Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with Matlab and Maple PDF
Preview Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with Matlab and Maple
“01-fm-i-iv-9780123846525” — 2010/12/8 — 15:29 — page 1 — #1 Traveling Wave Analysis of Partial Differential Equations “01-fm-i-iv-9780123846525” — 2010/12/8 — 15:29 — page 3 — #3 Traveling Wave Analysis of Partial Differential Equations Numerical and Analytical Methods (cid:114) with MATLAB and Maple™ Graham W. Griffiths CityUniversity,London,UK William E. Schiesser LehighUniversity,Bethlehem,PA,USA AMSTERDAM•BOSTON•HEIDELBERG•LONDON NEWYORK•OXFORD•PARIS•SANDIEGO SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO AcademicPressisanimprintofElsevier “01-copy-iv-9780123846525” — 2010/12/8 — 15:25 — page 4 — #1 AcademicPressisanimprintofElsevier 30CorporateDrive,Suite400,Burlington,MA01803,USA TheBoulevard,LangfordLane,Kidlington,Oxford,OX51GB,UK Copyright©2012ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicor mechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem,without permissioninwritingfromthepublisher.Detailsonhowtoseekpermission,furtherinformationaboutthe Publisher’spermissionspoliciesandourarrangementswithorganizationssuchastheCopyrightClearance CenterandtheCopyrightLicensingAgency,canbefoundatourWebsitewww.elsevier.com/permissions ThisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythePublisher(other thanasmaybenotedherein). MATLAB(cid:114)isatrademarkofTheMathWorks,Inc.andisusedwithpermission.TheMathWorksdoesnotwarrant theaccuracyofthetextorexercisesinthisbook.Thisbook’suseordiscussionofMATLAB(cid:114)softwareorrelated productsdoesnotconstituteendorsementorsponsorshipbyTheMathWorksofaparticularpedagogical approachorparticularuseoftheMATLAB(cid:114)software. Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperiencebroadenour understanding,changesinresearchmethods,professionalpractices,ormedicaltreatmentmaybecome necessary. Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluatingandusing anyinformation,methods,compounds,orexperimentsdescribedherein.Inusingsuchinformationormethods theyshouldbemindfuloftheirownsafetyandthesafetyofothers,includingpartiesforwhomtheyhavea professionalresponsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assumeany liabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproductsliability,negligenceor otherwise,orfromanyuseoroperationofanymethods,products,instructions,orideascontainedinthe materialherein. LibraryofCongressCataloging-in-PublicationData Applicationsubmitted BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary. ISBN:978-0-12-384652-5 ForinformationonallAcademicPresspublications visitourWebsiteatwww.elsevierdirect.com Typesetby:diacriTech,India PrintedintheUnitedStatesofAmerica 11 12 13 14 8 7 6 5 4 3 2 1 “02-ded-v-vi-9780123846525” — 2010/12/8 — 15:20 — page v — #1 Dedication Toourteachers,withrespectandappreciation. “04-pref-xi-xiv-9780123846525” — 2010/12/8 — 15:21 — page xi — #1 Preface Partialdifferentialequations(PDEs)havebeendevelopedandusedinscienceandengineer- ing for more than 200 years, yet they remain a very active area of research because of both theirroleinmathematicsandtheirapplicationtovirtuallyallareasofscienceandengineer- ing.Thisresearchhasbeenspurredbytherelativelyrecentdevelopmentofcomputersolution methods for PDEs. These have extended PDE applications such that we can now quantify broadareasofphysical,chemical,andbiologicalphenomena.ThecurrentdevelopmentofPDE solutionmethodsisanactiveareaofresearchthathasbenefitedgreatlyfromadvancesincom- puterhardwareandsoftware,andthegrowinginterestinaddressingPDEmodelsofincreasing complexity. Alargeclassofmodelsnowbeingactivelystudiedareofatypeandcomplexitysuchthat their solutions are usually beyond traditional mathematical analysis. Consequently, numeri- cal methods have to be employed. These