Springer.Singular.Perturbation.Theory.2005.(By.Laxxuss)

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SINGULARPERTURBATIONTHEORYMATHEMATICALANDANALYTICALTECHNIQUESWITHAPPLICATIONSTOENGINEERING MATHEMATICALANDANALYTICALTECHNIQUESWITHAPPLICATIONSTOENGINEERINGAlanJeffrey,ConsultingEditorPublished:InverseProblemsA.G.RammSingularPerturbationTheoryR.S.JohnsonForthcoming:MethodsforConstructingExactSolutionsofPartialDifferentialEquationswithApplicationsS.V.MeleshkoTheFastSolutionofBoundaryIntegralEquationsS.RjasanowandO.SteinbachStochasticDifferentialEquationswithApplicationsR.Situ SINGULARPERTURBATIONTHEORYMATHEMATICALANDANALYTICALTECHNIQUESWITHAPPLICATIONSTOENGINEERINGR.S.JOHNSONSpringer eBookISBN:0-387-23217-6PrintISBN:0-387-23200-1©2005SpringerScience+BusinessMedia,Inc.Print©2005SpringerScience+BusinessMedia,Inc.BostonAllrightsreservedNopartofthiseBookmaybereproducedortransmittedinanyformorbyanymeans,electronic,mechanical,recording,orotherwise,withoutwrittenconsentfromthePublisherCreatedintheUnitedStatesofAmericaVisitSpringer'seBookstoreat:http://ebooks.springerlink.comandtheSpringerGlobalWebsiteOnlineat:http://www.springeronline.com ToRos,whostill,afternearly40years,sometimeslistenswhenIextolthewondersofsingularperturbationtheory,fluidmechanicsorwaterwaves—usuallyonalongtrekinthemountains. Thispageintentionallyleftblank CONTENTSForewordxiPrefacexiii1.Mathematicalpreliminaries11.1Someintroductoryexamples21.2Notation101.3Asymptoticsequencesandasymptoticexpansions131.4Convergentseriesversusdivergentseries161.5Asymptoticexpansionswithaparameter201.6Uniformityorbreakdown221.7Intermediatevariablesandtheoverlapregion261.8Thematchingprinciple281.9Matchingwithlogarithmicterms321.10Compositeexpansions35FurtherReading40Exercises41 viiiContents2.Introductoryapplications472.1Rootsofequations472.2Integrationoffunctionsrepresentedbyasymptoticexpansions552.3Ordinarydifferentialequations:regularproblems592.4Ordinarydifferentialequations:simplesingularproblems662.5Scalingofdifferentialequations752.6Equationswhichexhibitaboundary-layerbehaviour802.7Whereistheboundarylayer?862.8Boundarylayersandtransitionlayers90FurtherReading103Exercises1043.Furtherapplications1153.1Aregularproblem1163.2SingularproblemsI1183.3SingularproblemsII1283.4Furtherapplicationstoordinarydifferentialequations139FurtherReading147Exercises1484.Themethodofmultiplescales1574.1Nearlylinearoscillations1574.2Nonlinearoscillators1654.3Applicationstoclassicalordinarydifferentialequations1684.4Applicationstopartialdifferentialequations1764.5Alimitationontheuseofthemethodofmultiplescales1834.6Boundary-layerproblems184FurtherReading188Exercises1885.Someworkedexamplesarisingfromphysicalproblems1975.1Mechanical&electricalsystems1985.2Celestialmechanics2195.3Physicsofparticlesandoflight2265.4Semi-andsuperconductors2355.5Fluidmechanics242 ix5.6Extremethermalprocesses2555.7Chemicalandbiochemicalreactions262Appendix:TheJacobianEllipticFunctions269AnswersandHints271References283SubjectIndex287 Thispageintentionallyleftblank FOREWORDTheimportanceofmathematicsinthestudyofproblemsarisingfromtherealworld,andtheincreasingsuccesswithwhichithasbeenusedtomodelsituationsrangingfromthepurelydeterministictothestochastic,iswellestablished.Thepurposeofthesetofvolumestowhichthepresentonebelongsistomakeavailableauthoritative,uptodate,andself-containedaccountsofsomeofthemostimportantandusefuloftheseanalyticalapproachesandtechniques.Eachvolumeprovidesadetailedintroductiontoaspecificsubjectareaofcurrentimportancethatissummarizedbelow,andthengoesbeyondthisbyreviewingrecentcontributions,andsoservingasavaluablereferencesource.Theprogressinapplicablemathematicshasbeenbroughtaboutbytheextensionanddevelopmentofmanyimportantanalyticalapproachesandtechniques,inareasbotholdandnew,frequentlyaidedbytheuseofcomputerswithoutwhichthesolutionofrealisticproblemswouldotherwisehavebeenimpossible.Acaseinpointistheanalyticaltechniqueofsingularperturbationtheorywhichhasalonghistory.Inrecentyearsithasbeenusedinmanydifferentways,anditsimportancehasbeenenhancedbyithavingbeenusedinvariousfieldstoderivesequencesofasymptoticapproximations,eachwithahigherorderofaccuracythanitspredecessor.Theseapproximationshave,inturn,providedabetterunderstandingofthesubjectandstimulatedthedevelopmentofnewmethodsforthenumericalsolutionofthehigherorderapproximations.Atypicalexampleofthistypeistobefoundinthegeneralstudyofnonlinearwavepropagationphenomenaastypifiedbythestudyofwaterwaves. xiiForewordElsewhere,aswiththeidentificationandemergenceofthestudyofinverseproblems,newanalyticalapproacheshavestimulatedthedevelopmentofnumericaltechniquesforthesolutionofthismajorclassofpracticalproblems.Suchworkdividesnaturallyintotwoparts,thefirstbeingtheidentificationandformulationofinverseproblems,thetheoryofill-posedproblemsandtheclassofone-dimensionalinverseproblems,andthesecondbeingthestudyandtheoryofmultidimensionalinverseproblems.Onoccasionsthedevelopmentofanalyticalresultsandtheirimplementationbycomputerhaveproceededinparallel,aswiththedevelopmentofthefastboundaryelementmethodsnecessaryforthenumericalsolutionofpartialdifferentialequationsinseveraldimensions.Thisworkhasbeenstimulatedbythestudyofboundaryinte-gralequations,whichinturnhasinvolvedthestudyofboundaryelements,collocationmethods,Galerkinmethods,iterativemethodsandothers,andthenontotheirim-plementationinthecaseoftheHelmholtzequation,theLaméequations,theStokesequations,andvariousotherequationsofphysicalsignificance.Amajordevelopmentinthetheoryofpartialdifferentialequationshasbeentheuseofgrouptheoreticmethodswhenseekingsolutions,andintheintroductionofthecomparativelynewmethodofdifferentialconstraints.Inadditiontotheusefulcontributionsmadebysuchstudiestotheunderstandingofthepropertiesofsolu-tions,andtotheidentificationandconstructionofnewanalyticalsolutionsforwellestablishedequations,theapproachhasalsobeenofvaluewhenseekingnumericalsolutions.Thisismainlybecauseofthewayinmanyspecialcases,aswithsimilaritysolutions,agrouptheoreticapproachcanenablethenumberofdimensionsoccurringinaphysicalproblemtobereduced,therebyresultinginasignificantsimplificationwhenseekinganumericalsolutioninseveraldimensions.Specialanalyticalsolutionsfoundinthiswayarealsoofvaluewhentestingtheaccuracyandefficiencyofnewnumericalschemes.Adifferentareainwhichsignificantanalyticaladvanceshavebeenachievedisinthefieldofstochasticdifferentialequations.Theseequationsarefindinganincreasingnumberofapplicationsinphysicalproblemsinvolvingrandomphenomena,andoth-ersthatareonlynowbeginningtoemerge,asishappeningwiththecurrentuseofstochasticmodelsinthefinancialworld.Themethodsusedinthestudyofstochasticdifferentialequationsdiffersomewhatfromthoseemployedintheapplicationsmen-tionedsofar,sincetheydependfortheirsuccessontheItocalculus,martingaletheoryandtheDoob-Meyerdecompositiontheorem,thedetailsofwhicharedevelopedasnecessaryinthevolumeonstochasticdifferentialequations.Thereare,ofcourse,othertopicsinadditiontothosementionedabovethatareofconsiderablepracticalimportance,andwhichhaveexperiencedsignificantdevelop-mentsinrecentyears,butaccountsofthesemustwaituntillater.AlanJeffreyUniversityofNewcastleNewcastleuponTyneUnitedKingdom PREFACEThetheoryofsingularperturbationshasbeenwithus,inoneformoranother,foralittleoveracentury(althoughtheterm‘singularperturbation’datesfromthe1940s).Thesubject,andthetechniquesassociatedwithit,haveevolvedoverthisperiodasaresponsetotheneedtofindapproximatesolutions(inananalyticalform)tocomplexproblems.Typically,suchproblemsareexpressedintermsofdifferentialequationswhichcontainatleastonesmallparameter,andtheycanariseinmanyfields:fluidmechanics,particlephysicsandcombustionprocesses,tonamebutthree.Theessentialhallmarkofasingularperturbationproblemisthatasimpleandstraightforwardapproximation(basedonthesmallnessoftheparameter)doesnotgiveanaccuratesolutionthroughoutthedomainofthatsolution.Perforce,thisleadstodifferentapproximationsbeingvalidindifferentpartsofthedomain(usuallyrequiringa‘scaling’ofthevariableswithrespecttotheparameter).Thisinturnhasledtotheimportantconceptsofbreakdown,matching,andsoon.MathematicalproblemsthatmakeextensiveuseofasmallparameterwereprobablyfirstdescribedbyJ.H.Poincaré(1854–1912)aspartofhisinvestigationsincelestialmechanics.