Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with Matlab and Maple

446 Pages·2011·7.26 MB·English

by  Graham W. Griffiths

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Traveling Wave Analysis of
Partial Differential
Equations
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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
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Dedication
Toourteachers,withrespectandappreciation.
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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.
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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
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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
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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
.
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Copyright©2012ElsevierInc.Allrightsreserved.
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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.