numerical methods, some of which are still being developed, require testing and validation. This is often achieved by studying PDEs that have knownexactanalyticalsolutions.Thedevelopmentofanalyticalsolutionsisalsoanactivearea of research, with many advances being reported recently, particularly for systems described bynonlinearPDEs.Thus,thedevelopmentofanalyticalsolutionsdirectlysupportsthedevel- opmentofnumericalmethodsbyprovidingaspectrumoftestproblemsthatcanbeusedto evaluatenumericalmethods. This book surveys some of these new developments in analytical and numerical meth- ods and is aimed at senior undergraduates, postgraduates, and professionals in the fields of engineering, mathematics, and the sciences. It relates these new developments through the expositionofaseriesoftravelingwavesolutionstocomplexPDEproblems.ThePDEsthathave beenselectedarelargelynamedinthesensethattheyaregenerallycloselylinkedtotheirorig- inalcontributors.ThesenamesusuallyreflectthefactthatthePDEsarewidelyrecognizedand areoffundamentalimportancetotheunderstandingofmanyapplicationareas.Eachchapter .followsthegeneralformat: ThePDEanditsassociatedauxiliaryconditions(initialconditions(ICs)andboundary . conditions(BCs))arestated. Aseriesofroutinesisdiscussedwithdetailedexplanationsofthecodeandhowitrelatesto thePDE.TheyarewritteninMatlabbuthavebeenspecificallyprogrammedsothatthey canbeeasilyconvertedtoequivalentroutinesinotherlanguages.Theroutineshavethe followingcommonfeatures: – Thenumericalprocedureisthemethodoflines(MOL)inwhichtheboundaryvalue (spatial)partialderivativesarereplacedwithalgebraicapproximations,inthepresent casefinitedifferences(FDs),althoughotherapproximationssuchasfiniteelements (FEs),finitevolumes(FVs),andspectralmethods(SMs)couldbeused.TheFD approximationsareimplementedinaseriesoflibraryroutines;thedetailsofhowthese routinesweredevelopedaregivenasanintroductiontofacilitatethedevelopmentof newroutinesthatmayberequiredforparticularPDEapplications. TravelingWaveAnalysisofPartialDifferentialEquations.DOI:10.1016/B978-0-12-384652-5.xxxxx-x xi Copyright©2011byElsevierInc.Allrightsreserved. “04-pref-xi-xiv-9780123846525” — 2010/12/8 — 15:21 — page xii — #2 xii Preface – Theresultingsystemofordinarydifferentialequations(ODEs)inaninitialvalue variable,typicallytimeinanapplication,isthenintegratednumericallyusinganinitial valueODEintegratorfromtheMatlablibrary. – Thedisplayednumericaloutputalsoincludestheanalyticalsolutionandthedifference betweenthenumericalandanalyticalsolutions.Theagreementbetweenthetwo solutionsisdisplayednumericallyandgraphicallyasawayofdemonstratingthe . validityofthenumericalmethods. AnanalyticalsolutionforthePDEisstated,includingareferencetotheoriginalsourceof thesolution,andinsomecases,averification(proof)ofthesolutionbysubstitutioninto . thePDEandauxiliaryconditions. Additionally,inseveralchapters,theanalyticalsolutionisderivedbyrelativelynew techniquessuchasthetanh-,exp-,Riccati-orfactorization-basedmethods.Thederivation iseitherbydirectapplicationoftheanalyticalmethodorthroughtheuseofthecomputer algebrasystem(CAS),Maple.WhereMapleisused,theassociatedcodeisincludedinthe textalongwithadescriptionofitsmainfunctionalelements.Thiscodeusually demonstratestheuseofournewMapleprocedures,whichimplementvariousanalytical methodsthataredescribedinthetext.GraphicaloutputfromtheseMapleapplicationsis provided,includinga2Danimation(tofacilitateinsightintoandunderstandingofthe solution)andaplotin3Dperspective.Mapleisalsousedinotherchapterstoconfirm analyticalsolutionsfromtheliterature.Whereappropriate,thecodeisprovidedinthemws . fileformataswellasthemwformatsothatitwillalsoruninearlyversionsofMaple. Theformoftheanalyticalsolutionisconsidered,withparticularemphasisontraveling waveanalysisbywhichthePDE(inanEulerianorfixedframe)isconvertedtoanODE(ina Lagrangianormovingframe).AnanalyticalsolutiontotheODEisthenderivedandthe . inversecoordinatetransformationisappliedtoprovideananalyticalsolutiontothePDE. AsecondapproachtoaPDEanalyticalsolution,themethodofresidualfunctions,isalso usedinsomeofthechapterstoderiveananalyticalsolutiontoaPDEthatiscloselyrelated . totheoriginalPDE. Thebasicapproachoftravelingwaveanalysis,wherebyaPDEistransformedtoan associatedODE,isalsoreversedintwochapters.ThesestartwithODEsthatarethen restatedasPDEsthatarefirstandsecondorderintheinitialvaluevariable.Theanalytical . solutiontotheinitialODEisthenprovidedasananalyticalsolutiontothePDE. ThestructureofthePDEisusuallyrevisitedbrieflywithregardtoitsform,suchaswhether itisfirstorsecondorderintheinitialvaluevariable,theorderoftheboundaryvalue derivatives,thefeaturesofnonlinearterms,andtheformoftheBCs.Inthisway,the intentionofthefinalsummaryistosuggestconceptsandcomputationalapproachesthat . canbeappliedinnewPDEapplications. Eachchapterconcludeswithadiscussionofthenumericalsolution,particularlyhowit conformstotheinitialstatementofthePDEanditsauxiliaryconditions;alsothe numericalsolutionisevaluatedwithregardtothemagnitudeoftheerrorsandhowthese . errorsmightbereducedthroughadditionalcomputation. InChapter2wediscussthelinearadvectionequation,oneofthesimplestPDEs,andshow thatsolutionsinvolvingsteepgradientsordiscontinuitiescanbedifficulttoachieve numerically.Wethenillustratehowfluxlimiterscanbeemployedtoimprovethefidelityof thenumericalsolution.Ashortappendixtothischapterisalsoincluded,whichbriefly . discussessomeofthebackgroundtotheideasbehindfluxlimiters. Ageneralappendixdetailsthetanh-,exp-,Ricatti-,directintegration-,and factorization-basedmethods.Mapleimplementation,bywayofnewlydeveloped procedures,isincludedforthetanh-,exp-andRicatti-basedanalyticalmethods.As “04-pref-xi-xiv-9780123846525” — 2010/12/8 — 15:21 — page xiii — #3 Preface xiii referredtoabove,thesegeneralfeaturesarethenreferencedforspecificapplicationsin appendicestoindividualchapters. InsummarythemajorfocusofthisbookisthenumericalMOLsolutionofPDEsandthe testingofnumericalmethodswithanalyticalsolutions,throughaseriesofapplications.The originoftheanalyticalsolutionsthroughtravelingwaveandresidualfunctionanalysisprovides aframeworkforthedevelopmentofanalyticalsolutionstononlinearPDEsthatarenowwidely reported in the literature. Also in selected chapters, procedures based on the tanh, exp, and Ricattimethodsthathaverecentlyreceivedmajorattentionareusedtoillustratethederivation of analytical solutions. References are provided where appropriate to additional information onthetechniquesandmethodsdeployed. Ourintentionistoprovideasetofsoftwaretoolsthatimplementnumericalandanalytical methodsthatcanbeappliedtoabroadspectrumofproblemsinPDEs.Theyarebasedonthe conceptofatravelingwaveandthecentralfeatureofthesemethodsisconversionofthesystem PDEstoODEs.Thediscussionislimitedtoone-dimensional(1D)PDEsandcomplementsour earlierbookACompendiumofPartialDifferentialEquationModels:MethodofLinesAnalysis withMatlab,CambridgeUniversityPress,2009. Finallyallthecodediscussedinthisbook,alongwithasetoftheMOLDSSlibraryroutines, isavailablefordownloadfromwww.pdecomp.net. GrahamW.Griffiths Nayland,Suffolk,UK WilliamE.Schiesser Bethlehem,PA,USA June1,2010 “05-ch01-001-006-9780123846525” — 2010/12/9 — 12:58 — page 1 — #1 1 Introduction to Traveling Wave Analysis Most applications of partial differential equations (PDEs) in science and engineering require numerical solutions, since the equations are typically too complicated, both in numberandform,toadmitanalyticalsolutions.However,numericalprocedures(meth- ods,algorithms)areavailabletocomputenumericalsolutionstomostproblems.Inthis book,weintroducethemethodoflines(MOL),ageneralnumericalprocedurethatcanbe appliedtoallthemajorclassesofPDEs. InordertotestMOLalgorithmsandsoftware,whichgivesussomeassurance,thatthe