(Thesmallparameter,inthiscontext,isusuallytheratiooftwomasses.)Althoughthemajorityoftheseproblemswerenotobviously‘singular’—andPoincarédidnotdwelluponthis—someare;forexample,oneistheearth-moon-spaceshipproblemmentionedinChapter2.Nevertheless,Poincarédidlaythefoundationsforthetechniquethatunderpinsourapproach:theuseofasymptoticexpansions.ThenotionofasingularperturbationproblemwasfirstevidentintheseminalworkofL.Prandtl(1874–1953)ontheviscousboundarylayer(1904).Here,thesmallparameteris xivPrefacetheinverseReynoldsnumberandtheequationsarebasedontheclassicalNavier-Stokesequationoffluidmechanics.Thisanalysis,coupledwithsmall-Reynolds-numberap-proximationsthatweredevelopedataboutthesametime(1910),preparedthegroundforacenturyofsingularperturbationworkinfluidmechanics.Butotherfieldsoverthecenturyalsomadeimportantcontributions,forexample:integrationofdifferentialequations,particularlyinthecontextofquantummechanics;thetheoryofnonlinearoscillations;controltheory;thetheoryofsemiconductors.Allthese,andmanyothers,havehelpedtodevelopthemathematicalstudyofsingularperturbationtheory,whichhas,fromthemid-1960s,beensupportedandmadepopularbyarangeofexcellenttextbooksandresearchpapers.Thesubjectisnowquitefamiliartopostgraduatestu-dentsinappliedmathematics(andrelatedareas)and,tosomeextent,toundergraduatestudentswhospecialiseinappliedmathematics.Indeed,itisanessentialtoolofthemodernappliedmathematician,physicistandengineer.Thisbookisbasedonmaterialthathasbeentaught,mainlybytheauthor,toMScandresearchstudentsinappliedmathematicsandengineeringmathematics,attheUniversityofNewcastleuponTyneoverthelastthirtyyears.However,thepresentationoftheintroductoryandbackgroundideasismoredetailedandcomprehensivethanhasbeenofferedinanyparticulartaughtcourse.Inaddition,therearemanymoreworkedexamplesandsetexercisesthanwouldbefoundinmosttaughtprogrammes.Thestyleadoptedthroughoutistoexplain,withexamples,theessentialtechniques,butwithoutdwellingonthemoreformalaspectsofproof,etcetera;thisisfortworeasons.Firstly,theaimofthistextistomakeallthematerialreadilyaccessibletothereaderwhowishestolearnandusetheideastohelpwithresearchproblemsandwho(inalllikelihood)doesnothaveastrongmathematicalbackground(orwhoisnotthatconcernedabouttheseniceties).Andsecondly,manyoftheresultsandsolutionsthatwepresentcannotberecasttoprovideanythingthatresemblesaroutineproofofexistenceorasymptoticcorrectness.Indeed,inmanycases,nosuchproofisavailable,butthereisoftenampleevidencethattheresultsarerelevant,usefulandprobablycorrect.Thistexthasbeenwritteninaformthatshouldenabletherelativelyinexperienced(ornew)workerinthefieldofsingularperturbationtheorytolearnandapplyalltheessentialideas.Tothisend,thetexthasbeendesignedasalearningtool(ratherthanareferencetext,forexample),andsocouldprovidethebasisforataughtcourse.Thenumerousexamplesandsetexercisesareintendedtoaidthisprocess.Althoughitisassumedthatthereaderisquiteunfamiliarwithsingularperturbationtheory,therearemanyoccasionsinthetextwhen,forexample,adifferentialequationneedstobesolved.Inmostcasesthesolution(andperhapsthemethodofsolution)arequoted,butsomereadersmaywishtoexplorethisaspectofmathematicalanalysis;therearemanygoodtextsthatdescribemethodsforsolving(standard)ordinaryandpartialdifferentialequations.However,ifthereadercanacceptthegivensolution,itwillenablethemainthemeofsingularperturbationtheorytoprogressmoresmoothly.Chapter1introducesallthemathematicalpreliminariesthatarerequiredforthestudyofsingularperturbationtheory.First,afewsimpleexamplesarepresentedthathighlightsomeofthedifficultiesthatcanarise,goingsomewaytowardsexplainingtheneedforthistheory.Thennotation,definitionsandtheprocedureoffinding