe[1,1]homotopy-Pad´eapproximants 336October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8328Y.-P.Liu√pω=[2((505−2123+224)4−22−376+2244−4922+384)]P[1,1]p/[(4−22)3/4((543−112)4−22−354+1542−192)],√(33−482−18+96)4−22−143+36−48(2−2)2δP=√,[1,1]4((242−48)4−22+73−18)the[2,2]homotopy-Pad´eapproximantsω=−[(207229112−2133996610+880938608−1782793366P[2,2]p+1656960004−331345922−29491200)4−22131197−106700+4451720−43305488+186308576p−4087621125+4494213123−197001216]/[416−82((4299411+16744249−171067447+593636485p−890122243+49250304)4−22−2184891121086+22445294−91960132+183929320−1682595844+325201922+29491200)],δ=−[(1935360−54835205+58072323−2460672−2711310P[2,2]+3021548−14053566+33476404−40250882+5824811p−6134409+25914247)4−22−3870720+99855362+4921344+65079846+112619525−108951844−117757443−3799912−12350411+44356410+12883209−22716008−53848327]/[2((2711310−3021548p+14053566−33476404+40250882−1935360)4−22+12350411−12883209+53848327−112619525+117757443−4921344)], 337October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations329andthe[3,3]homotopy-Pad´eapproximantsareomittedhereduetospacelimitations.Itwasfoundthatthe[2,2]and[3,3]homotopy-Pad´eapproximantsofthefrequencyωandthemeanofmotionδagreewell,asshowninFigs.8.3and8.4.10.980.96omega0.94[1,1]homotopy-padeapproximant[2,2]homotopy-padeapproximant[3,3]homotopy-padeapproximant0.92-0.500.51epsilonFig.8.3.TheHAM-Pad´eapproximantsofωinEq.(8.30)asafunctionofε.0-0.02-0.04delta-0.06-0.08[1,1]homotopy-padeapproximant[2,2]homotopy-padeapproximant[3,3]homotopy-padeapproximant-0.10.20.40.60.81epsilonFig.8.4.TheHAM-Pad´eapproximantsofδinEq.(8.30)asafunctionofε.Toshowtheaccuracyofthesesolutions,thecaseofε=1,forRand[29]isconsideredandpresentedasanexactsolutiontoEq.(8.30)forε= 338October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8330Y.-P.Liu1,whichhastheformU(t)=A+asn2(wt;m),asshownin[30].Forconvenienceofcomparison,δisassignedthevaluedeliveredbythepackage,namelyδ=−0.1181,thenA=1/2+δ=1/2−0.1181=0.3819.Followingthestepsshownin[30],{a=−0.9147,m=0.7269,w=0.5371}isobtained.Forthecaseofε=1,thereisnounknownparameterandthepackageautomaticallyderives(byhorder=12andsolorder=15)~=−0.4402,andapproximantsofωandδaswellasacomparisongraphfordifferentorderapproximationsofU(t)within820seconds.0.413th-orderHAMapproximation15th-orderHAMapproximationExactsolution0.20U-0.2-0.424681012tFig.8.5.ThecomparisonofU(t)forEq.(8.30)whenε=1,andtaking~=−0.4402.Table8.3.TheapproximationsofωandδinEq.(8.30)whenε=1andtaking~=−0.4402.iωδ10.8409−0.128120.8781−0.120530.8934−0.118040.8983−0.117650.8991−0.117860.8987−0.117970.8983−0.118180.8981−0.1181 339October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations331Table8.4.Thehomotopy-Pad´eapproximationsofωandδinEq.(8.30)whenε=1.iωδ10.9041−0.114920.8975−0.118430.8981−0.118040.8981−0.1181AsshowninFig.8.5,the13th-orderand15th-orderHAMapproxima-tionsofU(t)agreeverywellwiththeaboveexactsolution.Table8.3indicatesthatthefrequencyωaswellasthemeanofmotionδconvergequickly,andtheirhomotopy-Pad´eapproximantsconvergeevenfaster,asshowninTable8.4.8.5.3.Example3ThepackagealsoworksforrationalfunctionfwithrespecttoU,U˙andU¨.Forexample,considerafreeperiodicoscillatorgovernedbyλU+εUU˙2+2εU3U˙2+εU3+εU5U¨=−1234,(8.31)1+ε1U2+ε2U4subjecttoinitialconditionsU(0)=aandU˙(0)=0,whereλisanintegerthatmaytakevalues−1,0or1,aistheamplitudeandε1>0,ε2>0,ε3>0andε4>0arephysicalparameters.Obviously,Eq.(8.31)onlycontainsoddnonlinearity,sothemeanofmotionδequalszero.InChen’swork[31],theapproximatesolutionwasconstructedbyanewdifferentialtransformationmethod.Toshowtheeffectivenessoftheirsolutions,theauthorscomparedtheirsolutionsinfourdifferentparametermodeswithnumericalsolutionsobtainedusingthefourth-orderRunge–Kuttamethod;theirsolutionsagreedwellwithnumericalresults.Forthisexample,itissolvedusingthepackageinthefollowingthreecases. 340October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8332Y.-P.Liu8.5.3.1.AllphysicalparametersareunknownWhensettinghorder=3,theprogramautomaticallydeliversthe[1,1]homotopy-Pad´eapproximateexpressionofωwithin30seconds.However,theobtainedapproximateexpressionissocomplicatedthatitisomittedhereduetospacelimitations.8.5.3.2.Inthecaseofλ=1,ε=1,ε=1,ε=1anda=112223Inthiscase,thereisonlyoneunknownparameter4.Settinghorder=5,ittakesabout910secondsforthepackagetodeliverthe[1,1]homotopy-Pad´eapproximant√432ωP1,1=644+2304(2826214004+10406396904−21018406614−10283156607−7846822521)/[23(2892462004+10138711303444−21203800832−10191069819−7768422279)],44andthe[2,2]homotopy-Pad´eapproximantω=2(72160080229783054982000000013P2,2412+123830218897777573209640000004+77058516168389190627707250000114+134007102107481137948434740000104−767685927995469573412814008500948−47462630627251268468562529621504−898250996680388129498943631061574+538896756359369485912624869164164+60335045968270544233419404252763544+1341373249115401292388969994206094+16072790508675756222936201414383434 341October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations333+11468688337000976819551360862795124+470711032208488110485767396338454+8859722465244665645188353217170)/√[644+230(1479483256327424616440000001244+209549537893151903060560000011410+95652941057466754837609700004−69098271018605460645706800094−153990028938150585378469292900847−5161574431434277164919199389304−32765815090991422404506573135564+203331150195988774960283122790154+6346583360504481105299648696625443+88826808018003946126486006460854+700687483265063501357576387403624+31054957031114277730854788994574+626473417036090662183962398527)].Theabovetwoapproximationsagreeverywellfor4>0,asshowninFig.8.6.8.5.3.3.Inthecaseofa=λ=ε=ε=1andε=ε=134122Inthiscase,thereisnounknownparameterandtheprogramautomaticallygives(bysettinghorder=6,solorder=8)~=−0.3384,approximantsofωaswellasacomparisongraphfordifferentorderapproximationsofU(t)within2000seconds.AsshowninFig.8.7,the6th-orderand8th-order 342October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8334Y.-P.Liu2015omega10[1,1]homotopy-padeapproximant[2,2]homotopy-padeapproximant502004006008001000epsilon[4]Fig.8.6.Thehomotopy-Pad´eapproximantsofωinEq.(8.31)whena=λ=ε3=1,ε11=ε2=.2HAMapproximationsofU(t)agreeverywell.Meanwhilethesecurvesagreewellwiththenumericalimitatingcurveofthenumericalsolutionforthegivenvaluesa=λ=ε=ε=1,ε=ε=1andtheabsoluteerror10−7.34122Moreover,thefrequencyωalsoconvergesquickly,asshowninTables8.5and8.6.Itneedstobeemphasizedthatforthefourdifferentmodesoftheparametersprovidedin[31],theobtainedhomotopyanalysissolutionsalsoagreewellwiththenumericalsolutionobtainedusingtherkf45,whilethecomparisongraphisomittedduetospacelimitations.16th-oderHAMapproximation8th-orderHAMapproximationNumericalsolution0.5U0-0.5-1246810tFig.8.7.TheHAMapproximationsofU(t)forEq.(8.31)whena=λ=ε3=ε4=1,ε11=ε2=andtaking~=−0.3384.2 343October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations335Table8.5.Theith-orderHAMapproximationsofωinEq.(8.31)whena=λ=ε13=ε4=1,ε1=ε2=2andtaking~=−0.3384.iω11.285421.294231.294441.294551.2945Table8.6.The[i,i]homotopy-Pad´eapproximantsofωinEq.(8.31)whena=λ=ε13=ε4=1andε1=ε2=.2iω11.294421.29458.5.4.Example4ConsidertheJ.W.S.equationU¨(t)+U(t)=µ(1−βU˙(t)2)U˙(t),(8.32)withinitialconditionsU(0)=c1andU˙(0)=0,whereµ,βareunknownparameters.Therearetwounknownparametersinthissystem,ittakesabout114secondsfortheprogramtodeliver(byhorder=7)pp{{ω0=1,c1,0=2/3β},{ω0=1,c1,0=−2/3β}}.Byselectingthefirstsolution,itfurtheroutputs:the[1,1]homotopy-Pad´eapproximationµ2+32ωP1,1=2,3µ+32the[2,2]homotopy-Pad´eapproximation9µ6+16640µ2+960µ4+49152ωP2,2=45µ6+19712µ2+1920µ4+49152, 344October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8336Y.-P.Liuandthe[3,3]homotopy-Pad´eapproximationω=(−109231931392µ4−2382205288448µ2−3020544µ8P3,3−1970749440µ6−1175344644096+178848µ10+729µ12)/(−256206766080µ4−2455664328704µ2−73979136µ8−4347887616µ6−1175344644096+598752µ10+5103µ12).Astheabovehomotopy-Pad´eapproximantsofωarebutfunctionsofµ,thepackagealsoautomaticallyoutputsafiguretocompareapproximationsofωatdifferentorders,asshowninFig.8.8.Itcanbeseenthatthecurvesof[2,2]and[3,3]homotopy-Pad´eapproximantsofωagreewell.10.90.8omega0.7[1,1]homotopy-padeapproximant0.6[2,2]homotopy-padeapproximant[3,3]homotopy-padeapproximant0.5-6-4-20246muFig.8.8.Thehomotopy-Pad´eapproximantsofωinEq.(8.32)asafunctionofµ.Inaddition,inthecaseofµ=1,β=3,thereisnounknownpa-rameterintheinputsystemandthepackagetakesabout530secondstoautomaticallyoutput(byhorder=8andsolorder=11)~=−0.9453,theapproximantsofω,c1,andthecomparisongraphfordifferentorderapproximationsofU(t).AsshowninFig.8.9,the9th-orderand11th-orderHAMapproximationsofU(t)agreeverywell.Meanwhile,thesecurvesagreewellwiththenumericalimitatingcurveofthenumericalsolutionforthegivenvaluesµ=1,β=3,c=0.7193andtheabsoluteerror10−7.1Thefrequencyωandtheamplitudec1convergequicklyforthegivenvalue~=−0.9453,asshowninTables8.7and8.8. 345October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations3370.60.40.29th-orderHAMapproximation11th-orderHAMapproximationU’0Numericalsolution-0.2-0.4-0.6-0.6-0.4-0.200.20.40.6UFig.8.9.TheapproximationsofU(t)forEq.(8.32)whenµ=1,β=3andtaking~=−0.9453.Table8.7.Thehomotopyapproximationsofωandc1inEq.(8.32)whenµ=1,β=3bymeansof~=−0.9453.iωc111.00000.666720.94090.706130.94290.717940.94250.718750.94260.719660.94300.719470.94290.719380.94300.7193Table8.8.Thehomotopy-Pad´eapproximationsofωandc1inEq.(8.32)whenµ=1,β=3.iωc110.94290.723020.94260.719630.94270.719640.94300.719250.94300.7193 346October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8338Y.-P.Liu8.5.5.Example5ConsidertwodirectlycoupledvanderPoloscillators(U¨(t)−ε(1−U2(t))U˙(t)+U(t)=εµ(V(t)−U(t)),(8.33)V¨(t)−ε(1−V2(t))V˙(t)+p2V(t)=εµ(U(t)−V(t)),withinitialconditionsU(0)=a1,U˙(0)=b1,V(0)=c1,V˙(0)=0,(8.34)whereε,µandparephysicalparameters.Similartotheaboveexamples,thissystemissolvedbyconsideringthreedifferentcases.8.5.5.1.AllphysicalparametersareunknownInthiscasetherearethreeunknownparametersε,µandp,andthepackagetakesabout20secondstodeliver8groupsolutionsforω0,a1,0,b1,0andc1,0:{p=∓1,ω0=1,a1,0=2,b1,0=0,c1,0=2},{p=∓1,ω0=1,a1,0=−2,b1,0=0,c1,0=−2},√(8.35){p=∓1,ω0=1+2εµ,a1,0=2,b1,0=0,c1,0=−2},√{p=∓1,ω0=1+2εµ,a1,0=−2,b1,0=0,c1,0=2}.Aspappearsintheformofp2inEq.(8.33),itiseasytoverifythatifU(t)andV(t)aresolutionsofthesystems(8.33)and(8.34),then−U(t)and−V(t)arealsosolutionsforthesystem.Therefore,therearetwoessentiallydifferentsolutionsin(8.35)asfollows:{p=1,ω0=1,a1,0=2,b1,0=0,c1,0=2},√(8.36){p=1,ω0=1+2εµ,a1,0=−2,b1,0=0,c1,0=2}.Selectasolutionandinputitsorder,suchas2,thentheprogramfurtherdelivers:theparameterconstraintcondition:{p=1},the[1,1]homotopy-Pad´eapproximant√(ε2+64εµ+32)2εµ+1ωP1,1=2,(8.37)3ε+64εµ+32 347October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations339andthe[2,2]homotopy-Pad´eapproximantω=(16640ε2+66560ε3µ+960ε4+294912εµ+66560ε4µ2P2,2+589824ε2µ2+49152+393216ε3µ3+1920ε5µp663322+9ε)2εµ+1/(45ε+393216εµ+589824εµ+78848ε4µ2+78848ε3µ+294912εµ+3840ε5µ+19712ε2+49152+1920ε4).(8.38)8.5.5.2.Inthecaseofε=1/2From(8.36),itisknownthatthereisaparameterconstraintp=1ineachsolution.Therefore,lettingε=1/2,therewillbeonlyoneunknownparameterµcontainedintheinputsystem,thentheprogramautomaticallydelivers(byselectingthe2ndsolutionin(8.36)andsettinghorder=5)thefollowingresultswithin590seconds.Theparameterconstraintcondition:{p=1},the[1,1]homotopy-Pad´eapproximant√(128µ+129)µ+1ωP1,1=,(8.39)128µ+131andthe[2,2]homotopy-Pad´eapproximant√(9973504µ+9703424µ2+3415817+3145728µ3)µ+1ωP2,2=3468845+3145728µ3+9752576µ2+10075648µ.(8.40)Moreover,theprogramalsoprovidesafiguretocomparehomotopy-Pad´eapproximantsofωatdifferentorders,asshowninFig.8.10.Itindicatesthat[1,1]and[2,2]homotopy-Pad´eapproximantsofωagreeverywell.8.5.5.3.Inthecaseofε=1/2,µ=8Inthiscase,therearenounknownparametersintheinputsystem,forthereistheparameterconstraintp=1involvedineachsolution.Theprogramautomaticallyoutputs(bysettinghorder=7andsolorder=9)~=−0.498,approximantsofω,a1,b1andc1,aswellascomparisongraphsfordifferentorderapproximationsofU(t)andV(t)within755seconds. 348October24,201314:25WorldScientificReviewVolume-9inx6inAdvances/Chap.8340Y.-P.Liu302520omega15[1,1]homotopy-padeapproximant10[2,2]homotopy-padeapproximant502004006008001000muFig.8.10.Thehomotopy-Pad´eapproximantsofωinEq.(8.33)asafunctionofµ.22117th-orderHAMapproximation7th-orderHAMapproximation9th-orderHAMapproximationU’9th-orderHAMapproximationV’00NumericalsolutionNumericalsolution-1-1-2-2-6-4-20246-6-4-20246UVFig.8.11.ApproximationsofU(t),V(t)forEq.