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1、FuzzySetsandSystems189(2011)7491www.elsevier.com/locate/fssNumericalsolutionsoffuzzydifferentialequationsbyextendedRungeKutta-likeformulaeoforder4B.Ghazanfaria,∗,A.ShakeramibaDepartmentofMathematics,LorestanUniversity,68137-17133Khoramabad,IranbLorestanEducationDepar
2、tment,Khoramabad,IranReceived9September2009;receivedinrevisedform28June2011;accepted28June2011Availableonline19July2011AbstractInthispaper,weapplyanumericalalgorithmforsolvingthefuzzyfirstorderinitialvalueproblem,basedonextendedRungeKutta-likeformulaeoforder4.WeuseSei
3、kkalasderivative.Theelementarypropertiesofthisnewsolutionaregiven.WeusetheextendedRungeKutta-likeformulaeinordertoenhancetheorderofaccuracyofthesolutionsusingevaluationsofbothfandf,insteadoftheevaluationsoffonly.©2011ElsevierB.V.Allrightsreserved.Keywords:Fuzzynumbe
4、rs;Fuzzydifferentialequations;ExtendedRungeKutta-likemethods;Numericalsolution;Eulermethod1.IntroductionThetheoreticalframeworkoffuzzyinitialvalueproblems(FIVPs)hasbeenanactiveresearchfieldoverthelastfewyears.TheconceptoffuzzyderivativewasfirstintroducedbyChangandZadeh
5、in[4].ItwasfollowedupbyDuboisandPradein[6],whodefinedandusedtheextensionprinciple.AcomprehensiveapproachtoFIVPshasbeentheworkofSeikkala[19],andKaleva[12,13],especiallyinitsgeneralizedformgivenbyBuckleyandFeuring[2].Theirworkisimportantasitovercomestheexistenceofmultip
6、ledefinitionsofthederivativeoffuzzyfunctions,see[6,10,14,16,19].Also,[2]comparesvarioussolutionstothefuzzyinitialvalueproblemthatonemayobtainusingdifferentderivatives.Theresultsof[19]onacertaincategoryoffuzzydifferentialequations(FDEs)haveinspiredseveralauthorswhohave
7、appliednumericalmethodsforthesolutionoftheseequations.ThemostimportantcontributiononthesenumericalmethodsistheEulermethodprovidedbyMaetal.in[14].AbbasbandyandAllahviranlooin[1]developedfour-stageorderRungeKuttamethodsforaCauchyproblemwithafuzzyinitialvalue.Also,in[15
8、],theauthorsappliedRungeKuttamethodsforamoregeneralcategoryofproblems,andtheyprovedconvergencefors-stageRungeKuttamethods.Pedersona