xvasymptoticexpansions(basedonaparameter)aredescribed.Thenotionsofuniformityandbreakdownareintroduced,togetherwiththeimportantconceptsofscalingandmatching.Chapter2isdevotedtoroutineandstraightforwardapplicationsofthemethodsdevelopedinthepreviouschapter.Inparticular,wediscusshowtheseideascanbeusedtofindtherootsofequationsandhowtointegratefunctionsrepresentedbyanumberofmatchedasymptoticexpansions.Wethenturntothemostsignificantapplicationofthesemethods:thesolutionofdifferentialequations.Somesimpleregular(i.e.notsingular)problemsarediscussedfirst—theseareratherrareandofnogreatimportance—followedbyanumberofexamplesofsingularproblems,includingsomethatexhibitboundaryortransitionlayers.Theroleofscalingadifferentialequationisgivensomeprominence.InChapter3,thetechniquesofsingularperturbationtheoryareappliedtomoresophisticatedproblems,manyofwhicharisedirectlyfrom(orarebasedupon)im-portantexamplesinappliedmathematicsormathematicalphysics.Thuswelookatnonlinearwavepropagation,supersonicflowpastathinaerofoil,solutionsofLaplace’sequation,heattransfertoafluidflowingthroughapipeandanexampletakenfromgasdynamics.Alltheseareclassicalproblems,atsomelevel,andareintendedtoshowtheefficacyofthesetechniques.Thechapterconcludeswithsomeapplicationstoordinarydifferentialequations(suchasMathieu’sequation)andthen,asanextensionofsomeoftheideasalreadydeveloped,themethodofstrainedcoordinatesispresented.Oneofthemostgeneralandmostpowerfultechniquesinthearmouryofsingularperturbationtheoryisthemethodofmultiplescales.Thisisintroduced,explainedanddevelopedinChapter4,andthenappliedtoawidevarietyofproblems.Thesein-cludelinearandnonlinearoscillations,classicalordinarydifferentialequations(suchasMathieu’sequation—again—andequationswithturningpoints)andthepropagationofdispersivewaves.Finally,itisshownthatthemethodofmultiplescalescanbeusedtogreateffectinboundary-layerproblems(firstmentionedinChapter2).Thefinalchapterisdevotedtoacollectionofworkedexamplestakenfromawiderangeofsubjectareas.Itishopedthateachreaderwillfindsomethingofinteresthere,andthatthesewillshow—perhapsmoreclearlythananythingthathasgonebefore—therelevanceandpowerofsingularperturbationtheory.Evenifthereisnothingofimmediateinterest,thereaderwhowishestobecomemoreskilledwillfindtheseausefulsetofadditionalexamples.Thesearelistedundersevenheadings:mechanical&electricalsystems;celestialmechanics;physicsofparticles&light;semi-andsu-perconductors;fluidmechanics;extremethermalprocesses;chemical&biochemicalreactions.Throughoutthetext,workedexamplesareusedtoexplainanddescribetheideas,whicharereinforcedbythenumerousexercisesthatareprovidedattheendofeachofthefirstfourchapters.(TherearenosetexercisesinChapter5,buttheextensiveref-erencescanbeinvestigatedifmoreinformationisrequired.)AlsoattheendofeachofChapters1–4isasectionoffurtherreadingwhich,inconjunctionwiththereferencescitedinthebodyofthechapter,indicatewhererelevantreferencematerialcanbefound.Thereferences(alllistedattheendofthebook)containbothtextsandresearchpapers.Sectionsineachchapterarenumberedfollowingthedecimalpattern,and xviPrefaceequationsarenumberedaccordingtothechapterinwhichtheyappear;thusequation(2.3)isthethird(numbered)equationinChapter2.Theworkedexamplesfollowasimilarpattern(soE3.3isthethirdworkedexampleinChapter3)andeachisgivenatitleinordertohelpthereader—perhaps—toselectanappropriateoneforstudy;theendofaworkedexampleisdenotedbyahalf-lineacrossthepage.Thesetexercisesaresimilarlynumbered(soQ3.2isthesecondexerciseattheendofChapter3)and,again,eachisgivenatitle;theanswers(and,insomecases,hintsandintermediatesteps)aregivenattheendofthebook(whereA3.2istheanswertoQ3.2).Adetailedandcomprehensivesubjectindexisprovidedattheveryendofthetext.IwishtoputonrecordmythankstoProfessorAlanJeffreyforencouragingmetowritethistext,andtoKluwerAcademicPublishersfortheirsupportthroughout.Imustalsorecordmyheartfeltthankstoalltheauthorswhocamebeforeme(andmostarelistedintheReferences)because,withouttheirguidance,theselectionofmaterialforthistextwouldhavebeenimmeasurablymoredifficult.Ofcourse,whereIhavebasedanexampleonsomethingthatalreadyexists,asuitableacknowledge-mentisgiven,butIamsolelyresponsibleformyversionofit.Similarly,theclarityandaccuracyofthefiguresrestssolelywithme;theywereproducedeitherinWord(aswasthemaintext),orasoutputfromMaple,orusingSmartDraw. 