(8.33)whenε=1/2,µ=8bymeansof~=−0.498.AsshowninFig.8.11,the7th-orderand9th-orderHAMapproximationsofU(t)andV(t)agreeverywell.Meanwhile,bothagreeverywellwiththecorrespondingimitatingcurvesofthenumericalsolutionsforthegivenvaluesa1=1.9964,b1=−0.1234,c1=−1.9964andtheabsoluteerror10−7.Table8.9indicatesthatthehomotopyapproximantsofthefrequencyωandtheamplitudesa1,b1andc1convergeveryquicklywhensetting~=−0.498.ItisshownfromTable8.10thatthehomotopy-Pad´eapproximantsconvergeevenfasterthanthecorrespondinghomotopyapproximants. 349October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations341Table8.9.TheHAMapproximantsofω,a1,b1,c1inEq.(8.33)whenε=1/2,µ=8andtaking~=−0.498.iωa1b1c113.00002.0000−0.1245−2.000022.99642.0000−0.1248−2.000032.99641.9964−0.1234−1.996442.99641.9964−0.1234−1.9964Table8.10.TheHAM-Pad´eapproximantsofω,a1,b1,c1inEq.(8.33)whenε=1/2,µ=8.iωa1b1c112.99642.0000−0.1248−2.000022.99641.9964−0.1234−1.99642211U’0Obtainedfromthe2ndsolutionin(8.36)V’0Obtainedfromthe2ndsolutionin(8.36)Obtainedfromthe1stsolutionin(8.36)Obtainedfromthe1stsolutionin(8.36)-1-1-2-2-6-4-20246-6-4-20246UVFig.8.12.Differentapproximationsobtainedfromdifferentsolutionsin(8.36)whenε=1/2,µ=8.Toindicatethatthesolutionsobtainedfromthedifferentsolutionsin(8.36)aredifferent,U(t)andV(t)obtainedfromthe1stand2ndsolutionsin(8.36),respectively,werecompared.AsshowninFig.8.12,itcanbeseenthatthesolutionsobtainedfromdifferentsolutionsin(8.36)areindeeddifferent. 350October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8342Y.-P.Liu8.6.DiscussionsandconclusionsInthischapter,analgorithmandaMaplepackagearepresentedforau-tomatedreasoningofanalyticapproximatesolutionsofnonlinearoscillat-ingsystems.Thepackageisuser-friendlyasoneneedonlyinputasetofgoverningequationsandinitialconditions,andthenperiodicanalyticapproximationsarereturnedinashorttime.ThepackageNOPHhasthefollowingthreemerits:Firstly,itcanworkforhighlynonlinearoscillatingsystems;Secondly,ifthereareunknownparametersininputsystem,thepackageNOPHcanautomaticallyfilteroutthecorrespondingpossiblecon-straintconditionsontheseparameters;Thirdly,itmayconstructseveraldifferentsolutionssimultaneouslyfortheconsideredsystem.However,weshouldmentionthatthecurrentversionofthepackageNOPHonlyworksincaseswherefiinEq.(8.1)iseitherapolynomialorrationalfunction.The-oreticallyspeaking,ficouldrepresentmorecomplicatedfunctions,suchasapiecewisefunction,afunctioninvolvingabsolutevaluefunction,etc.Inaddition,forncoupledself-excitedoscillatingsystems,thenconvergence-controlparameters~i,i=1,...,n,canbeintroducedintheframeofHAM.Inthisway,thesolutionseriesmayconvergeevenfaster.Infuture,theseproblemswillbeinvestigatedandfurtherimprovestothepackageNOPHwillbemadesothatthenewerversionsofNOPHarevalidformorecomplicatedperiodicoscillations.AcknowledgmentThisworkispartiallysupportedbytheNationalNaturalScienceFounda-tionofChina(ApprovalNo.11071274).AppendixA.TheprocedurescontainedintheNOPHThepackageNOPHiscomposedofthesixdifferentprocedures:main,Deoper,Approxnth,solanaly,contihigherandprocpade.Weoutlineeachofthemasfollows:•main:Thisisthemainprocedure,itsmaintaskistoinitializerelevantvariablesandtocallotherprocedures. 351October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations343•Deoper:Themaintaskofthisprocedureistochangevariables,andtodefinelinearaswellasnonlinearoperators.Meanwhile,theinputequationisdecomposedtoextractdependentandindependentvariablesaswellasunknownparameters.•Approxnth:Inthisprocedure,followtheprocessandstepsofHAM,wecomputethefirsthordercomponents(includingui(τ),ωi−1,δi−1,i=1,...,horder)ofthesolutionseries.•solanaly:Inthisprocedure,firstnormalizeallsolutionexpressions.Ifthereareunknownparameterscontainedineqnorinicon,callthefunctionprocpadetocomputethehomotopy-Pad´eapproximationsofthefrequencyωaswellasthemeanofmotionδ.Ifthereisnounknownparametercontainedineqnorinicon,wecomputethesquaredresidualerroroftheinputequation,theobtainedresultisafunctiondenotedby∆(~,τ).Furthertoeliminateτbyintegrating∆withrespecttoτfrom0to2π,theobtainedresultisdenotedasΛ(~).BylookingfortheminimumvalueofthecommonlogarithmofΛ(~)todeterminethevalueof~.•contihigher:Themaintaskofthisprocedureistocontinuetogetmoreaccurateapproximationsoftherequiredsolutions.Itistobestressedthatthispartofthecalculationisfreeoftheconvergence-controlparameter~.Evidently,onlywhenequations(8.1)and(8.3)donotcontainanyparameters,thefunctioncontihigherwillbecalled.•procpade:Thisprocedureisacompleteimplementationofthehomotopy-Pad´etechnique.AppendixB.AsimpleuserguideToloadthepackageNOPH,oneproceedsasfollows:>restart:initializingMaple.>currentdir(“D:/ect001a”);settingcurrentworkdirectory.Notethatthesourceprogramandthesamplefileshouldbelocatedinthecurrentworkdirectory.>read“oscillation.mpl”:readingtheprogramfile. 352October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8344Y.-P.Liu>with(NOPH);loadingthepackageNOPH.InwhatfollowsthreedifferenttypesofexamplesaregiventoshowhowtousethepackageNOPH.B.1.Example1ConsiderDuffingoscillationequationU¨+U+αU3=0,(B.1)subjecttoinitialconditionsU(0)=a,U˙(0)=0.(B.2)Thisisasingleconservativeoscillatorwithjustoddnonlinearity.ItistobestressedthatthepackageNOPHcansolvesingleconservativeoscillatorcompletelyautomatically.Forthesystem(B.1)and(B.2),twodifferentcasesareconsideredtoshowthispoint.B.1.1.AllphysicalparametersareunknownInthiscase,therearetwounknownparametersαandainthesystem(B.1)and(B.2).Forthissystem,onecouldrunthemainprocedureasfollows:>main([diff(U(t),t$2)+U(t)+alpha∗U(t)3=0],[U(0)=a,D(U)(0)=0],5).Astherearemorethanoneunknownparametersintheinputsystem,solorderisomitted.horder=5,thegoverningequationshouldbeputinalist,asshouldinitialconditions.ThenNOPHdeliversthefollowingresults:Theinputequationandinitialconditionsare:∂23U(t)+U(t)+αU(t)=0,∂t2U(0)=a,D(U)(0)=0.ThenewEQandinitialconditionsinuwiththeform:∂2ω2u(τ)+u(τ)+αu(τ)3=0,∂τ2 353October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations345u(0)=a,D(u)(0)=0,where:τ=ωt,U(t)=u(τ).Ittakesabout2.715secondstocompletethefirstpartcomputation;√(279α2a4+768αa2+512)3αa2+4ωP1,1=2(285α2a4+768αa2+512)ω=(574118847α7a14+5452491348α6a12+22121717760α5a10P2,2+49711251456α4a8+66831974400α3a6+53758656512α2a4+23957864448αa2+4563402752)/(2(195618321α6a12+1584935424α5a10+5335944192α4a8+9556131840α3a6+9602662400α2a4+5133828096αa2√+1140850688)3αa2+4).Ittakesabout0.281secondstocompletethesecondpartcomputation.Thetotaltimeis2.996seconds.B.1.2.Inthecaseofα=5,a=1/2Inthiscase,thereisnounknownparameterinthesystem(B.1)and(B.2).Tosolvethesystems(B.1)and(B.2),onecouldrunthemainprocedureasfollows:>main([diff(U(t),t$2)+U(t)+5∗U(t)3=0],[U(0)=1/2,D(U)(0)=0],4,7).Here,horder=4,solorder=7.Itmeansthattheoptimalvalueof~isdeterminedbythefirst4componentsofsolutionseries.Thencontinuetocalculatefrom5to7orderbythesecondpartcomputationtogetmoreaccurateapproximants.Forthisexample,thepackageNOPHoutputsthefollowingresults:Theinputequationandinitialconditionsare:∂23U(t)+U(t)+5U(t)=0,∂t2U(0)=1/2,D(U)(0)=0. 354October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8346Y.-P.LiuThenewEQandinitialconditionsinuwiththeform:∂2ω2u(τ)+u(τ)+5u(τ)3=0,∂τ2u(0)=1/2,D(u)(0)=0,where:τ=ωt,U(t)=u(τ).Ittakesabout0.889secondstocompletethefirstpartcomputation.Theoptimalvalueof~is:~=−1.0.Thecomparisongraphbetween5th-orderand7th-orderapproximationsofU(t):where:n=7.TheHAMapproximantsofωandthecorrespondingrelativeerrors: 355October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations347iωerrors11.3919−−−21.38520.004931.38520.000041.38510.000051.38510.0000TheHAM-Pad´eapproximantsofωandthecorrespondingrelativeerrors:iωerrors11.3851−−−21.38510.0000Ittakesabout1.809secondstocompletethesecondpartcomputation.Thetotaltimeis5.912seconds.B.2.Example2ConsidertheVanderPolequationU¨+U=ε(1−U(t)2)U˙(t),(B.3)subjecttotheinitialconditionU(0)=c1,U˙(0)=0.(B.4)Thisisasingleself-excitedoscillator.Wesolveitbyconsideringtwodif-ferentcases. 356October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8348Y.-P.LiuB.2.1.AllphysicalparametersareunknownTosolvethesystems(B.3)and(B.4),onecanrunthemainprocedureasfollows:>main([diff(U(t),t$2)+U(t)=∗(1−U(t)2)∗diff(U(t),t)],[U(0)=c[1],D(U)(0)=0],7).Astherearemorethanoneunknownparametersintheinputequation,solorderisomitted.Theprogramjustperformthefirstpartcomputation.ThenthepackageNOPHdeliversthefollowingresults:Theinputequationandinitialconditionsare:∂2∂U(t)+U(t)−(1−U(t)2)U(t)=0,∂t2∂tU(0)=c1,D(U)(0)=0.ThenewEQandinitialconditionsinuwiththeform:∂2∂∂ω2u(τ)+u(τ)−ωu(τ)+ωu(τ)2u(τ)=0,∂τ2∂τ∂τu(0)=c1,D(u)(0)=0,where:τ=ωt,U(t)=u(τ).Thepossiblesolutionsare:[{ω0=1,c1,0=2},{ω0=1,c1,0=−2}].Pleaseselectonesolutionandinputitsorder:,suchasinput1,thentheNOPHfurthergivesout:The1stsolutionwasselected.Ittakesabout108.218secondstocompletethefirstpartcomputation.ε2+32ωP1,1=23ε+32 357October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations34949152+9ε6+16640ε2+960ε4ωP2,2=49152+45ε6+19712ε2+1920ε4ω=(−3020544ε8−1970749440ε6+729ε12−109231931392ε4P3,3−1175344644096−2382205288448ε2+178848ε10)/(−73979136ε8−4347887616ε6−256206766080ε4+5103ε12−1175344644096−2455664328704ε2+598752ε10)where:r=3.Ittakesabout5.179secondstocompletethesecondpartcomputation.Thetotaltimeis113.397seconds.B.2.2.Inthecaseof=2Inthiscase,nounknownparameterisinvolvedinthesystem(B.3)and(B.4).Onecouldrunthemainprocedureasfollows:>main([diff(U(t),t$2)+U(t)=2(1−U(t)2)∗diff(U(t),t)],[U(0)=c[1],D(U)(0)=0],12,14).theNOPHoutputsthefollowingresults:Theinputequationandinitialconditionsare:d2dU(t)+U(t)=2(1−U(t)2)U(t),dt2dt 358October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8350Y.-P.Liu[U(0)=c1,D(U)(0)=0].ThenewEQandinitialconditionsinuwiththeform:d2dd22u(τ)ω+u(τ)−2u(τ)ω+2u(τ)u(τ)ω=0,dτ2dτdτ[u(0)=c1,D(u)(0)=0]where:[U(t)=u(τ),τ=ωt].Thepossiblesolutionsare:[{ω0=1,c1,0=0},{ω0=1,c1,0=−2},{ω0=1,c1,0=2}].Pleaseselectonesolutionandinputitsorder:,suchasinput3,thentheNOPHfurtherdelivers:The3rdsolutionwasselected.Ittakesabout9603.695secondstocompletethefirstpartcomputation.Theoptimalvalueof~is:~=−0.6655.Thecomparisongraphbetween12th-orderand14th-orderapproximationsofU(t):where:n=14.TheHAMapproximantsofωandc1: 359October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations351iωc111.00002.000020.83362.000030.81951.769440.81561.691850.81941.740560.81921.782770.82151.801680.82341.805690.82451.8042100.82461.8029110.82451.8010120.82421.7988130.82381.7970140.82321.7964TheHAM-Pad´eapproximantsofωandc1:iωc110.81822.000020.81471.766530.81711.766540.82571.801250.82381.801760.82321.7957Ittakesabout860.540secondstocompletethesecondpartcomputation.Thetotaltimeis:10464.235seconds. 360October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8352Y.-P.LiuB.3.Example3ConsideratwodirectlycoupledvanderPoloscillatorU¨(t)−ε(1−U2(t))U˙(t)+U(t)=εµ(V(t)−U(t)),(B.5)V¨(t)−ε(1−V2(t))V˙(t)+p2V(t)=εµ(U(t)−V(t)),withinitialconditionsU(0)=a1,U˙(0)=b1,V(0)=c1,V˙(0)=0,(B.6)whereε,µ,pareunknownphysicalparameters.Thissystemisacoupledself-excitedsystem.Wesolveitbyconsideringthreedifferentcases.B.3.1.AllphysicalparametersareunknownTosolvethesystems(B.5)and(B.6),onecanrunthemainprocedureasfollows:>main([diff(U(t),t$2)−epsilon∗(1−U(t)2)∗diff(U(t),t)+U(t)=epsilon∗mu∗(V(t)−U(t)),diff(V(t),t$2)−epsilon∗(1−V(t)2)∗diff(V(t),t)+p2∗V(t)=epsilon∗mu∗(U(t)−V(t))],[U(0)=a[1],D(U)(0)=b[1],V(0)=c[1],D(V)(0)=0],5).Forthisexample,thepackageNOPHoutputsthefollowingresults:Theinputequationandinitialconditionsare:d2dU(t)−(1−U(t)2)U(t)+U(t)=µ(V(t)−U(t)),dt2dtd2dV(t)−(1−V(t)2)V(t)+p2V(t)=µ(U(t)−V(t)),dt2dtU(0)=a1,D(U)(0)=b1,V(0)=c1,D(V)(0)=0.ThenewEQandinitialconditionsinuwiththeform:d2du(t)−(1−u(t)2)u(t)+u(t)+µ(u(t)−u(t))=0,dt211dt1112d2du(t)−(1−u(t)2)u(t)+p2u(t)+µ(u(t)−u(t)),dt222dt2221 361October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations353u1(0)=a1,D(u1)(0)=b1,u2(0)=c1,D(u2)(0)=0.where:τ=ωt,U(t)=u1(τ),V(t)=u2(τ).Thenonlinearalgebraicequationsforai,j&bi,jare:4(a−ωb−ω2a+µa−µc)+ω(a2b+b3)=0,1,001,001,01,01,001,01,01,0−ωa3+4b−ωab2+4ωa+4µb−4ω2b=0,01,01,001,01,001,01,001,0µa+ω2c−µc−p2c=0,1,001,01,01,0−−4ωc+ωc3+4bµ=0.01,001,01,0Thevariablesandparameterslists,respectively:[ω0,a1,0,b1,0,c1,0],[,µ,p].ThenthepackageNOPHasks:“Pleaseinputadditionalconditions,{}meansnoadditionalconditions”.Suchasinput{b1,0=0}tosimplifytheabovenonlinearalgebraicequa-tions,orelsetheNOPHoutputstoomanysolutionsanditisinconvenientforustoselectasolutionandinputitsorder.ThenthepackageNOPHdelivers:Thereducednonlinearalgebraicequationsread:4(−ω2a+a+µa−µc)=0,01,01,01,01,0ωa(a2−4)=0,01,01,0µa+ω2c−µc−p2c=0,1,001,01,01,0ωc(−4+c2)=0.01,01,0Next,thepackageNOPHasks:Pleaseinputavarandparameterlisttobesolved:Ifwefurtherinput[p1,p2,p3,4,2,1],inwhichpimeanstheithelementintheaboveparameterlist,theintegerimeanstheithelementinthevariablelist.ThentheNOPHoutputs:Thevariablesandparameterslisttobesolved: 362October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8354Y.