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1、InternationalJournalofNon-LinearMechanics35(2000)37}43Acouplingmethodofahomotopytechniqueandaperturbationtechniquefornon-linearproblemsJi-HuanHe*ShanghaiUniversity,ShanghaiInstituteofAppliedMathematicsandMechanics,149YanchangRoad,Shanghai200072,People+sRepublicofChinaReceived6January1998;re
2、ceivedinrevisedform5June1998AbstractInthispaper,acouplingmethodofahomotopytechniqueandaperturbationtechniqueisproposedtosolvenon-linearproblems.Incontrasttothetraditionalperturbationmethods,theproposedmethoddoesnotrequireasmallparameterintheequation.Inthismethod,accordingtothehomotopytechni
3、que,ahomotopywithanimbeddingparameterp3[0,1]isconstructed,andtheimbeddingparameterisconsideredasa&&smallparameter''.Sotheproposedmethodcantakefulladvantageofthetraditionalperturbationmethods.Someexamplesaregiven.Theresultsrevealthatthenewmethodisverye!ectiveandsimple.(1999ElsevierScienceLtd
4、.Allrightsreserved.Keywords:Perturbationtechnique;Homotopy;Non-linearity1.Introductionstrictsapplicationsofperturbationtechniques,asiswellknown,anoverwhelmingmajorityofnon-lin-Untilrecently,non-linearanalyticaltechniquesearproblemshavenosmallparametersatall.Sec-forsolvingnon-linearproblemsh
5、avebeendomin-ondly,thedeterminationofsmallparametersseemsatedbytheperturbationmethods,whichhavetobeaspecialartrequiringspecialtechniques.Anfoundwideapplicationsinengineering.But,likeappropriatechoiceofsmallparametersleadstoothernon-linearanalyticaltechniques,perturba-idealresults,however,an
6、unsuitablechoiceofsmalltionmethodshavetheirownlimitations.Firstly,parametersresultsinbade!ects,sometimesalmostallperturbationmethodsarebasedonsmallseriously.Thirdly,evenifthereexistsuitablepara-parameterssothattheapproximatesolutionscanmeters,theapproximatesolutionssolvedbythebeexpressedina
7、seriesofsmallparameters.Thisperturbationmethodsarevalid,inmostcases,onlyso-calledsmallparameterassumptiongreatlyre-forthesmallvaluesoftheparameters.Itisobviousthatalltheselimitationscomefromthesmallpara-meterassumption.Soitisverynecessarytodevel-*Tel.:00