1.MATHEMATICALPRELIMINARIESBeforeweembarkonthestudyofsingularperturbationtheory,particularlyasitisrele-vanttothesolutionofdifferentialequations,anumberofintroductoryandbackgroundideasneedtobedeveloped.Weshalltaketheopportunity,first,todescribe(withoutbeingtoocarefulabouttheformalities)afewsimpleproblemsthat,itishoped,explaintheneedfortheapproachthatwepresentinthistext.Wediscusssomeelementarydif-ferentialequations(whichhavesimpleexactsolutions)andusethese—bothequationsandsolutions–tomotivateandhelptointroducesomeofthetechniquesthatweshallpresent.Althoughwewillwork,atthisstage,withequationswhichpossessknownsolutions,itiseasytomakesmallchangestothemwhichimmediatelypresentuswithequationswhichwecannotsolveexactly.Nevertheless,theapproximatemethodsthatwewilldeveloparegenerallystillapplicable;thuswewillbeabletotacklefarmoredifficultproblemswhichareoftenimportant,interestingandphysicallyrelevant.Manyequations,andtypically(butnotexclusively)wemeandifferentialequations,thatareencounteredin,forexample,scienceorengineeringorbiologyoreconomics,aretoodifficulttosolvebystandardmethods.Indeed,formanyofthem,itappearsthatthereisnorealisticchancethat,evenwithexceptionaleffort,skillandluck,theycouldeverbesolved.However,itisquitecommonforsuchequationstocontainparameterswhicharesmall;thetechniquesandideasthatweshallpresenthereaimtotakeadvantageofthisspecialproperty.Thesecond,andmoreimportantplaninthisfirstchapter,istointroducetheideas,definitionsandnotationthatprovidetheappropriatelanguageforourapproach.Thus 21.Mathematicalpreliminarieswewilldescribe:order,asymptoticsequences,asymptoticexpansions,expansionswithparameters,non-uniformitiesandbreakdown,matching.1.1SOMEINTRODUCTORYEXAMPLESWewillpresentfoursimpleordinarydifferentialequations–threesecond-orderandonefirst-order.Ineachcaseweareabletowritedowntheexactsolution,andwewillusethesetohelpustointerpretthedifficultiesthatweencounter.Eachequationwillcontainasmallparameter,whichwewillalwaystaketobepositive;theintentionistoobtain,directlyfromtheequation,anapproximatesolutionwhichisvalidforsmallE1.1AnoscillationproblemWeconsidertheconstantcoefficientequationwithx(0)=0,(wherethedotdenotesthederivativewithrespecttot);thisisaninitial-valueproblem.LetusassumethatthereisasolutionwhichcanbewrittenasapowerseriesinwhereeachoftheisnotafunctionofTheequation(1.1)thengiveswhereweagainuse,forconvenience,thedottodenotederivatives.Wewrite(1.3)intheformand,sincetheright-handsideispreciselyzero,allthemustvanish;thuswerequire(RememberthateachdoesnotdependonThetwoinitialconditionsgive 3and,usingthesameargumentasbefore,wemustchoosewherethe‘1’inthesecondconditionisaccommodatedby(Iftheinitialconditionswere,say,thenwewouldhavetoselectThusthefirstapproximationisrepresentedbytheproblemthegeneralsolutioniswhereAandBarearbitraryconstantswhich,tosatisfytheinitialconditions,musttakethevaluesA=1,B=0.ThesolutionisthereforeTheproblemforthesecondtermintheseriesbecomesThesolutionofthisequationrequirestheinclusionofaparticularintegral,whichhereisthecompletegeneralsolutionisthereforewhereCandDarearbitraryconstants.(Theparticularintegralcanbefoundbyanyoneofthestandardmethodse.g.variationofparameters,orsimplybytrial-and-error.)ThegivenconditionsthenrequirethatandD=0i.e.andsoourseriessolution,atthisstage,readsLetusnowreviewourresults.Theoriginaldifferentialequation,(1.1),shouldberecognisedastheharmonicoscillatorequationforalland,assuch,itpossessesbounded,periodicsolutions.Thefirstterminourseries,(1.5),certainlysatisfiesboththeseproperties,whereasthesecondfailsonbothcounts.Thustheseries,(1.7),alsofails:ourapproximation 41.MathematicalpreliminariesprocedurehasgeneratedasolutionwhichisnotperiodicandforwhichtheamplitudegrowswithoutboundasYettheexactsolutionissimplywhichiseasilyobtainedbyscalingoutthefactor,byworkingwithratherthant.(The‘e’subscripthereisusedtodenotetheexactsolution.)Itisnowanelementaryexercisetocheckthat(1.8)and(1.7)agree,inthesensethattheexpansionof(1.8),forsmallandfixedt,reproduces(1.7).(AfewexamplesofexpansionsaresetasexercisesinQ1.1,1.2.)Thisprocessimmediatelyhighlightsoneofourdifficulties,namely,takingfirstandthenallowingthisisaclassiccaseofanon-uniformlimitingprocessi.e.theanswerdependsontheorderinwhichthelimitsaretaken.