-P.Liu[,µ,p,c1,0,a1,0,ω0].Thepossiblesolutionsare:{p=1,ω0=1,a1,0=2,b1,0=0,c1,0=2},{p=1,ω0=1,a1,0=−2,b1,0=0,c1,0=−2},√{p=1,ω0=1+2εµ,a1,0=2,b1,0=0,c1,0=−2},√{p=1,ω0=1+2εµ,a1,0=−2,b1,0=0,c1,0=2},(B.7){p=−1,ω0=1,a1,0=2,b1,0=0,c1,0=2},{p=−1,ω0=1,a1,0=−2,b1,0=0,c1,0=−2},√{p=−1,ω0=1+2εµ,a1,0=2,b1,0=0,c1,0=−2},√{p=−1,ω0=1+2εµ,a1,0=−2,b1,0=0,c1,0=2}.Selectasolutionandinputitsorder:,suchas4,theprogramfur-therdelivers:The4thsolutionwasselected.Theparameterconstraintconditionis:{p=1},the[1,1]homotopy-Pad´eapproximant√(ε2+64εµ+32)2εµ+1ωP1,1=2,3ε+64εµ+32andthe[2,2]homotopy-Pad´eapproximantω=(16640ε2+66560ε3µ+960ε4+294912εµ+66560ε4µ2P2,2√+589824ε2µ2+49152+393216ε3µ3+1920ε5µ+9ε6)2εµ+1/(45ε6+393216ε3µ3+589824ε2µ2+78848ε4µ2+78848ε3µ+294912εµ+3840ε5µ+19712ε2+49152+1920ε4).B.3.2.Inthecaseofε=1/2From(B.7),itcanbeseenthatthereisparameterconstraintp=1ineachsolution.Therefore,lettingε=1/2,therewillbeonlyoneunknownparameterµcontainedininputsystem.Inthiscase,onecanrunthemainprocedureasfollows:>main([diff(U(t),t$2)−1/2∗(1−U(t)2)∗diff(U(t),t)+U(t)=1/2∗mu∗(V(t)−U(t)),diff(V(t),t$2)−1/2∗(1−V(t)2)∗diff(V(t),t)+p2∗V(t)=1/2∗mu∗(U(t)−V(t))],[U(0)=a[1],D(U)(0)=b[1],V(0)=c[1],D(V)(0)=0],5); 363October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations355Followingthesimilarstepsshowninthelastsubsection,theprogramautomaticallydelivers(byselectingthe4thsolutionin(B.7)andsettinghorder=5)thefollowingresultswithin590seconds.Theparameterconstraintconditionis:{p=1},the[1,1]homotopy-Pad´eapproximant√(128µ+129)µ+1ωP1,1=,128µ+131andthe[2,2]homotopy-Pad´eapproximant√(9973504µ+9703424µ2+3415817+3145728µ3)µ+1ωP2,2=3468845+3145728µ3+9752576µ2+10075648µ.Thecomparisongraphbetweenωp[1,1]andωp[2,2].B.3.3.Inthecaseofε=1/2,µ=8Inthiscase,thereisnounknownparameterininputsystem,forthereisparameterconstraintp=1involvedineachsolution.Onecanrunthemainprocedureasfollows:>main([diff(U(t),t$2)−1/2∗(1−U(t)2)∗diff(U(t),t)+U(t)=4∗(V(t)−U(t)),diff(V(t),t$2)−1/2∗(1−V(t)2)∗diff(V(t),t)+p2∗V(t)= 364October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8356Y.-P.Liu4∗(U(t)−V(t))],[U(0)=a[1],D(U)(0)=b[1],V(0)=c[1],D(V)(0)=0],7,9).Followingthesimilarstepsshowninthelastsubsection(byselectingthe1stsolution,i.e.,{p=1,ω0=1,a1,0=2,b1,0=0,c1,0=2}),theprogramautomaticallyoutputs~=−0.489,approximantsofω,a1,b1,c1,aswellascomparisongraphsfordifferentorderapproximationsofU(t)andV(t)within755seconds. 365October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations357Theapproximantsofω,a1,b1andc1:iωa1b1c111.00002.00000.02.000020.98472.0000−0.36672.000030.98471.9689−0.36921.968940.98471.9684−0.33301.968450.98471.9698−0.33221.969860.98471.9698−0.33411.9698ThePad´eapproximantsofω,a1,b1andc1:iωa1b1c110.98472.0000−0.36922.000020.98471.9698−0.33561.969830.98471.9698−0.33411.9698References[1]M.V.Dyke,PerturbationMethodsinFluidMechanics.TheParabolicPress,Stanford,CA(1975).[2]A.H.Nayfeh,IntroductiontoPerturbationTechniques.Wiley,NewYork(1981).[3]E.J.Hinch,Perturbationmethods.CambridgeUniversityPress,Cambridge(1991).[4]A.W.Bush,PerturbationMethodsforEngineersandScientists.CRCPress,BocaRoton,FL.(1992).[5]A.M.Lyapunov,Thegeneralproblemofthestabilityofmotion,Int.J.Control.55(3),531–534(1992).[6]A.V.Karmishin,A.I.Zhukov,andV.G.Kolosov,MethodsofDynam-icsCalculationandTestingforThin-walledStructures.Mashinostroyenie,Moscow(1990).[7]G.Adomian,Nonlinearstochasticdifferentialequations,J.Math.Anal.Appl.55(2),441–452(1976).[8]S.Liao,BeyondPerturbation:IntroductiontotheHomotopyAnalysisMethod.Chapman&Hall/CRCPress,BocaRaton(2003). 366October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8358Y.-P.Liu[9]S.Liao,HomotopyAnalysisMethodinNonlinearDifferentialEquation.Springer&HigherEducationPress,Heidelberg&Beijing(2012).[10]S.Liao,Theproposedhomotopyanalysistechniqueforthesolutionofnon-linearproblems,PhDthesis,ShanghaiJiaoTongUniversity(1992).[11]S.Liao,Auniformlyvalidanalyticsolutionoftwo-dimensionalviscousflowoverasemi-infiniteflatplate,J.FluidMech.385(1),101–128(1999).[12]S.LiaoandA.Campo,AnalyticsolutionsofthetemperaturedistributioninBlasiusviscousflowproblems,J.FluidMech.453,411–425(2002).[13]S.Liao,Onthehomotopyanalysismethodfornonlinearproblems,Appl.Math.Comput.147(2),499–513(2004).[14]S.Liao,Anewbranchofsolutionsofboundary-layerflowsoveranimperme-ablestretchedplate,Int.J.HeatMassTransfer.48(12),2529–2539(2005).[15]S.LiaoandY.Tan,Ageneralapproachtoobtainseriessolutionsofnonlineardifferentialequations,Stud.Appl.Math.119(4),297–354(2007).[16]S.Liao,Anewbranchofsolutionsofboundary-layerflowsoverapermeablestretchingplate,Int.J.Nonlin.Mech.42(6),819–830(2007).[17]T.HayatandM.Sajid,Onanalyticsolutionforthinfilmflowofafourthgradefluiddownaverticalcylinder,Phys.Lett.A.361(4),316–322(2007).[18]M.Sajid,T.Hayat,andS.Asghar,ComparisonbetweentheHAMandHPMsolutionsofthinfilmflowsofnon-Newtonianfluidsonamovingbelt,NonlinearDyn.50(1-2),27–35(2007).[19]S.Abbasbandy,Theapplicationofhomotopyanalysismethodtononlinearequationsarisinginheattransfer,Phys.Lett.A.360(1),109–113(2006).[20]S.Abbasbandy,TheapplicationofhomotopyanalysismethodtosolveageneralizedHirota–SatsumacoupledKdVequation,Phys.Lett.A.361(6),478–483(2007).[21]S.Abbasbandy,Homotopyanalysismethodforheatradiationequations,Int.Commun.HeatMassTransfer.34(3),380–387(2007).[22]Y.LiuandZ.Li,Homotopyanalysismethodfornonlineardifferentialequa-tionswithfractionalorders,Z.Naturforsch.A.63(5-6),241–247(2008).[23]Y.Liu,R.Yao,andZ.Li,Anapplicationofahomotopyanalysismethodtononlinearcomposites,J.Phys.A:Math.Theor.42(12),125205(2009).[24]Y.Liu,S.Liao,andZ.Li,Symboliccomputationofstronglynonlinearperiodicoscillations,JSymb.Comput.(2013).[25]W.J.Wu,MechanicalTheoremProvinginGeometries:BasicPrinciples.Springer-Verlag,NewYork(1994).[26]D.Wang,AnImplementationoftheCharacteristicSetMethodinMaple.Springer-Verlag,WienNewYork(1995).[27]D.Wang,EliminationPractice:SoftwareToolsandApplications.ImperialCollegePress,London(2004).[28]A.Bel´endez,G.Bernabeu,J.Franc´es,D.M´endez,andS.Marini,Anac-curateclosed-formapproximatesolutionforthequinticDuffingoscillatorequation,Math.Comput.Model.52(3),637–641(2010).[29]R.H.Rand,UsingComputerAlgebratoHandleEllipticFunctionsintheMethodofAveraging.vol.205,SymbolicComputationsandTheirImpactonMechanics,ASMEPVP(1992). 367October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.8HAM-BasedPackageNOPHforPeriodicOscillations359[30]H.Hu,Exactsolutionofaquadraticnonlinearoscillator,J.SoundVib.295(1),450–457(2006).[31]S.ChenandC.Chen,Applicationofthedifferentialtransformationmethodtothefreevibrationsofstronglynon-linearoscillators,NonlinearAnal.R.WorldAppl.10(2),881–888(2009). 368May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 369October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9Chapter9HAM-BasedMathematicaPackageBVPh2.0forNonlinearBoundaryValueProblemsYinlongZhao∗andShijunLiao†SchoolofNavalArchitecture,OceanandCivilEngineeringShanghaiJiaoTongUniversity,Shanghai200240,China∗zhaoyl@sjtu.edu.cn,†sjliao@sjtu.edu.cnWeissuethenewversionBVPh2.0oftheMathematicapackageBVPh,afreesoftwarebasedonthehomotopyanalysismethod(HAM)fornon-linearboundary-valueandeigenvalueproblems.Theaimofthepack-ageBVPhistodevelopakindofanalytictoolforasmanynonlinearboundaryvalueproblems(BVPs)aspossiblesuchthatmultipleso-lutionsofhighlynonlinearBVPscanbeconvenientlyfoundout,andthattheinfiniteintervalandsingularitiesofgoverningequationsand/orboundaryconditionsatmulti-pointscanbeeasilyresolved.Unlikeitspreviousversions,BVPh2.0worksforsystemsofcouplednonlinearor-dinarydifferentialequations.Itisuser-friendlyandfreeavailableon-line(http://numericaltank.sjtu.edu.cn/BVPh.htm).Differentfromnumericalpackages(suchasBVP4c),itisbasedontheidea“comput-ingnumericallywithfunctionsinsteadofnumbers”.Especially,un-likeotherpackages,theconvergenceofresultsgivenbytheBVPh2.0isguaranteedbymeansoftheso-calledconvergence-controlparameterintheframeofthehomotopyanalysismethod.Inthischapter,webrieflydescribehowtoinstallandusetheBVPh2.0withasimpleuser’sguide.Fivetypicalexamples(governedbyuptofourcoupledODEs)areusedtoillustratethevalidityoftheBVPh2.0,andthecorrespond-inginputdataoftheseexamplesfortheBVPh2.0arefreeavailableon-line(http://numericaltank.sjtu.edu.cn/BVPh.htm).TheBVPh2.0indeedprovidesuswithaneasy-to-usetooltoefficientlysolvevarioustypesofcoupledlinear/nonlinearordinarydifferentialequations.361 370October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9362Y.-L.ZhaoandS.-J.LiaoContents9.1.Introduction.....................................3629.2.InstallationoftheBVPh2.0............................3669.3.Illustrativeexample.................................3679.4.Briefmathematicalformulas............................3729.4.1.Boundaryvalueproblemsinafiniteinterval...............3729.4.2.Eigenvalue-likeproblemsinafiniteinterval................3759.4.3.Problemsinasemi-infiniteinterval....................3759.5.Approximationanditerationofsolutions.....................3769.5.1.Polynomials.................................3779.5.2.Trigonometricfunctions...........................3779.5.3.Hybrid-basefunctions............................3789.6.AsimpleuserguideoftheBVPh2.0.......................3809.6.1.Keymodules.................................3809.6.2.Controlparameters.............................3819.6.3.Input.....................................3839.6.4.Output....................................3849.6.5.Globalvariables...............................3849.7.Examples......................................3859.7.1.Example1:AsystemofODEsinfiniteinterval.............3859.7.2.Example2:AsystemofODEswithalgebraicpropertyatinfinity...3889.7.3.Example3:AsystemofODEswithanunknownparameter......3939.7.4.Example4:AsystemofODEsindifferentintervals...........3979.7.5.Example5:IterativesolutionsoftheGelfandequation.........4019.8.Conclusions.....................................405AppendixA.Codesforexamples............................405A.1.Samplecodestoruntheillustrativeexample...................406A.2.InputdataofBVPh2.0fortheillustrativeexample...............406A.3.InputdataofBVPh2.0forExample1......................408A.4.InputdataofBVPh2.0forExample2......................409A.5.InputdataofBVPh2.0forExample3......................411A.6.InputdataofBVPh2.0forExample4......................412A.7.InputdataofBVPh2.0forExample5......................414References.........................................4169.1.IntroductionOrdinarydifferentialequations(ODEs)arewidelyusedinmathematics,scienceandengineering.Theso-calledinitialvalueproblems(IVPs)specifysomerestrictionsonlyatasinglepoint.Thiskindofproblemsisoftensolvedbymeansofnumericalapproachbasedonintegration,suchastheRunge–Kuttamethod.However,inmanycases,asolutionisdescribedinamorecomplicatedway.Theso-calledboundaryvalueproblems(BVPs) 371October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems363mayspecifysomerestrictionsatmorethanonepoint.UnlikeIVPs,BVPsmighthavemultiplesolutions,ormaysatisfysomerestrictionsatinfinity.Hence,generallyspeaking,BVPsaremoredifficulttosolvethanIVPs.ThankstothedevelopmentofmathematicalsoftwaresuchasMaple,MathematicaandMATLAB,somepackagesaredevelopedtosolvenonlin-earBVPs,suchastheBVP4c,theChebfun,theNOPH,andtheBVPh.ThefamousMATLABpackageBVP4c[1,2]implementsacollocationmethod,insteadofashootingmethod.ThecollocationpolynomialprovidesaC1-continuoussolutionthatisfourth-orderaccurateuniformlyin[a,b].Meshselectionanderrorcontrolarebasedontheresidualofthecontinuousso-lution.However,itisnoteasyfortheBVP4ctoresolvethesingularityingoverningequationsand/orboundaryconditions.Besides,theBVP4cre-gardsaninfiniteintervalasakindofsingularityandreplacesitbyafiniteone:thisleadstotheadditionalinaccuracyanduncertaintyofsolutions.TheChebfun[3,4]isacollectionofalgorithmsonthe“chebfun”ob-jectswritteninMATLAB.Itaimstocombinethefeelofsymbolicswiththespeedofnumerics.ThebasisoftheChebfunisChebyshevexpan-sions,fastFouriestransform,baryentricinterpolationandsoon.Theidea“computingnumericallywithfunctionsinsteadofnumbers”[4]behindtheChebfunmakesithavethepotentialtohandleunboundeddomainsandsingularitiesinaneasyway.AlthoughlineardifferentialequationscanbesolvedinasinglestepbyChebfunwithoutiteration,onlyafewexam-plesfornonlineardifferentialequationsaregiven[3,4].Actually,ChebfunusesNewton’siterationtosolvenonlinearproblems.However,itiswellknownthattheconvergenceofNewton’siterationisstronglydependentuponinitialguessesandthusisnotguaranteed.Besides,Chebfunsearchesformultiplesolutionsofnonlineardifferentialequationsbyusingdifferentguessapproximations.However,itisnotveryclearhowtochoosethesedifferentguessapproximations.Basedonthehomotopyintopology,theso-calledhomotopyanalysismethod(HAM)wasproposedbyLiao[8–14]togainanalyticapproxima-tionsofhighlynonlinearproblems.TheHAMhassomeadvantagesoverothertraditionalanalyticapproximationmethods.First,unlikeperturba-tiontechniques,theHAMisindependentofsmall/largephysicalparam-eters,andthusisvalidinmoregeneralcases.Besides,differentfromall 372October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9364Y.