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1、http://www.paper.edu.cnAmodi¯edhomotopyanalysismethodandtheirapplicationtonon-homogeneousdi®erentialequationsMei-meiZhao¤y,ChaoLiSchoolofMathematicsandStatistics,LanzhouUniversityLanzhou,Gansu730000,P.R.ofChinaAbstractAmodi¯edhomotopyanalysismethod(HAM)ispres
2、entedforsolvingnonhomogeneousdi®erentialequations.Themainadvantageofthemodi¯edHAMisthatwecanacceleratetheconvergencerate,minimizeiterativetimes,accordingly,savethecomputationtimeandevaluatethee±ciency,ifwechoosetheproperdecompositionforthenonhomogeneoustermin
3、themodi¯edHAM.Examplesshowedthecapabilityofthemodi¯edHAM.KeywordsThehomotopyanalysismethod,Themodi¯edhomotopyanalysismethod,non-homogeneousdi®erentialequations.1IntroductionTheHAMisoneofthewell-knownmethodstosolvethenonlinearequationswhichwasexpressedbyLiao[1
4、-5]in1992,andstudiedbyalargenumberofresearcherssuchasAbbas-bandy[6]andD.D.Ganji[7].Thismethodhasbeensuccessfullyappliedtosolvemanytypesofnonlinearproblems[8-22].Theadvantageofthismethodisthat,itcontainsanauxiliaryparameter~whichprovidesuswithasimplewaytoadjus
5、tandcontroltheconvergenceregionofsolutionseries,ontheotherhand,wecanchoosefreedomtoselectrelatedinitialapproximation.Itisthepurposeofthepresentpapertointroduceanewmodi¯cationoftheHAM.Weapplyamodi¯edversionofHAMtoaclassnon-homogeneousequations,N(u(r))=f(r);r2•
6、;(1)¤CorrespondingAuthor.yE-mailaddress:yunyun1886358@163.com(M.Zhao).1http://www.paper.edu.cnwithboundaryconditions,B(u;@u=@n)=0;r2¡(2)whereNisnonlinearoperator,Bisaboundaryoperator,¡istheboundaryofthedomain•andf(r)isaknownanalyticfunction.Thenewmodi¯cationd
7、emonstratesarapidconvergenceoftheseriessolutionifwearedoingproperly,andthereforeithasbeenshownthattobecomputationallye±cientinseveralexamplesinapplied¯elds.Sometimes,themodi¯edalgorithmmaygivetheexactsolutionfornon-homogeneousdi®erentialequationsbyusingtwoite
8、rationsonly.Theobtainedresultsuggestthatthisnewlyimprovementtechniqueintroducesapowerfulimprovementforsolvingnon-homogeneousdi®erentialequations.2AnalysisofthemethodThePrinciplesoftheHAMa