(ExamplesofsimplelimitingprocessescanbefoundinQ1.4.)Clearly,anyapproximatemethodsthatwedevelopmustbeabletocopewiththistypeofbehaviour.So,forexample,ifitisknown(orexpected)thatbounded,periodicsolutionsexist,theapproachthatweadoptmustproduceasuitableapproximationtothissolution.Wehavetakensomecareinourdescriptionofthisfirstexamplebecause,atthisstage,theapproachandideasarenew;wewillpresenttheotherexampleswithslightlylessdetail.However,beforeweleavethisproblem,thereisonefurtherobservationtomake.Theoriginalequation,(1.1),canbesolvedeasilyanddirectly;anassociatedproblemmightbewithappropriateinitialdata.Thisdescribesanoscillatorforwhichthefrequencydependsonthevalueofx(t)atthatinstant—itisanonlinearproblem.Suchequationsaremuchmoredifficulttosolve;ourtechniqueshavegottobeabletomakesomeusefulheadwaywithequationslike(1.9).E1.2Afirst-orderequationWeconsidertheequationwithAgain,letusseekasolutionintheformandthenobtainor 5weusetheprimetodenotethederivative.ThuswerequirewiththeboundaryconditionsThesolutionforisimmediatelybutthisresultisclearlyunsatisfactory:thesolutionforgrowsexponentially,whereasthesolutionofequation(1.10)mustdecayfor(becausethenPer-hapsthenexttermintheserieswillcorrectthisbehaviourforlargeenoughx;wehaveThusandwerequireA=0;theseriessolutionsofaristhereforeHowever,thisisnoimprovement;now,forsufficientlylargex,thesecondtermdom-inatesandthesolutiongrowstowardsLetusattempttoclarifythesituationbyexaminingtheexactsolution.Wewriteequation(1.10)asthegeneralsolutionisthereforeand,withC=1tosatisfythegivenconditionatx=0,thisyieldsClearlytheseries,(1.12),isrecovereddirectlybyexpandingtheexactsolution,(1.13),inforfixedx,sothatweobtainEquallyclearly,thisprocedurewillgiveaverypoorapproximationforlargex;indeed,forxaboutthesizeoftheapproximationaltogetherfails.Aneatwaytoseethisistoredefinexasthisiscalledscalingandwillplayacrucialrôleinwhat 61.Mathematicalpreliminarieswedescribeinthistext.Ifwenowconsidersmall,forXfixed,thesizeofxisnowproportionaltoandtheresultsareverydifferent:indeed,inthisexample,wecannotevenwritedownasuitableapproximationof(1.14)forsmallTheexpressionin(1.14)attainsamaximumatX=1/2,andforlargerXthefunctiontendstozero.Weobservethatanytechniquesthatwedevelopmustbeabletohandlethissituation;indeed,thisexampleintroducestheimportantideathatthefunctionofinterestmaytakedifferent(approximate)formsfordifferentsizesofx.This,ultimately,isnotsurprising,butthesignificantingredienthereisthat‘differentsizes’aremeasuredintermsofthesmallparameter,Weshallbemorepreciseaboutthisconceptlater.E1.3Anothersimplesecond-orderequationThistimeweconsiderwith(Theuseofhere,ratherthanissimplyanalgebraicconvenience,aswillbecomeclear;obviouslyanysmallpositivenumbercouldberepresentedbyor—oranythingequivalent,suchasoretcetera.)Presumably—orsowewillassume—afirstapproximationtoequation(1.15),forsmallisjustbutthisproblemhasnosolution.ThegeneralsolutioniswhereAandBarethetwoarbitraryconstants,andnochoiceofthemcansatisfybothconditions.Inasense,thisisamoreworryingsituationthanthatpresentedbyeitherofthetwopreviousexamples:wecannotevengetstartedthistime.Theexactsolutionisandthedifficultiesareimmediatelyapparent:withxfixed,givesbutthenhowdoweaccommodatetheconditionatinfinity?Correspondingly,withandfixed,weobtainandnowhowcanweobtainthedependenceonAswecanreadilysee,totreatandxseparatelyisnotappropriatehere—weneedtoworkwithascaledversionofx(i.e.Thechoiceofsuchavariableavoidsthenon-uniformlimitingprocess:and 7E1.4Atwo-pointboundary-valueproblemOurfinalintroductoryexampleisprovidedbywithandgiven.Thisequationcontainstheparameterintwoplaces:multiplyingthehigherderivative,whichiscriticalhere(aswewillsee),andadjustingthecoefficientoftheotherderivativebyasmallamount.Thislatterappearanceoftheparameterisaltogetherunimportant—thecoefficientiscertainlyclosetounity—andservesonlytomakemoretransparentthecalculationsthatwepresent.Onceagain,wewillstartbyseekingasolutionwhichcanberepresentedbytheseriessothatweobtaintheshorthandnotationforderivativesisagainbeingemployed.Thuswehavethesetofdifferentialequationswithboundaryconditionswrittenaswhereandaregiven(butwewillassumethattheyarenotfunctionsofThegeneralsolutionforisbutitisnotatallclearhowwecandetermineA.Thedifficultythatwehaveinthisexampleisthatwemustapplytwoboundaryconditions,whichispatentlyimpossible(unlesssomespecialrequirementissatisfied).So,ifweuseweobtainif,byextremegoodfortune,wehavethenwealsosatisfythesecondboundarycondition(onx=1).Ofcourse,ingeneral,thiswillnotbethecase;letusproceedwiththeproblemforwhichThusthesolutionusingdoes 