-L.ZhaoandS.-J.Liaootheranalytictechniques,theHAMprovidesusaconvenientwaytoguar-anteetheconvergenceofseriessolution.Furthermore,theHAMprovidesextremelylargefreedomtochooseinitialguess,equation-typeandsolutionexpressionoflinearsub-problems.Itisfound[13,14]thatlotsofnonlinearBVPsinscience,engineeringandfinancecanbesolvedconvenientlybymeansoftheHAM,nomatterwhethertheintervalisfiniteornot.SincetheearlyHAMwasproposedbyLiao[9]in1992,theHAMhasbeendevelopedgreatlyintheoryandwidelyappliedtonumerousnonlinearproblemsinlotsoffields,withhundredsofpublications.So,itisneces-sarytodevelopaHAM-basedsoftwaretosimplifytheapplicationsoftheHAM.BasedontheHAM,aMathematicapackageBVPh1.0—fornonlin-earboundaryvalue/eigenvalueproblemswithsingularityand/ormultipointboundaryconditionswasissuedbyLiao[14]inMay2012,whichisfreeavail-ableonline(http://numericaltank.sjtu.edu.cn/BVPh.htm).ItsaimistodevelopakindofanalytictoolforasmanynonlinearBVPsaspossiblesuchthatmultiplesolutionsofhighlynonlinearBVPscanbeconvenientlyfoundout,andthattheinfiniteintervalandsingularitiesofgoverningequa-tionsand/orboundaryconditionsatmulti-pointscanbeeasilyresolved.AsillustratedbyLiao[14],theBVPh1.0isvalidforlotsofnonlinearBVPsandthusisausefultoolinpractice.Currently,basedontheHAM,theMaplepackageNOPH[5]forperiodi-callyoscillatingsystemsofcenterandlimitcycletypesisdeveloped,whichdeliversaccurateapproximationsoffrequency,meanofmotionandam-plitudeofoscillationautomatically.TheNOPHcombinesWu’seliminationmethodandthehomotopyanalysismethod(HAM).Itisfreeavailableon-line(http://numericaltank.sjtu.edu.cn/NOPH.htm).DifferentfromtheBVPh1.0,theNOPHisforperiodicoscillations.ThisillustratesthegeneralvalidityoftheHAMfornonlinearproblemsonceagain.ItisapitythattheBVPh1.0canonlydealwithBVPsofsingleordi-narydifferentialequation(ODE),say,itcannotsolvesystemsofcoupledODEs.InthischapterweissuethenewversionBVPh2.0,whichworksformanytypesofsystemsofcouplednonlinearordinarydifferentialequations(ODEs)infiniteand/orsemi-infiniteintervals.Besides,newalgorithmsareusedinsomemodulesofBVPh2.0.Asaresult,BVPh2.0ismuchfasterthanBVPh1.0inmostcases.Inthischapter,weillustratehowtouse 373October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems365BVPh2.0tosolvedifferentkindsofsystemsofcouplednonlinearODEs,includingasystemoftwocoupledODEsinfiniteinterval,asystemoftwocoupledODEsinsemi-infiniteinterval,asystemoftwocoupledODEswithalgebraicpropertyatinfinity,asystemofthreecoupledODEswithanun-knownparametertobedetermined,andasystemoffourcoupledODEsindifferentintervals.Thischapterisorganizedasfollows.In§9.2,weshowhowtoinstallthepackageBVPh2.0.In§9.3,wetakeanexampletoillustratehowtouseBVPh2.0indetail.In§9.4,webrieflydescribesomemathematicalformulasbasedonwhichtheBVPh2.0isdeveloped.In§9.5,theiterativetechniqueisillustratedfortwotypicalkindsofbase-functionsinfinitein-tervals.AsimpleuserguideofBVPh2.0isgivenin§9.6.Sometypicalexamplesaregivenin§9.7toillustratethepotentialoftheBVPh2.0andtoshowitsvalidity.In§9.8,somediscussionsaregiven.Thereaderissuggestedtoread§9.3atfirstforanillustrativeexample.Ifoneispuzzledwithsomeparameters,oneisencouragedtosearchforthemin§9.6,andreadthedetaileddescriptionthere.InordertocheckthevalidityandcorrectnessoftheBVPh2.0,wechooseallofourtypicalexamplesfrompublishedarticles,whichweresolvedei-theranalytically(bytheHAM)ornumericallybefore.Asillustratedinthischapter,thepackageBVPh2.0alwaysgivesresults,whichagreewellwiththepublishedones.ThevalidityoftheBVPh2.0ischeckedforeachexampleinthefollowingway:(1)thesquaredresidualerrorusuallydecreasestoaslowas10−10,anddecreasesatleast6ordersofmagnitude;(2)thesamephysicalquantitiesofinterestaregainedasthepublishedones;(3)analyticsolutionsgainedbyBVPh2.0agreewellwiththepublishedones.NotethatthepackageBVPh2.0isdevelopedwithMathematica7.0.AssomenewfeaturesofMathematica7.0areused,westronglyrecommendyoutousetheBVPh2.0inMathematica7.0orhigherversion,sinceitmightnotworkforsomelowerversionofMathematics. 374October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9366Y.-L.ZhaoandS.-J.Liao9.2.InstallationoftheBVPh2.0TheBVPhisafree/open-sourcesoftwarewritteninMathematicaforbound-aryvalueproblems(BVPs).ItsnewestversionBVPh2.0isavailableon-line(http://numericaltank.sjtu.edu.cn/BVPh.htm).Theinputdataforthischapter’sexamplescanalsobefoundthere.SincethecommandsofMathematicaaredesignedtobethesameondifferentoperatingsystems,thepackagewritteninMathematicaonWindowscanbeusedinMathe-maticaonotheroperatingsystems.ThesourcefileofthepackageBVPh2.0isBVPh2_0.m.TheeasiestwaytoloadthepackageBVPh2.0tosolveyourproblemistoputthefileBVPh2_0.mandtheinputdatafortheproblem,e.g.,Example.m,inthesamedirectory,thenopenanewnotebookfileandsaveditas,e.g.,runExample.nb,inthesamedirectoryandrunthefollowingcodes.(*Filename:runExample.nb*)ClearAll["Global‘*"];SetDirectory[ToFileName[Extract["FileName"/.NotebookInformation[EvaluationNotebook[]],{1},FrontEnd‘FileName]]];< 375October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems367Example.minthenotebookfilerunExample.nbareasfollowsonWindows.(*Filename:runExample.nb*)ClearAll["Global‘*"];<<"E:\\Package\\BVPh2_0.m";<<"E:\\Project\\Example\\Example.m";HereweassumethefileBVPh2_0.minthedirectoryE:\PackageandtheinputdataExample.minadifferentdirectoryE:\Project\Example.TheabovesamplecodesmaylooklikethefollowingonUnix(*Filename:runExample.nb*)ClearAll["Global‘*"];<<"/home/user/Package/BVPh2_0.m";<<"/home/user/Example/Example.m";HereweassumeBVPh2_0.minthedirectory/home/user/Package/andtheinputdataExample.minadifferentdirectory/home/user/Example.Fromnowon,wewillassumethatthepackageBVPh2.0hasbeensuccessfullyloadedsothatthemodulesinthepackageareavailable.Inthenextsection,wewilluseanillustrativeexampletoshowhowtowritetheinputdataandhowtogettheapproximationsbyBVPh2.0.9.3.IllustrativeexampleConsiderasystemofODEs[6]00002000002000f−(f)+ff+2λg+β[2fff−ff]=0,(9.1)0000000200g−fg+fg−2λf+β[2ffg−fg]=0,(9.2)subjecttof0(0)=1,f(0)=0,g(0)=0,(9.3)f0(∞)=0,g(∞)=0,(9.4)whereλisrotationparameter,βisviscoelasticparameter,andtheprimeindicatesthedifferentiationwithrespecttoη.Thissystemmodelstwo-dimensionalflowofanupperconvectedMaxwellfluidinarotatingframe.Sajid[6]hassolveditbytheHAM. 376October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9368Y.-L.ZhaoandS.-J.LiaoTosolvethisproblembyBVPh2.0,wehavetoinputthedifferentialequations,boundaryconditions,initialguessesandconvergence-controlpa-rameters.Thedifferentialequations(9.1)and(9.2)canbecodedasfollowsTypeEQ=1;NumEQ=2;f[1,z_,{f_,g_},Lambda_]:=D[f,{z,3}]-D[f,z]^2+f*D[f,{z,2}]+2*la*g+beta*(2*f*D[f,z]*D[f,{z,2}]-f^2*D[f,{z,3}]);f[2,z_,{f_,g_},Lambda_]:=D[g,{z,2}]-D[f,z]*g+f*D[g,z]-2*la*D[f,z]+beta*(2*f*D[f,z]*D[g,z]-f^2*D[g,{z,2}]);HereTypeEQcontrolsthetypeofgoverningequations:TypeEQ=1cor-respondstoasystemofODEswithoutanunknowntobedetermined,TypeEQ=2correspondstoasystemofODEswithanunknown,Lambda,tobedetermined.Sincealltheparametersintheproblemwillbegiven,wesetTypeEQto1.NotethatweusethedelayedassignmentSetDelayed(:=)inMathematicatodefinetheseODEstoavoidtheevaluationwhentheassignmentismade.Theboundaryconditions(9.3)and(9.4)aredefinedinasemi-infiniteinterval,from0to+∞.TheyarecodedasNumBC=5;BC[1,z_,{f_,g_}]:=(D[f,z]-1)/.z->0;BC[2,z_,{f_,g_}]:=f/.z->0;BC[3,z_,{f_,g_}]:=g/.z->0;BC[4,z_,{f_,g_}]:=D[f,z]/.z->infinity;BC[5,z_,{f_,g_}]:=g/.z->infinity;HereNumBCisthenumberofboundaryconditionsoftheproblem.Forthisproblem,wehave5boundaryconditions,soNumBCissetto5.Thesymbolinfinityisintroducedinourpackagetodenote∞.Whenanexpressioncontainsinfinity,thelimitoftheexpressioniscomputedaszapproaches∞.Thedelayedassignment(:=)isalsousedtoavoidtheevaluationwhentheassignmentismade—thesamereasonasdefiningthedifferentialequations. 377October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems369Foramulti-layerproblem,thedifferentialequationsinthesystemarenotnecessarilyinthesameinterval(seeExample4in§9.7).Hence,wehavetogiveeachequationitssolutioninterval.Tomeasuretheaccuracyoftheapproximatesolutions,wehavetocomputethesquaredresidualerroroverthecorrespondingsolutioninterval.Inpractice,whenthedifferentialequationisdefinedinasemi-infiniteinterval,wesimplytruncatetheinfiniteintervaltoafiniteintervaltocomputethesquaredresidualerror,oritwilltakealotofcomputationtime.Forthisproblem,thesolutionintervalforeachequationandtheintegralintervalforthesquaredresidualerroraredefinedaszL[1]=0;zR[1]=infinity;zL[2]=0;zR[2]=infinity;zRintegral[1]=10;zRintegral[2]=10;HerezL[k](orzR[k])istheleft(orright)endpointofthesolutioninter-valforthekthequationf[k,z,{f,g},Lambda].AndzLintegral[k](orzRintegral[k])istheleft(orright)endpointoftheintegralintervaltocomputethesquaredresidualerrorforthekthequation.IfthevalueofzL[k](orzR[k])isafinitenumber,zLintegral[k](orzRintegral[k])issettothesamevalueautomatically.However,ifanyofthemcontainsthesymbolinfinity,wehavetosetthecorrespondingendpointoftheintegralintervaltoafinitevalue.ThatiswhywewriteexplicitlyzRintegral[1]=10andzRintegral[2]=10.Forthisproblem,thesquaredresidualisinte-gratedovertherange[0,10]forbothequations.TheauxiliarylinearoperatorsforthisproblemarechosenasL1=32∂−∂,L=∂−1,whicharecodedas∂η3∂η2∂η2L[1,u_]:=D[u,{z,3}]-D[u,z];L[2,u_]:=D[u,{z,2}]-u;HereL[k,u]istheauxiliarylinearoperatorcorrespondingtothekthequa-tion.Notethati)ηistheindependentvariableinthedifferentialequations 378October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9370Y.-L.ZhaoandS.-J.Liao(9.1)and(9.2),whilezistheuniversalindependentvariableinthepackageBVPh2.0;ii)thedelayedassignmentSetDelayed(:=)isusedtodefinetheoperator;iii)uisaformalparameter.Forthisproblem,theinitialguessesaref=1−e−zandg=ze−z.00TheyarecodedasU[1,0]=1-Exp[-z];U[2,0]=alpha*z*Exp[-z];Herealphaisanintroducedconvergence-controlparameterthatwillbedeterminedlater.U[k,0]istheinitialguessofthekthequation.NotethatU[k,0]andu[k,0]areusuallythesameinthepackageBVPh2.0.Wewanttosolvethisproblemwhenthephysicalparametersβ=1/5andλ=1/10.Thesetwoparametersarecodedasbeta=1/5;la=1/10;Sofar,wehavedefinedalltheinputofthisproblemproperly,excepttheconvergence-controlparameterc0[k]andalpha.Usually,theoptimalvaluesoftheconvergence-controlparametersareobtainedbyminimizingtheaveragedsquaredresidualerror.Forthisproblem,wegettheapproxi-mateoptimalvaluesofc0[1],c0[2]andalphabyminimizingthesquaredresidualerrorofthe3rd-orderapproximationasGetOptiVar[3,{},{c0[1],c0[2],alpha}];ThefirstparameterofGetOptiVardenoteswhichorderapproximationisused.Here3meansthe3rd-orderapproximationisused.Thesecondparameterdenotesalistofconstraintsusedintheoptimization.WhenthesecondparameterofGetOptiVarisanemptylist,itmeanstheaver-agedsquaredresidualisminimizedwithoutanyconstraint.Hereweaddnoconstraintstominimizetheaveragedsquaredresidualerror.Thethirdparameterisalistofthevariablestobeoptimized.Herewewanttoopti-mizec0[1],c0[2]andalpha.Aftersomecomputation,itgivestheopti-mizedconvergence-controlparametersc0[1]=-1.26906,c0[2]=-1.19418 379October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems371andalpha=-0.0657063.NowwecanuseBVPh[1,10]togetthe10th-orderapproximation.Ifwearenotsatisfiedwiththeaccu-racyofthe10th-orderapproximation,wecanuseBVPh[11,20],insteadofBVPh[1,20],toget20th-orderapproximationorhigherorderapproxima-tion.Thekth-orderapproximationoftheithdifferentialequationisstoredinU[i,k].WecanusePlot[{U[1,20],U[2,20]},{z,0,10},AxesLabel->{"\[Eta]",""},PlotStyle->{{Thin,Red},{Dashed,Blue}}]toplotthe20th-orderapproximatesolution,whichisshowninFig.9.1.Theaccuracyofthekth-orderapproximationismeasuredbytheaver-agedsquaredresidual.WecanuseListLogPlot[Table[{2*i,ErrTotal[2*i]},{i,1,10}],Joined->True,Mesh->All,PlotRange->{{2,20},{10^(-15),10^(-5)}},AxesLabel->{"m","error"}];toplotthecurveofthetotalerrorversustheorderofapproximation,whichisshowninFig.9.2.NotethatErrTotal[k]storesthetotalerrorofthesystemwhenthekth-orderapproximationisused,whileErr[k]isalistthatstorestheerrorforeachODEinthesystemwhenthekth-orderap-proximationisused. 