methodsarecorrect,weutilizeanalytical(exact)solutionsforcomparisonwiththenumer- icalsolutions.Inthesubsequentdiscussion,wepresenttwodistinctmethodologieswith regard to the derivation of analytical solutions that have been widely used and reported extensively.Theyare(1)thetravelingwavemethodand(2)theresidualfunctionmethod. The approach we have followed is, for each chapter, to illustrate the use of these meth- odsthroughexampleapplications.Thus,typicallyaMOLnumericalsolutionispresented foranimportantPDEthroughaparalleldiscussionoftheequationsandMatlabroutines. Theparticularfocusoftheapplicationisemphasized,e.g.,calculationofthePDEspatial derivatives, implementation of the boundary conditions, and extension of the applica- tiontoothercasesthatrequireanumericalsolution.Inaddition,ananalyticalsolutionis derivedusingeithertheassociatedtravelingwaveorresidualfunctionmethod.Thetrav- elingwaveanalyticalsolutionsarederivedwithMapleproceduresandscriptsthatarealso presented. Through this approach, we hope to convey the essence of MOL and analyti- cal analysis as applied to a series of applications that illustrate a spectrum of important conceptsanddetails. Westartwithabriefintroductiontothemethodsoftravelingwavesolutionsandresid- ualfunctionstoprovideanalyticalsolutionsthatcanbeusedtotestthenumericalMOL procedures. Traveling Wave Solutions WeconsiderageneralPDE (cid:32) (cid:33) ∂u ∂u ∂2u ∂2u ∂3u =f u, , , , ,... (1.1) ∂t ∂x ∂x2 ∂x∂t ∂x2∂t . TravelingWaveAnalysisofPartialDifferentialEquations DOI:10.1016/B978-0-12-384652-5.00001-7 1 Copyright©2012ElsevierInc.Allrightsreserved. “05-ch01-001-006-9780123846525” — 2010/12/9 — 12:58 — page 2 — #2 2 TRAVELINGWAVEANALYSISOFPARTIALDIFFERENTIALEQUATIONS whichcanbeanalyzedthroughachangeofvariablesu(x,t)=U(ξ),whereξ =ξ(x,t)isa functiontobespecified.Then,eq.(1.1)canbewrittenas ∂u dU ∂ξ (cid:18) dU ∂ξ ∂ (cid:18)dU ∂ξ(cid:19) ∂ (cid:18)dU ∂ξ(cid:19) ∂ (cid:20) ∂ (cid:18)dU ∂ξ(cid:19)(cid:21) (cid:19) = =f U, , , , ,... ∂t dξ ∂t dξ ∂x ∂x dξ ∂x ∂t dξ ∂x ∂t ∂x dξ ∂x (cid:32) dU ∂ξ dU(cid:18)∂2ξ(cid:19) d2U(cid:18)∂ξ(cid:19)2 dU(cid:18) ∂2ξ (cid:19) d2U(cid:18)∂ξ(cid:19)(cid:18)∂ξ(cid:19) (cid:33) =f U, , + , + ,... (1.2) dξ ∂x dξ ∂x2 dξ2 ∂x dξ ∂x∂t dξ2 ∂x ∂t ∂ξ ∂ξ Forthelinearcaseξ(x,t)=k(x−ct),thepartialderivativesineq.(1.2)are =−kc, = ∂t ∂x ∂2ξ ∂2ξ k, = =···=0.Thiscase(ξ(x,t)=k(x−ct))isgenerallytermedatravelingwave, ∂x2 ∂x∂t sinceitcorrespondstoalineartranslationalongthex axiswithrespecttot;k andc are arbitraryconstantsgenerallytermedthewavenumberandwavevelocity,respectively.For thiscase,eq.(1.2)reducesto (cid:32) (cid:33) dU dU d2U d2U d3U (−kc) =f U,k ,k2 ,−k2c ,−k3c ,... dξ dξ dξ2 dξ2 dξ3 orincanonicalform (cid:32) (cid:33) dU dU d2U d3U =f U, , , ,... (1.3) dξ dξ dξ2 dξ3 where the constants c and k are included in f. Equation (1.3) is an ordinary differential equation(ODE)inξ (whichillustratesaprincipaladvantageofatravelingwavesolution, i.e., a PDE is reduced to an ODE). If a solution to eq. (1.3), U(ξ), can be found, then the solutiontoeq.(1.1)followsasu(x,t)=U(ξ).Theextensiontootherderivativesineq.(1.1), ∂3u ∂3u ∂4u suchas , , ,...,followsinthesamewayastheprecedinganalysis. ∂x3 ∂x∂t2 ∂x4 The solution process for eq. (1.3) is often based on the auxiliary conditions that the dependent variable and its first, second, and higher spatial derivatives tend to zero as ξ →∞,i.e., dU(ξ→±∞) d2U(ξ→±∞) U(ξ→±∞)=0, =0, =0,...,etc. (1.4) dξ dξ2 Consequently,constantsofintegrationproducedduringthesolutionofeq.(1.3)aretaken aszero. Analyticalsolutionsofeq.(1.3)havetypicallybeenachievedusingmanyapproaches. We discuss in detail the following methods in the main Appendix and give examples throughoutthevariouschapters: . Directintegrationmethod,seeappendix3of[6],whichappliesstandardcalculus techniquestotransformtheproblemintoonethatcanbeintegrated.
Description:The list of books you might like
