Homotopy analysis method in nonlinear differential equations PDF
Preview Homotopy analysis method in nonlinear differential equations
Shijun Liao Homotopy Analysis Method in Nonlinear Differential Equations Shijun Liao Homotopy Analysis Method in Nonlinear Differential Equations With 127 figures Author ShijunLiao ShanghaiJiaoTongUniversity Shanghai200030,China Email:[email protected] ISBN978-7-04-032298-9 HigherEducationPress,Beijing ISBN978-3-642-25131-3 ISBN978-3-642-25132-0(eBook) SpringerHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber:2011941367 (cid:2)c HigherEducationPress,BeijingandSpringer-VerlagBerlinHeidelberg2012 Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames, registerednames, trademarks, etc. inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To my mother,wife and daughter Preface Itiswell-knownthatperturbationandasymptoticapproximationsofnonlinearprob- lems often break down as nonlinearity becomes strong. Therefore, they are only validforweaklynonlinearordinarydifferentialequations(ODEs)andpartialdiffer- entialequations(PDEs)ingeneral. Thehomotopyanalysismethod(HAM)isananalyticapproximationmethodfor highly nonlinear problems, proposed by the author in 1992. Unlike perturbation techniques,theHAMisindependentofanysmall/largephysicalparametersatall: onecanalwaystransferanonlinearproblemintoaninfinitenumberoflinearsub- problemsbymeansoftheHAM.Secondly,differentfromallofotheranalytictech- niques, the HAM provides us a convenient way to guarantee the convergence of solutionseriessothatitisvalidevenifnonlinearitybecomesratherstrong.Besides, basedonthehomotopyintopology,itprovidesusextremelylargefreedomtochoose equationtypeoflinearsub-problems,basefunctionofsolution,initialguessandso on,sothatcomplicatednonlinearODEsandPDEscanoftenbesolvedinasimple way. Finally, the HAM logically contains some traditional methods such as Lya- punov’ssmallartificialmethod,Adomiandecompositionmethod,theδ-expansion method,andeventheEulertransform,sothatithasthegreatgenerality.Therefore, the HAMprovidesus a usefultool tosolvehighlynonlinearproblemsin science, financeandengineering. This book consists of three parts. In Part I, the basic ideas of the HAM, espe- cially its theoretical modificationsand developments,are described, includingthe optimal HAM approaches, the theorems about the so-called homotopy-derivative operator and the different types of deformationequations, the methods to control andaccelerateconvergence,therelationshiptoEulertransform,andsoon. In Part II, inspirited by so manysuccessful applicationsof the HAM in differ- entfieldsandalsobytheabilityof“computingwithfunctionsinsteadofnumbers” ofcomputeralgebrasystemlikeMathematicaandMaple,aMathematicapackage BVPh(version1.0)isdevelopedbytheauthorintheframeoftheHAMfornonlinear boundary-valueproblems.Adozenofexamplesareusedtoillustrateitsvalidityfor highlynonlinearODEswithsingularity,multiplesolutionsandmultipointboundary conditionsineitherafiniteoraninfiniteinterval,andevenforsometypesofnon- viii Preface linearPDEs.Asanopenresource,theBVPh1.0isgiveninthisbookwithasimple usersguideandisfreeavailableonline. In Part III, we illustrate that the HAM can be used to solve some complicated highlynonlinearPDEs soas to enrichanddeepenourunderstandingsaboutthese interesting nonlinear problems. For example, By means of the HAM, an explicit analytic approximation of the optimal exercise boundary of American put option wasgained,whichisoftenvalidforacoupleofdecadespriortoexpiry,whereasthe asymptoticandperturbationformulasarevalidonlyforacoupleofdaysorweeks ingeneral.AMathematicacodebasedonsuchkindofexplicitformulaisgivenin thisbookforbusinessmentogainaccurateresultsinafewseconds.Inaddition,by meansoftheHAM,thewave-resonancecriterionofarbitrarynumberoftraveling gravity waves was found, for the first time, which logically contains the famous Phillips’criterionforfourwaveswithsmallamplitude. Alloftheseshowtheoriginality,validityandgeneralityoftheHAMforhighly nonlinearproblemsinscience,financeandengineering. AllMathematicacodesandtheirinputdatafilesaregivenintheappendixesof thisbookandavailable(Accessed25Nov2011,willbeupdatedinthefuture)either at http://numericaltank.sjtu.edu.cn/HAM.htm orat http://numericaltank.sjtu.edu.cn/BVPh.htm Thisbookissuitableforresearchersandpostgraduatesinappliedmathematics, physics,financeandengineering,whoareinterestedinhighlynonlinearODEsand PDEs. Iwouldliketoexpressmygratitudetomycollaboratorsfortheirvaluablediscus- sionsandcommunications,andtomypostgraduatesfortheirhardworking.Thanks toNaturalScienceFoundationofChinaforthefinancialsupport. Iwouldliketoexpressmysincerethankstomyparents,wifeanddaughterfor theirlove,encouragementandsupportinthepast20years. Shanghai,China March2011 ShijunLiao Contents PartI BasicIdeasandTheorems 1 Introduction .................................................. 3 1.1 Motivationandpurpose ..................................... 3 1.2 Characteristicofhomotopyanalysismethod .................... 5 1.3 Outline ................................................... 6 References..................................................... 8 2 BasicIdeasoftheHomotopyAnalysisMethod .................... 15 2.1 Conceptofhomotopy ....................................... 15 2.2 Example2.1:generalizedNewtonianiterationformula ........... 19 2.3 Example2.2:nonlinearoscillation ............................ 26 2.3.1 Analysisofthesolutioncharacteristic ................... 26 2.3.2 Mathematicalformulations ............................ 31 2.3.3 Convergenceofhomotopy-seriessolution ............... 38 2.3.4 Essenceoftheconvergence-controlparameterc ......... 48 0 2.3.5 Convergenceaccelerationbyhomotopy-Pade´technique ... 56 2.3.6 Convergenceaccelerationbyoptimalinitial approximation ...................................... 59 2.3.7 Convergenceaccelerationbyiteration ................... 63 2.3.8 Flexibilityonthechoiceofauxiliarylinearoperator ....... 69 2.4 Concludingremarksanddiscussions .......................... 75 Appendix2.1 Derivationofδ in(2.57) ........................... 79 n Appendix2.2 Derivationof(2.55)bythe2ndapproach.............. 80 Appendix2.3 ProofofTheorem2.3 .............................. 82 Appendix2.4 Mathematicacode(withoutiteration)forExample2.2... 83 Appendix2.5 Mathematicacode(withiteration)forExample2.2...... 87 Problems ...................................................... 92 References..................................................... 92 x Contents 3 OptimalHomotopyAnalysisMethod ............................ 95 3.1 Introduction ............................................... 95 3.2 Anillustrativedescription ...................................101 3.2.1 Basicideas .........................................101 3.2.2 Differenttypesofoptimalmethods .....................104 3.3 Systematicdescription ......................................117 3.4 Concludingremarksanddiscussions ..........................121 Appendix3.1 MathematicacodeforBlasiusflow ...................122 Problems ......................................................126 References.....................................................127 4 SystematicDescriptionsandRelatedTheorems ...................131 4.1 Briefframeofthehomotopyanalysismethod...................131 4.2 Propertiesofhomotopy-derivative ............................133 4.3 Deformationequations ......................................148 4.3.1 Abriefhistory.......................................148 4.3.2 High-orderdeformationequations ......................153 4.3.3 Examples ..........................................165 4.4 Convergencetheorems ......................................168 4.5 Solutionexpression ........................................173 4.5.1 Choiceofinitialapproximation ........................175 4.5.2 Choiceofauxiliarylinearoperator .....................176 4.6 Convergencecontrolandacceleration .........................179 4.6.1 Optimalconvergence-controlparameter .................180 4.6.2 Optimalinitialapproximation .........................181 4.6.3 Homotopy-iterationtechnique .........................181 4.6.4 Homotopy-Pade´technique ............................182 4.7 Discussionsandopenquestions...............................183 References.....................................................185 5 RelationshiptoEulerTransform ................................189 5.1 Introduction ...............................................189 5.2 GeneralizedTaylorseries ...................................190 5.3 Homotopytransform .......................................210 5.4 RelationbetweenhomotopyanalysismethodandEuler transform .................................................215 5.5 Concludingremarks ........................................219 References.....................................................220 6 SomeMethodsBasedontheHAM ...............................223 6.1 Abriefhistoryofthehomotopyanalysismethod ................223 6.2 Homotopyperturbationmethod...............................225 6.3 Optimalhomotopyasymptoticmethod.........................228 6.4 Spectralhomotopyanalysismethod ...........................230 6.5 Generalizedboundaryelementmethod.........................230 6.6 Generalizedscaledboundaryfiniteelementmethod..............231 Contents xi 6.7 Predictorhomotopyanalysismethod ..........................232 References.....................................................233 PartII MathematicaPackageBVPhandItsApplications 7 MathematicaPackageBVPh ....................................239 7.1 Introduction ...............................................239 7.1.1 Scope..............................................242 7.1.2 Briefmathematicalformulas...........................243 7.1.3 Choiceofbasefunctionandinitialguess ................247 7.1.4 Choiceoftheauxiliarylinearoperator ..................250 7.1.5 Choiceoftheauxiliaryfunction ........................252 7.1.6 Choiceoftheconvergence-controlparameterc ..........253 0 7.2 Approximationanditerationofsolutions.......................254 7.2.1 Polynomials ........................................255 7.2.2 Trigonometricfunctions ..............................256 7.2.3 Hybrid-basefunctions ................................257 7.3 AsimpleusersguideoftheBVPh1.0 .........................260 7.3.1 Keymodules........................................260 7.3.2 Controlparameters...................................261 7.3.3 Input...............................................263 7.3.4 Output .............................................264 7.3.5 Globalvariables .....................................264 Appendix7.1 MathematicapackageBVPh(version1.0) .............265 References.....................................................278 8 NonlinearBoundary-valueProblemswithMultipleSolutions .......285 8.1 Introduction ...............................................285 8.2 Briefmathematicalformulas .................................286 8.3 Examples .................................................289 8.3.1 Nonlineardiffusion-reactionmodel .....................289 8.3.2 Athree-pointnonlinearboundary-valueproblem .........296 8.3.3 Channelflowswithmultiplesolutions ..................301 8.4 Concludingremarks ........................................306 Appendix8.1 InputdataofBVPhforExample8.3.1 ................307 Appendix8.2 InputdataofBVPhforExample8.3.2 ................309 Appendix8.3 InputdataofBVPhforExample8.3.3 ................310 Problems ......................................................312 References.....................................................313 9 NonlinearEigenvalueEquationswithVaryingCoefficients .........315 9.1 Introduction ...............................................315 9.2 Briefmathematicalformulas .................................317 9.3 Examples .................................................322 9.3.1 Non-uniformbeamactedbyaxialload ..................322 9.3.2 Gelfandequation ....................................333
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