380October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9372Y.-L.ZhaoandS.-J.Liao0.80.60.40.2Η246810Fig.9.1.Thecurveoff(z)(solid)andg(z)(dashed)fortheillustrativeexamplewhenβ=1/5,λ=1/10.error10-510-710-910-1110-1310-15m5101520Fig.9.2.Totalerrorvs.orderofapproximationfortheillustrativeexamplewhenβ=1/5,λ=1/10.9.4.BriefmathematicalformulasTheBVPh2.0isbasedontheHAM.ItisanextensionofBVPh1.0tosystemsofODEs.Herethemathematicalformulasarebrieflydescribed.9.4.1.BoundaryvalueproblemsinafiniteintervalConsiderasystemofNumEQordinarydifferentialequations(ODEs),Fi[z,u1,u2,···]=0,z∈[zLi,zRi],16i6NumEQ,(9.5) 381October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems373subjecttotheNumBClinearboundaryconditionsBk[z,u1,u2,···]=γk,16k6NumBC,(9.6)whereFiisanonlineardifferentialoperator,Bkisalineardifferentialop-erator,zistheindependentvariable,ui(z)isasmoothfunction,NumEQisapositiveinteger,γk,zLiandzRiareconstants,respectively.Thebound-aryconditionsmaybedefinedatmultipointsincludingthetwoendpoints.Assumethatatleastonesolutionexists,andthatallsolutionsaresmooth.Letq∈[0,1]denoteanembeddingparameter,ui,0(z)aninitialguessofthesolutionui(z),respectively.IntheframeoftheHAM,weconstructsuchacontinuousdeformationφi(z;q)that,asqincreasesfrom0to1,φi(z;q)variescontinuouslyfromtheinitialguessui,0(z)tothetruesolutionui(z)of(9.5)and(9.6).Suchkindofcontinuousdeformationsaregovernedbytheso-calledzeroth-orderdeformationequations(1−q)Li[φi(z;q)−ui,0(z)]=qc0,iHi(z)Fi[φ1(z;q),φ2(z;q),···],z∈[zLi,zRi],q∈[0,1],16i6NumEQ(9.7)whereLiisanauxiliarylinearoperator,c0,iistheso-calledconvergence-controlparameter,Hi(z)isanauxiliaryfunction,correspondingtotheithgoverningequationin(9.5),respectively.NotethattheHAMprovidesusextremelylargefreedomtochoosetheauxiliarylinearoperatorLi,theconvergence-controlparameterc0,iandtheauxiliaryfunctionHi(z),16i6NumEQ.Assumethatallofthemareproperlychosensothatthehomotopy-MaclaurinseriesX+∞kφi(z;q)=ui,0(z)+ui,k(x)q(9.8)k=1absolutelyconvergesatq=1,where1∂mφ(z;q)ui,m=Dm[φ(z;q)]=.(9.9)m!∂qmq=0Here,Dmiscalledthemth-orderhomotopy-derivativeoperator[8].Then,wehavetheso-calledhomotopy-seriessolutionX+∞ui(z)=ui,0+ui,m(z),(9.10)m=1 382October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9374Y.-L.ZhaoandS.-J.Liaowhereui,misgovernedbytheso-calledmth-orderdeformationequationLi[ui,m(z)−χmui,m−1(z)]=c0,iHi(z)δi,m−1(z),z∈[zLi,zRi],(9.11)subjecttotheNumBChomogeneouslinearboundaryconditionsBk[z,u1,u2,···]=0,16k6NumBC,(9.12)where(0,m≤1,χm=(9.13)1,m>1,andδi,m−1(z)=Dm−1{Fi[φ1(z;q),φ2(z;q),···]}("#)mX−1mX−1=DFu(z)qi,u(x)qi,···(9.14)m−1i1,i2,ii=0i=0canbeeasilyobtainedbymeansofthetheoremsprovedinRef.8.Themth-orderapproximationofui(z)isgivenbyXmui(x)≈Ui,m(x)=ui,k(z).(9.15)k=0Tomeasuretheaccuracyofthemth-orderapproximationsUi,m,16i6NumEQ,theaveragedsquaredresidualerrorforthesystem(9.5)isdefinedasNumEQRzRintegral[i]2XzLintegral[i]|Fi[U1,m,U2,m,···]|dzEm=.(9.16)zRintegral[i]−zLintegral[i]i=1HerezLintegral[i]andzRintegral[i]aretwoendpointsoftheintegralintervaloverwhichthesquaredresidualoftheithgoverningequationisintegrated.Theoreticallyspeaking,thesmallerEm,themoreaccuratethemth-orderapproximationUi,m(z).Giventheinitialguessui,0(z),theauxiliarylinearoperatorLiandtheauxiliaryfunctionHi(z),thesquaredresidualerrorEmisdependentontheconvergence-controlparametersc0,i,i=1,2,···,NumEQ.Hence,theoptimalvaluesofc0,icanbedeterminedbytheminimumofEmatsomeproperorderm.Notethati)forboundaryvalueproblemswithoutanunknowntobedetermined,weshouldsetthecontrolparameterTypeEQ=1fortheBVPh2.0;ii)theintervals[zLi,zRi]arenotnecessarilythesame,sothepackagecansolvemulti-layerproblemswithoutanymodification. 383October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems3759.4.2.Eigenvalue-likeproblemsinafiniteintervalConsiderasystemofNumEQordinarydifferentialequations(ODEs)Fi[z,u1,u2,···,λ]=0,z∈[zLi,zRi],16i6NumEQ,(9.17)subjecttotheNumBClinearboundaryconditions,Bk[z,u1,u2,···]=γk,16k6NumBC.(9.18)Notethatthereisanunknownparameterλinthesystem(9.17),andtheboundaryconditions(9.18)includeanadditionalboundaryconditiondeterminingtheunknownλ.ThisadditionalboundaryconditionisthezerothboundaryconditioninBVPh1.0.However,itistreatedthesameastheotherboundaryconditionsinBVPh2.0,thatis,anyoneoftheboundaryconditionsin(9.18)canbetheadditionalboundaryconditionprovidedthatNumEQ−1istheorderofthesystem(9.17).Allrelatedformulasarethesameasthosegivenin§9.4.1,exceptthatadeformationΛ(q)isalsoconstructedsuchthatΛ(q)variescontinuouslyfromtheinitialguessλ0toλasqincreasesfrom0to1.Notethat,whentheBVPh2.0isusedtosolveeigenvalue-likeproblemsinafiniteinterval,weshouldsetTypeEQ=2.9.4.3.Problemsinasemi-infiniteintervalWhentherightendpointzRiofthesolutionintervalin(9.5)is+∞,thegoverningequationisdefinedinasemi-infiniteinterval[zLi,+∞].TheBVPhcansolvethiskindofproblemwithouttruncatingthedomain.ThisfeatureisquitedifferentfromBVP4c,whichregardsasemi-infiniteintervalasakindofsingularityandreplacesitbyafiniteone.Allrelatedformulasarethesameasthosegivenin§9.4.1,exceptthatthefiniteintervalsz∈[zLi,zRi]arereplacedbythesemi-infiniteonesz∈[zLi,∞].However,completelydifferentinitialguessesui,0(z)andaux-iliarylinearoperatorsLiareusedforthiskindofproblems,becausetheirsolutionsareexpressedbycompletelydifferentbasefunctions.Notethati)whentheBVPh2.0isusedtosolveproblemsinasemi-infiniteinterval,thesymbolinfinityisusedtodenote∞ininputdata;ii)theboundaryconditionsatinfinityareconsideredasalimitingprocess; 384October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9376Y.-L.ZhaoandS.-J.Liaoiii)togettheerroroftheapproximations,wehavetoprovidefiniteintervalsoverwhichthesquaredresidualsareintegrated,asthenumericalintegrationmethodisusedinBVPh2.0;iv)theintegralintervalforsquaredresidualerrorcanbesetthroughzLintegral[i]andzRintegral[i].Forexample,zLintegral[i]=0andzRintegral[i]=10meansthesquaredresidualoftheithgoverningequationisintegratedover[0,10].9.5.ApproximationanditerationofsolutionsAlthoughthehigh-orderdeformationequations(9.11)arealwayslinear,theyarestillnoteasytosolveingeneral,becausetheright-handsidetermδi,m−1mayberathercomplicated.Tosolvethisproblem,wecanapproxi-mateδi,m−1intheformXNtδi,m−1≈bkek(z)k=0whereekdenotesthebase-function,Ntdenotesthenumberoftruncatedterms,thecoefficientbkisuniquelydeterminedbyδi,m−1(z)andthebase-functionsek(z).Tofurtheracceleratetherateofconvergence,wecanemploytheitera-tionapproachbyusingtheMth-orderapproximationasanewinitialguessu∗(z),i.e.,i,0XMu(z)+u(z)→u∗(z).i,0i,mi,0m=1Theaboveexpressionprovidesustheso-calledMth-orderiterationfor-mula.Inthisway,theconvergenceofthehomotopy-seriescanbegreatlyaccelerated.TheiterationapproachoftheBVPh2.0iscurrentlypossibleonlywhenthebasefunctionsareofpolynomials,trigonometricfunctionsandhybrid-basefunctions.Weconsiderheretheapproximationofasmoothfunctionf(z)inthesethreedifferentkindsofbasefunctions.Theapproximationofafunctioninasemi-infiniteintervalisproposedinRef.20intheframeofHAM. 385October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems3779.5.1.PolynomialsItiswellknownthatasmoothfunctionf(z)inthefiniteintervalz∈[−1,1]canbewellapproximatedbytheChebyshevpolynomialsXNta0f(x)≈+akTk(z),(9.19)2k=1whereTk(z)isthekthChebyshevpolynomialofthefirstkind,NtdenotesthenumberofChebyshevpolynomials,andNXt+12ak=f(xi)Tk(xi),Nt+1i=1NXt+1π(i−1)πk(i−1)2=fcos2cos2.Nt+1Nt+1Nt+1i=1WhenChebyshevpolynomialisusedtoapproximatetheboundaryvalue/eigenvalueproblemsinafiniteintervalz∈[a,b]bymeansoftheBVPh2.0,weshouldsetTypeL=1,ApproxQ=1.9.5.2.TrigonometricfunctionsItiswellknownthattheFourierseriesaX+∞nπznπz0+ancos+bnsin2aan=1ofacontinuousfunctionf(z)inafiniteintervalz∈(−a,a)convergestof(z)intheintervalz∈(−a,a),whereZaZa1nπt1nπtan=f(t)cosdt,bn=f(t)sindt.a−aaa−aaForacontinuousfunctionf(z)in[0,a],wecandefinef(z)=f(−z)inz∈[−a,0)anditsFourierseriesreadsX+∞a0nπzf(z)=+ancos.(9.20)2an=1Alternatively,wecandefinef(z)=−f(−z)inz∈[−a,0)anditsFourierseriesreadsX+∞nπzf(z)∼bnsin.(9.21)an=1 386October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9378Y.-L.ZhaoandS.-J.LiaoNotethatin(9.21),wewrite”∼”insteadof”=”,becauseitholdsonlyiff(0)=f(a)=0isassumed.Inpractice,wehavethefollowingapproximationsXNta0nπzf(z)≈+ancos(9.22)2an=1orXNtnπzf(z)≈bnsin,(9.23)an=1whereNtdenotesthenumberoftruncatedterms.Whentheabove-mentionedtrigonometricfunctionsareusedtosolveboundaryvalue/eigenvalueproblemsinafiniteintervalz∈[0,a]bymeansoftheBVPh2.0,weshouldsetTypeL=2,ApproxQ=1andHYBRID=0withTypeBase=1fortheoddexpression(9.23)andTypeBase=2fortheevenexpression(9.22),respectively.9.5.3.Hybrid-basefunctionsNotethatthefirst-orderderivativeoftheevenFourierseries(9.20)equalstozeroatthetwoendpointsz=0andz=a,buttheoriginalfunctionf(z)mayhavearbitraryvaluesoff0(0)andf0(a).So,incaseoff0(0)6=0and/orf0(a)6=0,onehadtousemanytermsoftheevenFourierseries(9.20)soastoobtainanaccurateapproximationnearthetwoendpoints.Toovercomethisdisadvantage,wefirstexpressf(z)bysuchacombinationf(z)≈Y(z)+w(z),(9.24)wheref0(0)+f0(a)πzY(z)=f0(0)z−z2cos(9.25)2aaandthenapproximatew(z)=f(z)−Y(z)bytheevenFourierseriesXNta¯0nπzw(z)≈+a¯ncos(9.26)2an=1withtheFouriercoefficientZa2nπta¯n=[f(t)−Y(t)]cosdt.a0a 387October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems379HereNtisthenumberoftruncatedterms.NotethatY(z)in(9.25)satisfies0000Y(0)=f(0),Y(a)=f(a)sothatw0(0)=w0(a)=0.Therefore,weoftenneedafewtermsoftheevenFourierseriestoaccuratelyapproximatew(z).Notealsothatboththetrigonometricandpolynomialfunctionsareusedin(9.24)toapproximatef(z).Itisfoundthat,bymeansofsuchkindofapproximationsbasedonhybrid-basefunctions,oneoftenneedsmuchlesstermstoapproximateagivensmoothfunctionf(z)in[0,a]thanthetraditionalFourierseries.Alternatively,foracontinuousfunctionf(z)in[0,a],wecanusef(a)−f(0)Y(z)=f(0)+z(9.27)aorf(0)+f(a)f(0)−f(a)πzY(z)=+cos,(9.28)22aandapproximatew(z)bytheoddFourierseriesXNtnπzw(z)≈¯bnsin,(9.29)an=1whereZa2nπt¯bn=[f(t)−Y(t)]sindt.a0aNotethatY(z)in(9.27)and(9.28)satisfiesY(0)=f(0),Y(a)=f(a)sothatw(0)=w(a)=0.Itissuggestedtousethehybrid-baseapproximation(9.24)with(9.25)foranevenfunctionf(z),and(9.24)with(9.27)or(9.28)foranoddfunc-tionf(z),respectively.Iff(z)isneitheranoddnorevenfunction,bothofthemwork.Whentheabove-mentionedhybrid-baseapproximationisused,wehaveevenlargerfreedomtochoosetheinitialguessu0(z).Forexample,fora2nd-orderboundaryvalue/eigenvalueprobleminafiniteintervalz∈[0,a],wemaychoosesuchaninitialguessintheformκπzκπzu0(z)=B0+B1cos+B2sin(9.30)aa 388October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9380Y.-L.ZhaoandS.-J.Liaoor2κπzu0(z)=(B0+B1z+B2z)cos,(9.31)awhereB0,B1andB2aredeterminedbylinearboundaryconditions,κisapositiveinteger.Whentheabove-mentionedtrigonometricfunctionsareusedtosolveboundaryvalue/eigenvalueproblemsinafiniteintervalz∈[0,a]bymeansoftheBVPh2.0,weshouldsetTypeL=2,ApproxQ=1andHYBRID=1withTypeBase=1fortheoddexpression(9.29)andTypeBase=2fortheevenexpression(9.26),respectively.9.6.AsimpleuserguideoftheBVPh2.0Inthissection,wewilltakeaglanceattheMathematicapackageBVPh2.0.9.6.1.KeymodulesBVPhThemoduleBVPh[k_,m_]givesthekthtomth-orderhomotopyapproximationsofasystemofordinarydifferentialequations(ODEs)subjecttosomeboundaryconditions.Thesystemmayhaveanunknownparameter(whenTypeEQ=2)ormaynothaveanunknownparameter(whenTypeEQ=1).Itisthebasicmodule.Forexample,BVPh[1,10]givesthe1st-orderto10th-orderhomotopy-approximations.Thereafter,BVPh[11,15]furthergivesthe11th-orderto15th-orderhomotopy-approximations.Forproblemswithanunknownparameter,theinitialguessoftheunknownparameterisdeterminedbyanalgebraicequation.Thus,iftherearemorethanoneinitialguessesoftheunknownparam-eter,itisrequiredtochooseoneamongthembyinputtinganinteger,suchas1or2,correspondingtothe1storthe2ndinitialguessoftheunknownparameter,respectively.iterThemoduleiter[k_,m_,r_]providesushomotopyapproxima-tionsofthekthtomthiterationbymeansoftherth-orderiterationfor-mula.ItcallsthebasicmoduleBVPh.Forexample,iter[1,10,3]giveshomotopy-approximationsofthe1stto10thiterationbythe3rd-orderiterationformula.Furthermore,iter[11,20,3]givesthehomotopy-approximationsofthe11thto20thiterations.Theiterationstopswhen 389October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems381theaveragedsquaredresidualerrorofthesystemislessthanacriticalvalueErrReq,whosedefaultis10−20.GetErrThemoduleGetErr[k_]givestheaveragedsquaredresidualer-rorofthegoverningequationatthekth-orderhomotopy-approximationgainedbythemoduleBVPh.Notethat,errer[i,k]providestheresidualoftheithgoverningequationatthekth-orderhomotopy-approximationgainedbyBVPh,andErrTotal[k]givesthetotalav-eragedsquaredresidualerrorofthesystematthekth-orderhomotopy-aprroximationgainedbyBVPh.hpThemodulehp[f_,m_,n_]givesthe[m,n]homotopy-pad´eapprox-imationofalistofthehomotopy-approximationsf,wheref[[i+1]]denotestheith-orderhomotopy-approximationofthesamegoverningequation.GetBCThemoduleGetBC[i_,k_]givestheithboundaryconditionofthekth-orderdeformationequation.9.6.2.ControlparametersTypeEQAcontrolparameterforthetypeofgoverningequations:TypeEQ=1correspondingtoanonlinearproblemwithoutanunknownparameter,TypeEQ=2correspondstoanonlinearproblemwithanunknownparameter(calledeigenvalueproblemoreigenvalue-likeprob-lem),respectively.TypeLAcontrolparameterforthetypeofauxiliarylinearoperator:TypeL=1correspondstopolynomialapproximation,andTypeL=2correspondstoatrigonometricapproximationorahybrid-baseapprox-imation,respectively.ApproxQAcontrolparameterforapproximationofsolutions.WhenApproxQ=1,theright-handsidetermofallhigher-orderdeformationequationsareapproximatedbyChebyshevpolynomials(9.19),trigono-metricfunctionsin§9.5.2,orbythehybrid-basefunctionsin§9.5.3.WhenApproxQ=0,thereisnosuchkindofapproximation.WhenTypeL=2,ApproxQ=1isvalidonlyforproblemsinafiniteintervalz∈[0,a],wherea>0isaconstant.HYBRIDAcontrolparameterforthehybrid-basefunctions.WhenHYBRID=1,hybrid-basefunctionsareemployedinapproximation. 