81.MathematicalpreliminariesnotsatisfyandthesolutiondoesnotsatisfyIndeed,wehavenowayofknowingwhich,ifeither,iscorrect;thusthereislittletobegainedbysolvingtheproblem:(Wenotethat,sincewemusthaveandthenthereis,ex-ceptionally,asolutionofthecompleteproblem:forButwestilldonotknowAsinourpreviousexamples,letusconstructandexaminetheexactsolution.Equa-tion(1.16)isasecondorder,constantcoefficient,ordinarydifferentialequationandsowemayseekasolutionintheformi.e.Thegeneralsolutionisthereforeand,imposingthetwoboundaryconditions,thisbecomes(WecannoteherethatthecontributionfromthetermisabsentinthespecialcaseweproceedwiththeproblemforwhichThissolution,(1.22),isdefinedforandwithletusselectanyand,forthisxfixed,allow(wheredenotestendingtozerothroughthepositivenumbers).Weobservethatthetermsandvanishrapidlyinthislimit,leavingthisisourapproximatesolutiongivenin(1.20).(Someexamplesthatexploretherelativesizesofexp(x)andln(x)canbefoundinQ1.5.)Thusoneofthepossibleoptionsforintroducedabove,isindeedcorrect.However,thissolutionis,asalreadynoted,incorrectonx=0(although,ofcourse,Thedifficultyisplainlywiththetermforanyx>0fixed,asthisvanishesexponentially,butonx=0thistakesthevalue1(one).Inordertoexaminetherôleofthisterm,asweneedtoretainit(butnottorestrictourselvestox=0);as 9wehaveseeninearlierexamples,asuitablerescalingofxisuseful.Inthiscasewesetandsoobtainandnow,foranyXfixed,aswehaveThisisasecond,anddifferent,approximationtovalidforxswhichareproportionaltonotethatonX=0,(1.25)givesthevaluewhichisthecorrectboundaryvalue.Insummary,therefore,wehave(from(1.23))and(from(1.25))Thesetwotogetherconstituteanapproximationtotheexactsolution,eachvalidforanappropriatesizeofx.Further,thesetwoexpressionspossessthecomfortingpropertythattheydescribeasmooth—notdiscontinuous—transitionfromonetotheother,inthefollowingsense.Theapproximation(1.26)isnotvalidforsmallx,butasxdecreaseswehave(whichwealreadyknowisincorrectbecausecorrespondingly,(1.27)isnotvalidforlargebutweseethatresults(1.28)and(1.29)agreeprecisely.Thisisclearlydemonstratedinfigure1,wherewehaveplottedtheexactsolutionfor(asanexample)i.e.forvariousAsdecreases,thedramaticallydifferentbehavioursforxnottoosmall,andxsmall,areveryevident.(Notethatthesolutionforxnottoosmallis 101.MathematicalpreliminariesFigure1.Plotofforwiththemaximumvalueattained(e)ismarkedonthey-axis.Inthesefoursimpleexamples,wehavedescribedsomedifficultiesthatareencounteredwhenweattempttoconstructapproximatesolutions,validasdirectlyfromgivendifferentialequations;anumberofotherexamplesofequationswithexactsolutionscanbefoundinQ1.3.Wemustnowturntothediscussionoftheideasthatwillallowasystematicstudyofsuchproblems.Inparticular,wefirstlookatthenotationthatwillhelpustobepreciseabouttheexpansionsthatwewritedown.1.2NOTATIONWeneedanotationwhichwillaccuratelydescribethebehaviourofafunctioninalimit.Toaccomplishthis,considerafunctionf(x)andalimithereamaybeanyfinitevalue(andapproachedeitherfromtheleftortheright)orinfinite.Further,itisconvenienttocomparef(x)againstanother,simpler,function,g(x);wecallg(x)agaugefunction.Thethreedefinitions,andassociatednotation,thatweintroduceare 11basedontheresultoffindingthelimitWeconsiderthreecasesinturn.(a)Little-ohWewriteifthelimit,(1.31),isL=0;wesaythat‘fislittle-ohofgasClearly,thispropertyofafunctiondoesnotprovideveryusefulinformation;essentiallyallitsaysisthatf(x)issmallerthang(x)(asSo,forexample,wehavebutalsoandItisanelementaryexercisetoshowthateachsatisfythedefinitionL=0from(1.31),byusingfamiliarideasthataretypicallyinvokedinstandard‘limit’problems.Forexample,thelastexampleaboveinvolvesconfirmingthatthelimitiszero.(Notethat,intheaboveexamples,thegaugefunc-tionwhichisanon-zeroconstantisconventionallytakentobeg(x)=1;notealsothatthelimitunderconsiderationshouldalwaysbequoted,oratleastunderstood.)