390October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9382Y.-L.ZhaoandS.-J.LiaoWhenHYBRID=0,trigonometricfunctionswithoutpolynomialsareemployedinapproximation.Thisparameterisusuallyusedinconjunc-tionwithTypeBase,andisvalidonlywhenTypeL=2andApproxQ=1forproblemsinafiniteintervalz∈[0,a].TypeBaseAcontrolparameterforthetypeofFourierseriesapproxi-mation:TypeBase=1correspondstotheoddFourierapproximation(9.23)or(9.29),TypeBase=2correspondstotheevenFourierapprox-imation(9.22)or(9.26),respectively.ThisparameterisusuallyusedinconjunctionwithHYBRID,andisvalidonlywhenTypeL=2andApproxQ=1forproblemsinafiniteintervalz∈[0,a].NtruncatedAcontrolparametertodeterminethenumberoftruncatedtermsusedtoapproximatetheright-handsideofhigher-orderdeforma-tionequations.ThelargerNtruncated,thebettertheapproximations,butthemoreCPUtime.ItisvalidonlywhenApproxQ=1.Thedefaultis10.NtermMaxApositiveintegerusedinthemoduletruncated,whichig-noresallpolynomialtermswhoseorderishigherthanNtermMax.Thedefaultis90.ErrReqAcriticalvalueoftheaveragedsquaredresidualerrorofgovern-ingequationstostopthecomputation.Thedefaultis10−20.NgetErrApositiveintegerusedinthemoduleBVPh.TheaveragedsquaredresidualerrorofgoverningequationsiscalculatedwhentheorderofapproximationisdivisiblebyNgetErr.Thedefaultis2.NintegralNumberofdiscretepointswithequalspace,whichareusedtonumericallycalculatetheintegralofafunction.Itisusedinthemoduleint.Thedefaultis50.ComplexQAcontrolparameterforcomplexvariables.ComplexQ=1correspondstotheexistenceofcomplexvariablesingoverningequa-tionsand/orboundaryconditions.ComplexQ=0correspondstothenonexistenceofsuchkindofcomplexvariables.Thedefaultis0.FLOATAcontrolparameterforfloating-pointcomputation.WhenFLOAT=1,floating-pointnumbersareemployedincomputation.WhenFLOAT=0,rationalnumbersareemployedincomputation.Thedefaultis1. 391October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems3839.6.3.InputNumEQThenumberofgoverningequations.f[i,z,{u,···},lambda]Theithgoverningequationwithorwithoutanunknownparameter,correspondingtoFi[z,u,···]orFi[z,u,···,λ]ineitherafiniteintervalz∈[a,b]orasemi-infiniteintervalz∈[b,+∞),whereaandbareboundedconstants.Notethattheformalparameterlambdadenotestheunknownparameterλtobedetermined,ortheeigenvalueλforeigenvalueproblems,buthasnomeaningatallforproblemswithoutanunknownparameterλ,ornon-eigenvalueproblems.NumBCThenumberofboundaryconditions.BC[k,z,{u,···}]ThekthboundaryconditioncorrespondingtoBk[z,u,···],where1≤k≤NumBC.Notethatthesymbolinfinitydenotes∞inboundaryconditions.U[i,0]TheinitialguessUi,0(z),i.e.,ui,0(z).c0[i]Theconvergence-controlparameterc0,i,correspondingtotheithgoverningequation.H[i,z]Theauxiliaryfunctioncorrespondingtotheithgoverningequation.ThedefaultisH[i_,z_]:=1.L[i,f]Theauxiliarylinearoperatorcorrespondingtotheithgovern-ingequation.zL[i]Theleftendpointoftheintervalofthesolutioncorrespondingtotheithgoverningequation.Notethatintervalsofthesolutionsarenotnecessarilythesame,especiallyformulti-layerflowproblem.zR[i]Therightendpointoftheintervalofthesolutioncorrespondingtotheithgoverningequation.zLintegral[i]Theleftendpointoftheintegralintervaltocomputetheaveragedsquaredresidualerroroftheithgoverningequation.Whentheleftendpointofthesolutionintervalfortheithgoverningequationisafinitenumber,zLintegral[i]isautomaticallysettozL[i].Otherwise,theuserhastospecifythevalueofzLintegral[i].zRintegral[i]Therightendpointoftheintegralintervaltocomputetheaveragedsquaredresidualerroroftheithgoverningequation.Whentherightendpointofthesolutionintervalfortheithgoverningequa-tionisafinitenumber,zRintegral[i]isautomaticallysettozR[i].Otherwise,theuserhastospecifythevalueofzRintegral[i]. 392October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9384Y.-L.ZhaoandS.-J.Liao9.6.4.OutputU[i,k]Thekth-orderhomotopy-approximationofthesolutiontotheithgoverningequationgivenbythebasicmoduleBVPh.V[i,k]Thekth-iterationhomotopy-approximationofthesolutiontotheithgoverningequationgivenbytheiterationmoduleiter.Lambda[k]Thekth-orderhomotopy-approximationoftheeigenvalueλortheunknownparameterλgivenbythebasicmoduleBVPh.LAMBDA[k]Thekth-iterationhomotopy-approximationoftheeigenvalueλortheunknownparameterλgivenbytheiterationmoduleiter.error[i,k]Theresidualoftheithgoverningequationgivenbythekth-orderhomotopy-approximation(obtainedbythebasicmoduleBVPh).Err[k]Alistoftheaveragedsquaredresidualerrorofeachgoverningequationgivenbythekth-orderhomotopy-approximation(obtainedbythebasicmoduleBVPh).ErrTotal[k]Thetotaloftheaveragedsquaredresidualerrorforeachgoverningequationgivenbythekth-orderhomotopy-approximation(obtainedbythebasicmoduleBVPh).ERR[k]Alistoftheaveragedsquaredresidualerrorofeachgoverningequationgivenbythekth-iterationhomotopy-approximation(obtainedbytheiterationmoduleiter).ERRTotal[k]Thetotaloftheaveragedsquaredresidualerrorforeachgoverningequationgivenbythekth-iterationhomotopy-approximation(obtainedbytheiterationmoduleiter).9.6.5.GlobalvariablesAllcontrolparametersandoutputvariablesmentionedaboveareglobal.Besidesthese,thefollowingvariablesandparametersarealsoglobal.zTheindependentvariablez.u[i,k]Thesolutiontothekth-orderdeformationequationoftheithgoverningequation.lambda[k]Aconstantvariable,correspondingtoλk.delta[i,k]Afunctiondependentuponz,correspondingtotheright-handsidetermδi,k(z)inthehigh-orderdeformationequation.L[i,w]TheithauxiliarylinearoperatorLiappliedtow. 393October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems385−1Linv[i,f]TheinverseoperatorofLi,correspondingtoLi,appliedtof.sNumApositiveintegertodeterminewhichinitialguessλ0ischosenwhentherearemultiplesolutionsofλ0.9.7.ExamplesMoreexamplesaregiveninthissectiontoshowtheusagethepackageBVPh2.0.9.7.1.Example1:AsystemofODEsinfiniteintervalConsiderasystemofcoupledODEs[7](1+K)f0000−ReMf00+2Reff000−Kg00=0,(9.32)K0000001+g−ReK[2g−f]+Re[2fg−fg]=0,(9.33)2subjecttof(0)=0,f(1)=0,f0(1)=1,f00(0)=0,(9.34)g(1)=0,g(0)=0,(9.35)whereKistheratioofviscosities,ReistheReynoldsnumberandMistheHartmannumber.Hayat[7]hassolvedthisproblembytheHAM.HerewesolvethisproblembyBVPh2.0.SincetherearetwoODEsinsystem(9.32)–(9.33)withoutanunknowntobedetermined,wehaveNumEQ=2andTypeEQ=1.ThesystemisinputasTypeEQ=1;NumEQ=2;f[1,z_,{f_,g_},Lambda_]:=(1+K)*D[f,{z,4}]-Rey*M*D[f,{z,2}]+2*Rey*f*D[f,{z,3}]-K*D[g,{z,2}];f[2,z_,{f_,g_},Lambda_]:=(1+K/2)*D[g,{z,2}]-Rey*K*(2*g-D[f,{z,2}])+Rey*(2*f*D[g,z]-D[f,z]*g);Thesixboundaryconditionsaredefinedas 394October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9386Y.-L.ZhaoandS.-J.LiaoNumBC=6;BC[1,z_,{f_,g_}]:=f/.z->0;BC[2,z_,{f_,g_}]:=f/.z->1;BC[3,z_,{f_,g_}]:=(D[f,z]-1)/.z->1;BC[4,z_,{f_,g_}]:=D[f,{z,2}]/.z->0;BC[5,z_,{f_,g_}]:=g/.z->1;BC[6,z_,{f_,g_}]:=g/.z->0;NowletusinputthesolutionintervalszL[1]=0;zR[1]=1;zL[2]=0;zR[2]=1;Sinceallthesolutionintervalsarefinite,wedonothavetospecifytheintegralintervaltocomputetheaveragedsquaredresidualerror.Theinitialguessesarechosenasf=(z3−z)/2andg=0.Theyare00inputasU[1,0]=(z^3-z)/2;U[2,0]=0;42TheauxiliarylinearoperatorsarechosenasL=∂andL=∂.1∂z42∂z2TheyaredefinedasL[1,u_]:=D[u,{z,4}];L[2,u_]:=D[u,{z,2}];NotethatweusethedelayedassignmentSetDelayed(:=)todefinetheselinearoperators.Withoutlossofgenerality,letusconsiderthecasewhenRe=M=2andK=1/2.ThesephysicalparametersareinputasRey=M=2;K=1/2;Atthistime,wehaveinputallthedataforthisproblem,excepttheconvergence-controlparametersc0[k].Hayat[7]chosetheconvergence- 395October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems387controlparametersc0[1]=c0[2]=-0.7through~-curve.Hereweminimizetheaveragedsquaredresidualerrorofthe4th-orderapproximationstogetoptimalvaluesforc0[k]GetOptiVar[4,{},{c0[1],c0[2]}];Theconvergence-controlparametersc0[1]andc0[2]arefoundtobeabout−0.5825and−0.721452,respectively.ThenwecallthemainmoduleBVPhtogetthe20th-orderapproxima-tionsBVPh[1,20];The20th-orderapproximationsarestoredinU[i,20],i=1,2,whilethecorrespondingaveragedsquaredresidualerrorofthesystemisErrTotal[20].WecanusePlot[{U[1,20],U[2,20]},{z,0,1},AxesLabel->{"z",""},PlotStyle->{{Thin,Red},{Dashed,Blue}},PlotRange->{{0,1},{-0.2,0.2}}]toplotthe20th-orderapproximations,whichisshowninFig.9.3.ThisfigureagreeswithHayat’sFigs.9and12whenM=2,Re=2andK=0.5.The20th-orderapproximationsgivethevaluesoff00(1)=3.61076396287andg0(1)=−0.738463496789,whicharethesamewithHayat’sresult[7].ThetotalerrorErrTotal[k]ofthesystemforeverytwoorderofapproximationsisplottedbythecommandListLogPlot[Table[{2i,ErrTotal[2*i]},{i,1,10}],Joined->True,Mesh->All,PlotRange->{{2,20},{10^(-34),1}},AxesLabel->{"m","error"}]inFig.9.4.Wecanseefromitthattheerrordecreasesbeautifully. 396October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9388Y.-L.ZhaoandS.-J.Liao0.20.10.0z0.20.40.60.81.0-0.1-0.2Fig.9.3.Thecurveoff(z)(solid),g(z)(dashed)forExample1.error10-410-1010-1610-2210-2810-34m5101520Fig.9.4.Totalerrorvs.orderofapproximationforExample1.9.7.2.Example2:AsystemofODEswithalgebraicpropertyatinfinityConsiderasetoftwocouplednonlineardifferentialequations[13]f000(η)+θ(η)−f02=0,(9.36)000θ(η)=3σf(η)θ(η),(9.37)subjecttof(0)=f0(0)=0,θ(0)=1,f0(+∞)=θ(+∞)=0,(9.38)wheretheprimedenotesdifferentiationwithrespecttothesimilarityvari-ableη,σisthePrandtlnumber,f(η)andθ(η)relatetothevelocityprofile 397October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems389andtemperaturedistributionoftheboundarylayer,respectively.Liao[13]employedtheHAMtosolvethissystemanalytically.NowweusetheBVPh2.0tosolveit.Underthetransformation0ξ=1+λη,F(ξ)=f(η),S(ξ)=θ(η),(9.39)Eqs.(9.36)and(9.37)becomeλ2F00(ξ)+S(ξ)−F2(ξ)=0,(9.40)λ2S00(ξ)=3σF(ξ)S(ξ),(9.41)subjecttoF(1)=0,S(1)=1,F(+∞)=S(+∞)=0.(9.42)SincetherearetwoODEsinsystem(9.40)–(9.41)withoutanunknowntobedetermined,wehaveNumEQ=2andTypeEQ=1.ThisnewsystemisdefinedasTypeEQ=1;NumEQ=2;f[1,z_,{F_,S_},Lambda_]:=la^2*D[F,{z,2}]+S-F^2;f[2,z_,{F_,S_},Lambda_]:=la^2*D[S,{z,2}]-3*sigma*F*S;Thefourboundaryconditions(9.42)aredefinedasNumBC=4;BC[1,z_,{F_,S_}]:=F/.z->1;BC[2,z_,{F_,S_}]:=(G-1)/.z->1;BC[3,z_,{F_,S_}]:=F/.z->infinity;BC[4,z_,{F_,S_}]:=G/.z->infinity;NowletusinputthesolutionintervalsandintegralintervalstocomputeaveragedsquaredresidualerrorzL[1]=1;zR[1]=infinity;zL[2]=1;zR[2]=infinity;zRintegral[1]=10;zRintegral[2]=10; 398October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9390Y.-L.ZhaoandS.-J.LiaoTheinitialguessesarechosenasF=γ(ξ−2−ξ−3),S=ξ−4,andthey00areinputasU[1,0]=gamma*(z^(-2)-z^(-3));U[2,0]=z^(-4);ξ2ξ2TheauxiliarylinearoperatorsareL=∂+∂andL=∂+∂,F3∂ξ2∂ξS5∂ξ2∂ξwhicharedefinedasL[1,u_]:=D[u,{z,2}]*z/3+D[u,z];L[2,u_]:=D[u,{z,2}]*z/5+D[u,z];Withoutlossofgenerality,letusconsiderthecasewhenσ=1,γ=3andλ=1/3.Weusethesameconvergence-controlparametersc0[1]=c0[2]=−1/2asLiao[13].Thesephysicalparametersandthecontrolparametersc0[k]aredefinedassigma=1;gamma=3;la=1/3;c0[1]=-1/2;c0[2]=-1/2;ThenwecallthemainmoduleBVPhBVPh[1,20];togetthe20th-orderapproximation.Ifwearenotsatisfiedwiththeaccu-racyofthe20th-orderapproximation,wecanuseBVPh[21,40],insteadofBVPh[1,40],toget40th-orderapproximationorhigherorderapproxima-tion.NotethatU[1,40]andU[2,40]arethe40th-orderapproximationsofthetransformedsystem(9.40),(9.41)and(9.42).Toplotthecurveofthe40th-orderapproximationsfortheoriginalproblem,wefirstreplacezwith1+ληtoobtainthe40th-orderapproximationsforf0(η)andg(η),thenplotthecurvewewant.ThisisdoneinMathematicabythefollowingcommand 399October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems391trans={z->1+la*\[Eta]};Plot[Evaluate[{U[1,40],U[2,40]}/.trans],{\[Eta],0,10},PlotRange->{{0,10},{0,1}},AxesLabel->{"\[Eta]",""},PlotStyle->{{Thin,Red},{Dashed,Blue}}]andthecurveisshowninFig.9.5.Heretrans={z->1+la*\[Eta]}isthecorrespondingtransformation,and\[Eta]isthesymbolηinMathematica.ThetotalerrorErrTotal[k]ofthetransformedsystemforeverytwoorderapproximationsisplottedinFig.9.6bythefollowingcommandListLogPlot[Table[{2i,ErrTotal[2*i]},{i,1,20}],Joined->True,Mesh->All,PlotRange->{{2,40},{10^(-10),0.01}},AxesLabel->{"m","error"}]NotethatErrTotal[k]notonlymeasurestheaccuracyofthekth-orderapproximationsforthetransformedproblem,butalsomeasuresthecorre-spondingapproximationsfortheoriginalproblem.1.00.80.60.40.20.0Η0246810Fig.9.5.Thecurveoff0(η)(solid)andθ(η)(dashed)forExample2. 400October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9392Y.-L.ZhaoandS.-J.Liaoerror0.0110-410-610-810-10m510152025303540Fig.9.6.Totalerrorvs.orderofapproximationforExample2.The40th-orderapproximationsoff00(0)andg00(0)are0.693268and−0.769879,respectively.Kuiken’snumericalresultisf00(0)≈0.693212andg0(0)≈−0.769861.Togetmoreaccurateresult,wehavetwochoices.OneistocallthemoduleBVPhtogethigherorderapproximationasbefore,theotheristoapplythePad´eapproximationtothecurrentapproximations.Thelatterisdonebycallingthemodulehpasfollowshp[Table[D[(U[1,i]/.trans),\[Eta]]/.\[Eta]->0,{i,0,40}],20,20]hp[Table[D[(U[2,i]/.trans),\[Eta]]/.\[Eta]->0,{i,0,40}],20,20]whichgive0.693212and−0.769861,the[20,20]homotopy-Pad´eapproxi-mationsoff00(0)andg0(0),respectively.Notethatwecancomparethecurveof2nth-orderapproximationandthe[n,n]homotopy-Pad´eapproximationinasimpleandefficientway.HerewecompareU[1,40]andthe[20,20]homotopy-Pad´eapproximationofU[1,i],i=0···40,intheMathematicabythefollowingcommand.Plot[{U[1,40]/.trans,hp[Table[U[1,i]/.trans,{i,0,40}],20,20]},{\[Eta],0,10},PlotRange->Full,AxesLabel->{"\[Eta]",""},PlotStyle->{{Thin,Red},{Dashed,Blue}} 401October24,201314:55WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems3930.350.300.250.200.150.100.05Η246810Fig.9.7.Thecurveof40th-orderapproximationoff0(η)(solid)and[20,20]homotopy-Pad´eapproximationsoff0(η)(dashed)forExample2.ThecomparisonisshowninFig.9.7.Fromitwecanseethatthetwoarealmostthesame.Thisvalidatetheconvergenceoftheapproximationstosomeextent.Theabovecommandisveryefficient,becausethePlotcom-mandinMathematicafirstsubstitutethesamplepointsintotheexpressionandthenappliesthehptoalistofnumericalvalues,ratherthanappliesthehptoalistofexpressionsandthensubstitutethesamplepointsintotheresultingexpression.9.7.3.Example3:AsystemofODEswithanunknownparameterConsiderasystemofODEs[15]00U+(GrPr)θ−Nrφ+σ=0,(9.43)000002θ+Nbθφ+Nt(θ)+Nb+Nt−U=0,(9.44)00Nt00φ+θ−LeU=0,(9.45)NbsubjecttoU(−1)=U(1)=0,θ(−1)=θ(1)=0,φ(−1)=φ(1)=0,(9.46)withanadditionalconditionZ1UdY=RePr,(9.47)0 402October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9394Y.-L.ZhaoandS.-J.LiaowhereGristheGrashofnumber,PrthePrandtlnumber,Nrthebuoyancyratio,σthepressureparameter,NbtheBrownianmotionparameter,Ntthethermophoresisparameter,LetheLewisnumber,andRetheReynoldsnumber.Alloftheaboveparameterswillbegivenforaspecialcaseexceptσ,whichistobedeterminedfromthesystem.Xu[15]solvedthisproblembytheHAM.HerewesolvethisproblembyBVPh2.0.SincetherearethreeODEsinsystem(9.43)–(9.45)withanunknownσtobedetermined,wehaveNumEQ=3andTypeEQ=2.ThesystemisinputasTypeEQ=2;NumEQ=3;f[1,z_,{f_,g_,s_},sigma_]:=D[f,{z,2}]+Gr*Pr*g-Nr*s+sigma;f[2,z_,{f_,g_,s_},sigma_]:=D[g,{z,2}]+Nb*D[g,z]*D[s,z]+Nt*(D[g,z])^2-f;f[3,z_,{f_,g_,s_},sigma_]:=D[s,{z,2}]+Nt/Nb*D[f,{z,2}]-Le*f;Thesevenboundaryconditions,includingtheadditionalcondition(9.47),aredefinedasNumBC=7;BC[1,z_,{f_,g_,s_}]:=f/.z->-1;BC[2,z_,{f_,g_,s_}]:=f/.z->1;BC[3,z_,{f_,g_,s_}]:=g/.z->-1;BC[4,z_,{f_,g_,s_}]:=g/.z->1;BC[5,z_,{f_,g_,s_}]:=s/.z->-1;BC[6,z_,{f_,g_,s_}]:=s/.z->1;BC[7,z_,{f_,g_,s_}]:=Integrate[f,{z,0,1}]-Ra*Pr;NowletusinputthesolutionintervalszL[1]=-1;zR[1]=1;zL[2]=-1;zR[2]=1;zL[3]=-1;zR[3]=1;Sinceallthesolutionintervalsarefinite,wedonothavetospecifythe 403October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems395integralintervaltocomputetheaveragedsquaredresidualerror.TheinitialguessesarechosenasU=−3(−25+)z2/2+5(−15+0112)z4/2,θ=(1−z2)andφ=(1−z2),where,andare10203123constantstobeoptimized.TheyareinputasU[1,0]=eps1-3/2*(-25+4eps1)z^2+5/2*(-15+2eps1)*z^4;U[2,0]=eps2*(1-z^2);U[3,0]=eps3*(1-z^2);2TheauxiliarylinearoperatorsarechosenasL=L=L=∂.123∂Y2TheyaredefinedasL[1,u_]:=D[u,{z,2}];L[2,u_]:=D[u,{z,2}];L[3,u_]:=D[u,{z,2}];NotethatweusethedelayedassignmentSetDelayed(:=)todefinetheselinearoperators.Withoutlossofgenerality,letusconsiderthecasewhenNr=3/20,Nt=Nb=1/20,Le=10,Gr=5,Pr=1,andRe=5.ThesephysicalparametersareinputasNr=3/20;Nt=1/20;Nb=1/20;Le=10;Gr=5;Pr=1;Ra=5;Atthistime,wehaveinputallthedataforthisproblem,excepttheconvergence-controlparametersc0[k],eps1,eps2andeps3.Weminimizetheaveragedsquaredresidualerrorofthe3th-orderapproximationstogetoptimalvaluesfortheseparametersbythemoduleGetOptiVarasfollowc0[1]=c0[2]=c0[3]=h;GetOptiVar[3,{},{eps1,eps2,eps3,h}];Notethatweputconstraintsc0[1]=c0[2]=c0[3]onc0[1],c0[2]andc0[3]tosimplifythecomputation.Thereisnoconstraintoneps1,eps2andeps3. 404October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9396Y.-L.ZhaoandS.-J.LiaoAftersomecomputation,wegetoptimalvaluesforalltheconvergence-controlparametersc0[1]=c0[2]=c0[3]≈−0.769452,eps1≈7.56408,eps2≈−2.58887andeps3≈−30.0044.NowwecanuseBVPh[1,10]togetthe10th-orderapproximation.Ifwearenotsatisfiedwiththeac-curacyofthe10th-orderapproximation,wecanuseBVPh[11,20]toget20th-orderapproximationorhigherorderapproximation.5z-1.0-0.50.51.0-5-10-15-20-25Fig.9.8.ThecurveofU(solid),θ(dashed)andφ(z)(dotdashed)forExample3.The20th-orderapproximationsofU,θandφarestoredinU[1,20],U[2,20]andU[3,20],the20th-orderapproximationofσisstoredinLambda[19],whilethecorrespondingaveragedsquaredresidualerrorofthesystemisstoredinErrTotal[20].Lambda[19]isabout18.272555944,whichisthesamewithXu’sresult[15].The20th-orderapproximationsareplottedinFig.9.8.ThetotalerrorofthesystemforeverytwoorderofapproximationsisplottedinFig.9.9. 405October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems397error110-410-810-1210-1610-20m5101520Fig.9.9.Totalerrorvs.orderofapproximationforExample3.9.7.4.Example4:AsystemofODEsindifferentintervalsConsideratwo-phaseflow[16]:(i)Region1d2uGr1+sin(φ)θ1=P,(9.48)dy2Re22dθ1du1+PrEc=0,(9.49)dy2dy(ii)Region2d2uGrnbh2M2h2h22+sin(φ)θ2−u2=P,(9.50)dy2Reλλλ222dθ2λdu2h222+EcPr+EcPrMu2=0,(9.51)dyλTdyλTsubjecttou1(1)=1,θ1(1)=1,(9.52)u1(0)=u2(0),θ1(0)=θ2(0),(9.53)0λ00λT0u1(0)=u2(0),θ1(0)=θ2(0),(9.54)hhu2(−1)=0,θ2(−1)=0,(9.55)whereGristheGrashofnumber,EcistheEckertnumber,PristhePrandtlnumber,ReistheReynoldsnumber,MistheHartmannnumberandP 406October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9398Y.-L.ZhaoandS.-J.Liaoisthedimensionlesspressuregradient.Thismodeldescribesatwo-fluidmagnetohydrodynamicPoiseuille–Couetteflowandheattransferinanin-clinedchannel.Umavathi[16]investigatethismodelanalyticallybyregularperturbationmethodandnumericallybyfinitedifferencetechnique.TheBVPh2.0cansolvethisproblem(9.48)–(9.55)directlywithoutdifficulty.Sincealltheparametersinthesystemwillbegiven,wehaveNumEQ=4andTypeEQ=1.ThesystemisinputasTypeEQ=1;NumEQ=4;f[1,z_,{u1_,s1_,u2_,s2_},Lambda_]:=D[u1,{z,2}]+Gr/Ra*Sin[phi]*s1-P;f[2,z_,{u1_,s1_,u2_,s2_},Lambda_]:=D[s1,{z,2}]+Pr*Ec*(D[s1,z])^2;f[3,z_,{u1_,s1_,u2_,s2_},Lambda_]:=D[u2,{z,2}]-h^2/lamb*P+Gr/Ra*Sin[phi]*n*b*h^2/lamb*s2-M^2*h^2/lamb*u2;f[4,z_,{u1_,s1_,u2_,s2_},lambda_]:=D[s2,{z,2}]+Pr*Ec*lamb/lambT*D[u2,z]^2+Pr*Ec*h^2/lambT*M^2*u2^2;Theeightboundaryconditions(9.52)–(9.55)aredefinedasNumBC=8;BC[1,z_,{u1_,s1_,u2_,s2_}]:=(u1-1)/.z->1;BC[2,z_,{u1_,s1_,u2_,s2_}]:=(u1-u2)/.z->0;BC[3,z_,{u1_,s1_,u2_,s2_}]:=u2/.z->-1;BC[4,z_,{u1_,s1_,u2_,s2_}]:=(D[u1,z]-D[u2,z]*lamb/h)/.z->0;BC[5,z_,{u1_,s1_,u2_,s2_}]:=(s1-1)/.z->1;BC[6,z_,{u1_,s1_,u2_,s2_}]:=(s1-s2)/.z->0;BC[7,z_,{u1_,s1_,u2_,s2_}]:=s2/.z->-1;BC[8,z_,{u1_,s1_,u2_,s2_}]:=(D[s1,z]-D[s2,z]*lambT/h)/.z->0;NowletusinputthesolutionintervalszL[1]=0;zR[1]=1;(*u1*)zL[2]=0;zR[2]=1;(*s1*)zL[3]=-1;zR[3]=0;(*u2*)zL[4]=-1;zR[4]=0;(*s2*) 407October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems399Notethatthesolutionintervalsarenotthesame.Sinceallthesolutionintervalsarefinite,wedonothavetospecifytheintegralintervaltocomputetheaveragedsquaredresidualerror.Theinitialguessesarechosenasu=λ(z−z2)+1,θ=zλT+(1−1,0h1,0hλT)z2,u=1+zandθ=z+z2.Theyareinputash2,02,0U[1,0]=(z-z^2)*lamb/h+1;(*u1*)U[2,0]=z*lambT/h+(1-lambT/h)*z^2;(*s1*)U[3,0]=1+z;(*u2*)U[4,0]=z^2+z;(*s2*)2TheauxiliarylinearoperatorsarechosenasL=L=L=L=∂.1234∂y2TheyaredefinedasL[1,u_]:=D[u,{z,2}];L[2,u_]:=D[u,{z,2}];L[3,u_]:=D[u,{z,2}];L[4,u_]:=D[u,{z,2}];NotethatweusethedelayedassignmentSetDelayed(:=)todefinetheselinearoperators,andzistheindependentvariableinthepackage.Withoutlossofgenerality,letusconsiderthecasewhenPr=7/10,Ec=1/100,P=−5,b=1,n=1,Re=1,M=2,Gr=5,h=1,λ=1,λT=1,andφ=π/6.ThesephysicalparametersareinputasP=-5;b=1;n=1;Ra=1;M=2;Gr=5;lamb=1;lambT=1;h=1;phi=Pi/6;Pr=7/10;Ec=1/100;Atthistime,wehaveinputallthedataforthisproblem,excepttheconvergence-controlparametersc0[k].Weminimizetheaveragedsquaredresidualerrorofthe4th-orderapproximationstoobtainoptimalvaluesforc0[k]bythecommand 408October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9400Y.-L.ZhaoandS.-J.LiaoGetOptiVar[4,{},{c0[1],c0[2],c0[3],c0[4]}];NotethatthesecondparameterofGetOptiVarisanemptylist,whichmeansthatwegivenoconstraintontheconvergence-controlparametersc0[k].Aftersometime,weobtaintheoptimalvaluesforc0[k],whichreadsc0[1]≈−0.898166,c0[2]≈−0.946828,c0[3]≈−0.780946andc0[4]≈−1.12363.ThenwecallthemainmoduleBVPhtogetthe30th-orderapproximationsBVPh[1,30];The30th-orderapproximationsforu1,θ1,u2,θ2arestoredinU[1,30],U[2,30],U[3,30]andU[4,30],respectively,whilethecorrespondingaver-agedsquaredresidualerrorofthesystemisErrTotal[30].The30th-orderapproximationsareplottedinFig.9.10.Thevalueofθ(y)agreeswithUma-vathi’sresult[16](blackdots),asshowninFig.9.10.The30th-orderap-proximationofθ(y)givestheheattransferrateNu=θ0(1)=0.8860625+1andNu=θ0(1)=1.122312,whichagreeswithNu=0.88606and−2+Nu−=1.12230inUmavathi’s[16]Table3.ThetotalerrorErrTotal[k]ofthesystemforeverytwoorderofapproximationsisplottedinFig.9.11.3.02.52.01.51.00.5z-1.0-0.50.00.51.0Fig.9.10.Thecurveofu(y)(solid)andθ(y)(dashed)forExample4.Theblackdotsarethevaluesforθ(y)obtainedbyUmavathi[16]. 409October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems401error0.110-510-910-1310-1710-21m51015202530Fig.9.11.Totalerrorvs.orderofapproximationforExample4.9.7.5.Example5:IterativesolutionsoftheGelfandequationWhentheproblemisdefinedinafiniteinterval,theBVPh1.0cansolveitusinganiterativeapproach.TheBVPh2.0hasinheritedthisfeature.However,therearesomeminordifferencesintheinput.ConsidertheGelfandequation[17–19]u000u0u+(K−1)+λe=0,u(0)=u(1)=0,(9.56)zwheretheprimedenotesthedifferentiationwithrespecttoz,K≥1isaconstant,u(z)andλdenoteeigenfunctionandeigenvalue,respectively.FollowingLiao[14],anadditionalboundaryconditionu(0)=A(9.57)isaddedtodistinguishdifferenteigenfunctions.TosolvethisproblembyBVPh2.0,wehavetoinputthedifferentialequations,boundaryconditions,andinitialguesses.SincetheproblemisasingleODEwithanunknownλtobedetermined,wesetNumEQ=1andTypeEQ=2.ThedifferentialequationcanbecodedasfollowsTypeEQ=2;NumEQ=1;f[1,z_,{u_},lambda_]:=D[u,{z,2}]+(K-1)*D[u,z]/z+lambda*Exp[u]; 410October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9402Y.-L.ZhaoandS.-J.LiaoThethreeboundaryconditions,includingtheadditionalcondition(9.57),aredefinedasNumBC=3;BC[1,z_,{u_}]:=(u-A)/.z->0;BC[2,z_,{u_}]:=D[u,z]/.z->0;BC[3,z_,{u_}]:=u/.z->1;NowletusinputthesolutionintervalszL[1]=0;zR[1]=1;Sincethesolutionintervalisfinite,wedonothavetospecifytheintegralintervaltocomputetheaveragedsquaredresidualerror.TheinitialguessischosenasU=A[1+cos(πz)],whichisinputinto02MathematicaasU[1,0]=A/2*(1+Cos[Pi*z]);2 2TheauxiliarylinearoperatorischosenasL=∂+π,whichis∂z2adefinedinMathematicaasL[1,f_]:=D[f,{z,2}]+Pi^2*f;NotethatweusethedelayedassignmentSetDelayed(:=)todefinethelinearoperator.Withoutlossofgenerality,letusconsiderthecasewhenA=1andK=2.ThephysicalparametersareinputasK=2;A=1;Becausewewanttoapproximatetheright-handsidesusingthehybrid-basefunctionanduseaniterativeapproachtogettheapproximations,thecontrolparametersinBVPh2.0aremodifiedto 411October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems403TypeL=2;HYBRID=1;(*hybrid-basefunctions*)TypeBase=2;(*evenFourierseries*)ApproxQ=1;Ntruncated=30;HereTypeL=2,HYBRID=1andApproxQ=1togethermeanthattheright-handsidesofallhigh-orderdeformationequationsisapproximatedbythehybrid-baseapproximations.TypeBase=2meanstheevenex-pression(9.26)isused(TypeBase=1alsoappliestothisproblem).Ntruncated=20meansNt=30.Atthistime,wehaveinputallthedataforthisproblem,excepttheconvergence-controlparameterc0[1].Togetoptimalc0[1],weminimizetheaveragedsquaredresidualerrorofthe6th-orderapproximations.ThisisdoneinBVPh2.0bycallingthemoduleGetOptiVarGetOptiVar[6,{},{c0[1]}];Aftersomecomputation,wegettheoptimalvaluefortheconvergence-controlparameterc0[1]=−0.522418···.Nowwecanusethe3rd-orderiterationHAMapproachiter[1,6,3]togetthedesiredapproximation.Here6meanstheiterationtimes.Afterabout40seconds,the6thiterationgivestheeigenvalue1.90921,whichisthesamewithLiao’sresult[14].ThekthiterationapproximationsofuandλarestoredinV[1,k],andLAMBDA[k],whilethecorrespondingaveragedsquaredresidualerrorisstoredinERRTotal[k].The6thiterationapproximationisplottedinFig.9.12byPlot[V[1,6],{z,0,1},AxesLabel->{"z","u(z)"}] 412October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9404Y.-L.ZhaoandS.-J.LiaouHzL1.00.80.60.40.2z0.20.40.60.81.0Fig.9.12.Thecurveoftheeigenfunctionu(z)correspondingtotheeigenvalueλ=1.90921whenA=1andK=2forExample5.ThetotalerrorERRTotal[k]oftheproblemforeachiterationisplottedinFig.9.13bythecommandListLogPlot[Table[{i,ERRTotal[i]},{i,1,6}],PlotRange->{{1,6},{10^-10,0.01}},Joined->True,Mesh->All,AxesLabel->{"m","error"}]error0.0110-410-610-810-10m123456Fig.9.13.Totalerrorforeachiterationvs.iterationtimesmforExample5. 413October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems4059.8.ConclusionsThehomotopyanalysismethod(HAM)hasbeensuccessfullyappliedtosolvelotsofnonlinearproblemsinscienceandengineering.BasedontheHAM,theMathematicapackageBVPh1.0wasissuedbyLiaoinMay,2012.TheaimofBVPhistoprovideananalytictoolforasmanynonlinearBVPsaspossibleintheframeworkoftheHAM.However,theBVPh1.0canonlydealwithproblemsofasingleODE.UnlikeBVPh1.0,thenewversionBVPh2.0worksformanytypesofsys-temsofcouplednonlinearODEs.Inthischapter,webrieflydescribehowtoinstallandusetheBVPh2.0.FivetypicalexamplesareemployedtodemonstratethevalidityofBVPh2.0,includingasystemoftwocoupledODEsinfiniteinterval,asystemoftwocoupledODEsinsemi-infiniteinterval,asystemoftwocoupledODEswithalgebraicpropertyatin-finity,asystemofthreecoupledODEswithanunknownparametertobedetermined,andasystemoffourcoupledODEsindifferentinter-vals.Besides,newalgorithmsareusedinsomemodulesofBVPh2.0.Hence,BVPh2.0ismuchfasterthanBVPh1.0inmostcases.ThepackageBVPh2.0andallinputdatafortheseexamplesarefreeavailableonlineathttp://numericaltank.sjtu.edu.cn/BVPh.htm.Itiswellknownthattheiterativemethodcangainaccurateapprox-imationsmoreefficientlybymeansoftheHAM.TheBVPh2.0hasalsoinheritedthefeatureofBVPh1.0tosolvetheproblemsinfiniteintervalusingtheiterativemethod.Forproblemsinsemi-infiniteinterval,anitera-tivemethodfortwotypicalkindsofbasedfunctionsisproposed[20].ThisiterativeapproachwillbeemployedinthefutureversionofBVPh.AppendixA.CodesforexamplesHerearetheinputdataforalltheexamplesinthischapter.Notethatthelistingswithoutanend-of-linesemicoloniswrappedtofitthepagewidth.However,ifyoubreakthelongcommandintentionallyinMathematica,itwillrunasmulti-linecommandsandmaynotworkasyouexpected. 414October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9406Y.-L.ZhaoandS.-J.LiaoA.1.Samplecodestoruntheillustrativeexample(*Filename:runIllustrative.nb*)(*1.Clearallglobalvariables*)ClearAll["Global‘*"];(*2.ReadinthepackageBVPh2.0*)<<"E:\\Package\\BVPh2_0.m"(*3.Setthecurrentworkingdirectoryto*)(*"thecurrentdirectory"*)SetDirectory[ToFileName[Extract["FileName"/.NotebookInformation[EvaluationNotebook[]],{1},FrontEnd‘FileName]]];(*4.Readinyourinputdataincurrentdirectory*)(*NotethatthetwofilesrunIllustrative.nband*)(*Illustrative.mareinthecurrentdirectory*)< 415October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems407D[g,{z,2}]-D[f,z]*g+f*D[g,z]-2*la*D[f,z]+beta*(2*f*D[f,z]*D[g,z]-f^2*D[g,{z,2}]);(*DefineBoundaryconditions*)NumBC=5;BC[1,z_,{f_,g_}]:=(D[f,z]-1)/.z->0;BC[2,z_,{f_,g_}]:=f/.z->0;BC[3,z_,{f_,g_}]:=g/.z->0;BC[4,z_,{f_,g_}]:=D[f,z]/.z->infinity;BC[5,z_,{f_,g_}]:=g/.z->infinity;(*solutionintervalandintegralintervalforerror*)zL[1]=0;zR[1]=infinity;zL[2]=0;zR[2]=infinity;zRintegral[1]=10;zRintegral[2]=10;(*Defineinitialguess*)U[1,0]=1-Exp[-z];U[2,0]=ahpha*z*Exp[-z];(*Definetheauxiliarylinearoperator*)L[1,u_]:=D[u,{z,3}]-D[u,z];L[2,u_]:=D[u,{z,2}]-u;(*Definephysicalparameters*)beta=1/5;la=1/10;(*Printinputdata*)PrintInput[{f[z],g[z]}];(*Getoptimalc0*) 416October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9408Y.-L.ZhaoandS.-J.LiaoGetOptiVar[3,{},{c0[1],c0[2],alpha}];(*Gain10th-orderHAMapproximation*)BVPh[1,10];A.3.InputdataofBVPh2.0forExample1(*Filename:Example1.m*)Print["Theinputfile",$InputFileName,"isloaded!"];(*ModifycontrolparametersinBVPhifnecessary*)ErrReq=10^-30;(*Definethegoverningequation*)TypeEQ=1;NumEQ=2;f[1,z_,{f_,g_},Lambda_]:=(1+K)*D[f,{z,4}]-Rey*M*D[f,{z,2}]+2*Rey*f*D[f,{z,3}]-K*D[g,{z,2}];f[2,z_,{f_,g_},Lambda_]:=(1+K/2)*D[g,{z,2}]-Rey*K*(2*g-D[f,{z,2}])+Rey*(2*f*D[g,z]-D[f,z]*g);(*DefineBoundaryconditions*)NumBC=6;BC[1,z_,{f_,g_}]:=f/.z->0;BC[2,z_,{f_,g_}]:=f/.z->1;BC[3,z_,{f_,g_}]:=(D[f,z]-1)/.z->1;BC[4,z_,{f_,g_}]:=D[f,{z,2}]/.z->0;BC[5,z_,{f_,g_}]:=g/.z->1;BC[6,z_,{f_,g_}]:=g/.z->0;(*solutionintervalandintegralintervalforerror*)zL[1]=0;zR[1]=1;zL[2]=0;zR[2]=1; 417October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems409(*Defineinitialguess*)U[1,0]=(z^3-z)/2;U[2,0]=0;(*Definetheauxiliarylinearoperator*)L[1,u_]:=D[u,{z,4}];L[2,u_]:=D[u,{z,2}];(*Definephysicalparameters*)Rey=M=2;K=1/2;(*Printinputdata*)PrintInput[{f[z],g[z]}];(*Getoptimalc0*)GetOptiVar[4,{},{c0[1],c0[2]}];(*Gain20th-orderHAMapproximation*)BVPh[1,20];A.4.InputdataofBVPh2.0forExample2(*Filename:Example2.m*)Print["Theinputfile",$InputFileName,"isloaded!"];(*ModifycontrolparametersinBVPhifnecessary*)(*Definethegoverningequation*)TypeEQ=1;NumEQ=2;f[1,z_,{F_,S_},Lambda_]:=la^2*D[F,{z,2}]+S-F^2;f[2,z_,{F_,S_},Lambda_]:=la^2*D[S,{z,2}]-3*sigma*F*S; 418October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9410Y.-L.ZhaoandS.-J.Liao(*DefineBoundaryconditions*)NumBC=4;BC[1,z_,{F_,S_}]:=F/.z->1;BC[2,z_,{F_,S_}]:=(G-1)/.z->1;BC[3,z_,{F_,S_}]:=F/.z->infinity;BC[4,z_,{F_,S_}]:=G/.z->infinity;(*solutionintervalandintegralintervalforerror*)zL[1]=1;zR[1]=infinity;zL[2]=1;zR[2]=infinity;zRintegral[1]=10;zRintegral[2]=10;(*Defineinitialguess*)U[1,0]=gamma*(z^(-2)-z^(-3));U[2,0]=z^(-4);(*Definestheauxiliarylinearoperator*)L[1,u_]:=D[u,{z,2}]*z/3+D[u,z];L[2,u_]:=D[u,{z,2}]*z/5+D[u,z];(*Definephysicalandcontrolparameters*)sigma=1;gamma=3;la=1/3;c0[1]=-1/2;c0[2]=-1/2;(*Printinputdata*)PrintInput[{f[z],g[z]}];(*Gain20th-orderHAMapproximation*)BVPh[1,20]; 419October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems411A.5.InputdataofBVPh2.0forExample3(*Filename:Example3.m*)Print["Theinputfile",$InputFileName,"isloaded!"];(*ModifycontrolparametersinBVPhifnecessary*)(*Definethegoverningequation*)TypeEQ=2;NumEQ=3;f[1,z_,{f_,g_,s_},sigma_]:=D[f,{z,2}]+Gr*Pr*g-Nr*s+sigma;f[2,z_,{f_,g_,s_},sigma_]:=D[g,{z,2}]+Nb*D[g,z]*D[s,z]+Nt*(D[g,z])^2-f;f[3,z_,{f_,g_,s_},sigma_]:=D[s,{z,2}]+Nt/Nb*D[f,{z,2}]-Le*f;(*Defineboundaryconditions*)NumBC=7;BC[1,z_,{f_,g_,s_}]:=f/.z->-1;BC[2,z_,{f_,g_,s_}]:=f/.z->1;BC[3,z_,{f_,g_,s_}]:=g/.z->-1;BC[4,z_,{f_,g_,s_}]:=g/.z->1;BC[5,z_,{f_,g_,s_}]:=s/.z->-1;BC[6,z_,{f_,g_,s_}]:=s/.z->1;BC[7,z_,{f_,g_,s_}]:=Integrate[f,{z,0,1}]-Ra*Pr;(*Definesolutioninterval*)zL[1]=-1;zR[1]=1;zL[2]=-1;zR[2]=1;zL[3]=-1;zR[3]=1; 420October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9412Y.-L.ZhaoandS.-J.Liao(*Definestheauxiliarylinearoperator*)L[1,u_]:=D[u,{z,2}];L[2,u_]:=D[u,{z,2}];L[3,u_]:=D[u,{z,2}];(*Definephysicalparameters*)Nr=1/5;Nt=1/20;Nb=1/20;Le=10;Gr=5;Pr=1;Ra=5;(*Defineinitialguess*)U[1,0]=eps1-3/2*(-25+4eps1)z^2+5/2*(-15+2eps1)*z^4;U[2,0]=eps2*(1-z^2);U[3,0]=eps3*(1-z^2);(*Printinputdata*)PrintInput[{f[z],g[z],s[z]}];(*Getoptimalconvergence-controlparameters*)c0[1]=c0[2]=c0[3]=h;GetOptiVar[3,{},{eps1,eps2,eps3,h}];(*Gain10th-orderHAMapproximation*)BVPh[1,20];A.6.InputdataofBVPh2.0forExample4(*Filename:Example4.m*)Print["Theinputfile",$InputFileName,"isloaded!"];(*ModifycontrolparametersinBVPhifnecessary*) 421October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems413(*Definethegoverningequation*)TypeEQ=1;NumEQ=4;f[1,z_,{u1_,s1_,u2_,s2_},Lambda_]:=D[u1,{z,2}]+Gr/Ra*Sin[phi]*s1-P;f[2,z_,{u1_,s1_,u2_,s2_},Lambda_]:=D[s1,{z,2}]+Pr*Ec*(D[s1,z])^2;f[3,z_,{u1_,s1_,u2_,s2_},Lambda_]:=D[u2,{z,2}]-h^2/lamb*P+Gr/Ra*Sin[phi]*n*b*h^2/lamb*s2-M^2*h^2/lamb*u2;f[4,z_,{u1_,s1_,u2_,s2_},lambda_]:=D[s2,{z,2}]+Pr*Ec*lamb/lambT*D[u2,z]^2+Pr*Ec*h^2/lambT*M^2*u2^2;(*DefineBoundaryconditions*)NumBC=8;BC[1,z_,{u1_,s1_,u2_,s2_}]:=(u1-1)/.z->1;BC[2,z_,{u1_,s1_,u2_,s2_}]:=(u1-u2)/.z->0;BC[3,z_,{u1_,s1_,u2_,s2_}]:=u2/.z->-1;BC[4,z_,{u1_,s1_,u2_,s2_}]:=(D[u1,z]-D[u2,z]/m/h)/.z->0;BC[5,z_,{u1_,s1_,u2_,s2_}]:=(s1-1)/.z->1;BC[6,z_,{u1_,s1_,u2_,s2_}]:=(s1-s2)/.z->0;BC[7,z_,{u1_,s1_,u2_,s2_}]:=s2/.z->-1;BC[8,z_,{u1_,s1_,u2_,s2_}]:=(D[s1,z]-D[s2,z]/K/h)/.z->0;(*Definesolutioninterval*)zL[1]=0;(*u1*)zR[1]=1;zL[2]=0;(*s1*)zR[2]=1;zL[3]=-1;(*u2*)zR[3]=0;zL[4]=-1;(*s2*) 422October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9414Y.-L.ZhaoandS.-J.LiaozR[4]=0;(*Defineinitialguess*)U[1,0]=(z-z^2)*lamb/h+1;(*u1*)U[2,0]=z*lambT/h+(1-lambT/h)*z^2;(*s1*)U[3,0]=1+z;(*u2*)U[4,0]=z^2+z;(*s2*)(*Definetheauxiliarylinearoperator*)L[1,u_]:=D[u,{z,2}];L[2,u_]:=D[u,{z,2}];L[3,u_]:=D[u,{z,2}];L[4,u_]:=D[u,{z,2}];(*Definephysicalparameters*)P=-5;b=1;n=1;Ra=1;M=2;Gr=5;lamb=1;lambT=1;h=1;phi=Pi/6;Pr=7/10;Ec=1/100;(*Printinputdata*)PrintInput[{u1[z],s1[z],u2[z],s2[z]}];(*Getoptimalc0*)GetOptiVar[4,{},{c0[1],c0[2],c0[3],c0[4]}](*Gain10th-orderHAMapproximation*)BVPh[1,30];A.7.InputdataofBVPh2.0forExample5(*Filename:Example5.m*)Print["Theinputfile",$InputFileName,"isloaded!"]; 423October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9HAM-BasedPackageBVPh2.0forNonlinearBoundaryValueProblems415(*ModifycontrolparametersinBVPhifnecessary*)TypeL=2;HYBRID=1;(*hybrid-basefunctions*)TypeBase=2;(*evenFourierseries*)ApproxQ=1;Ntruncated=20;(*Definethegoverningequation*)TypeEQ=2;NumEQ=1;f[1,z_,{u_},lambda_]:=D[u,{z,2}]+(K-1)*D[u,z]/z+lambda*Exp[u];(*DefineBoundaryconditions*)NumBC=3;BC[1,z_,{u_}]:=(u-A)/.z->0;BC[2,z_,{u_}]:=D[u,z]/.z->0;BC[3,z_,{u_}]:=u/.z->1;(*Definesolutioninterval*)zL[1]=0;zR[1]=1;(*Defineinitialguess*)U[1,0]=A/2*(1+Cos[Pi*z]);(*Definetheauxiliarylinearoperator*)L[1,f_]:=D[f,{z,2}]+Pi^2*f;(*Definephysicalparameters*)K=2;A=1;(*Printinputdata*) 424October24,201310:44WorldScientificReviewVolume-9inx6inAdvances/Chap.9416Y.-L.ZhaoandS.-J.LiaoPrintInput[{u[z]}];(*Getoptimalc0*)GetOptiVar[6,{},c0[1]];(*Printinputdata*)PrintInput[{u[z]}];(*Use3rd-orderiterationapproach*)iter[1,6,3]AcknowledgmentThisworkispartlysupportedbyNationalNaturalScienceFoundationofChinaunderGrantNo.11272209andStateKeyLaboratoryofOceanEngineeringunderGrantNo.GKZD010056.References[1]L.F.Shampine,I.GladwellandS.Thompson,SolvingODEswithMATLAB.CambridgeUniversityPress,Cambridge(2003).[2]J.Kierzenka,L.F.Shampine,ABVPsolverbasedonresidualcontrolandtheMatlabPSE,ACMTOMS.27(3),299–316(2001).[3]Z.BattlesandL.N.Trefethen,AnextensionofMatlabtocontinuousfunc-tionsandoperators,SIAMJ.Sci.Comput.25(5),1743–1770(2004).[4]L.N.Trefethen,Computingnumericallywithfunctionsinsteadofnumbers,Math.Comput.Sci.1,9–19(2007).[5]Y.P.Liu,S.J.LiaoandZ.B.Li,Symboliccomputationofstronglynonlinearperiodicoscillations,J.Symb.Comput.55,72–95(2013).[6]M.Sajid,Z.Iqbal,T.HayatandS.Obaidat,SeriessolutionforrotatingflowofanupperconvectedMaxwellfluidoverastrtchingsheet,Commun.Theor.Phys.56(4),740–744(2011).[7]T.Hayat,M.NawaandA.A.Hendi,HeattransferanalysisonaxisymmetricMHDflowofamicropolarfluidbetweentheradiallystretchingsheets,J.Mech.27(4),607–617(2011).[8]S.J.Liao,Notesonthehomotopyanalysismethod:Somedefinitionsandtheorems,Commun.NonlinearSci.Numer.Simulat.14(4),983–997(2009).[9]S.J.Liao,Proposedhomotopyanalysistechniquesforthesolutionofnon-linearproblem.Ph.D.thesis,ShanghaiJiaoTongUniversity(1992).[10]S.J.Liao,Auniformlyvalidanalyticsolutionoftwo-dimensionalviscousflowoverasemi-infiniteflatplate,J.FluidMech.385,101–128(1999). 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