(b)Big-ohWewriteifthelimit,(1.31),isfiniteandnon-zero;thistimewesaythat‘fisbig-ohofgasorsimply‘fisordergasAsexamples,weofferbutalso 121.Mathematicalpreliminariesfinallybut(Little-ohandbig-oh–oandO—areusuallycalledtheLandausymbols.)(c)AsymptoticallyequaltoorbehaveslikeFinally,wewriteifthelimitL,in(1.31),ispreciselyL=1;thenwesaythat‘fisasymptoticallyequaltogasor‘fbehaveslikegasSomeexamplesareandthenwemayalsowriteFinally,itisnotunusualtouse‘=’inplaceof‘~’,butinconjunctionwithameasureoftheerror.So,with‘~’,‘O’and‘o’asdefinedabove,wewriteorbutsuchstatementsshouldberegardedasnomorethanequivalentstosomeofthestatementsgivenearlier.Someexercisesthatuseo,Oand~aregiveninQ1.6,1.7and1.8.WeshouldcommentthatotherdefinitionsexistforO,forexample,althoughwhatwehavepresentedis,webelieve,themoststraightforwardandmostdirectlyuseful.Analternative,inparticular,istodefinef(x)=O[g(x)]asifpositiveconstantsCandRs.t.ourlimitdefinitionfollowsdirectlyfromthis. 131.3ASYMPTOTICSEQUENCESANDASYMPTOTICEXPANSIONSFirstwerecallexample(1.32),whichepitomisestheideathatwewillnowgeneralise.Wealreadyhaveandthisprocedurecanbecontinued,so(andthecorrectnessofthisfollowsdirectlyfromtheMaclaurinexpansionofsin(3x)).Theresultin(1.33),anditscontinuation,producesprogressivelybetterapproximationstosin(3x),inthatwemaywriteandthenAteachstage,weperforma‘variesas’calculation(asin(1.33),viathedefinitionof‘~’);inthisexamplewehaveusedthesetofgaugefunctionsforn=0,1,2,....;suchasetiscalledanasymptoticsequence.Inordertoproceed,weneedtodefineageneralsetoffunctionswhichconstituteanasymptoticsequence.Definition(asymptoticsequence)Thesetoffunctionsisanasymptoticsequenceasifforeveryn.Asexamples,wehave(Ineachcase,itissimplyamatterofconfirmingthatSomefurtherexamplesaregiveninQ1.9. 141.MathematicalpreliminariesNow,withrespecttoanasymptoticsequence(thatis,usingthechosensequence),wemaywritedownasetofterms,suchas(1.34);thisiscalledanasymptoticexpansion.Wenowgiveaformaldefinitionofanasymptoticexpansion(whichisusuallycreditedtoHenriPoincaré(1854–1912)).Definition(asymptoticexpansion)Theseriesoftermswrittenaswheretheareconstants,isanasymptoticexpansionoff(x),withrespecttotheasymptoticsequenceif,foreveryIfthisexpansionexists,itisuniqueinthatthecoefficients,arecompletelydetermined.Therearesomecommentsthatweshouldaddinordertomakeclearwhatthisdefi-nitionsaysandimplies—andwhatitdoesnot.First,givenonlyafunctionandalimitofinterest(i.e.f(x)andtheasymp-toticexpansionisnotunique;itisunique(ifitexists—weshallcommentonthisshortly)onlyiftheasymptoticsequenceisalsoprescribed.Toseethatthisisthecase,letusconsiderourfunctionsin(3x)again;wewilldemonstratethatthiscanberepre-sented,asinanynumberofdifferentways,bychoosingdifferentasymptoticsequences(although,presumably,wewouldwishtousethesequencewhichisthesimplest).So,forexample,indeed,thislastexample,isafamiliaridentityforsin(3x).(Anothersimpleexampleofthisnon-uniquenessisdiscussedinQ1.10.)So,givenafunctionandthelimit,weneedtoselectanappropriateasymptoticsequence—appropriatebecause,forsomechoices,theasymptoticexpansiondoesnotexist. 15Toseethis,letusconsiderthefunctionsin(3x)again,thelimitandtheasymptoticsequenceThefirstterminsuchanexpansion,ifitexists,willbeaconstant(correspondington=0);butinthislimit,sotheconstantiszero.Perhapsthefirsttermisproportionaltoforsomen>0;thusweexamineIfwearetohave(forsomenandsomeconstantc),thenthislimitistobeL=1.However,thislimitdoesnotexist—itisinfinite—foreveryn>0.Henceweareunabletorepresentsin(3x),aswiththeasymptoticsequenceproposed(whichmanyreaderswillfindself-evident,essentiallybecausesin(3x)~3xasIfeveryintheasymptoticexpansioniseitherzeroorisundefined,thentheexpansiondoesnotexist.Letustakethisonestepfurther;ifwehaveafunction,alimitandanappropriateasymptoticsequence,thenthecoefficients,areunique.Thisisreadilydemonstrated.Fromthedefinitionofanasymptoticexpansion,wehaveconsiderandtakethelimittogivewhichdetermineseachFinally,thetermsshouldnotberegardedortreatedasaseriesinanyconventionalway.Thisnotationissimplyashorthandforasequenceof‘variesas’calculations(asin(1.33),forexample);atnostageinourdiscussionhavewewrittenthatthesearethefamiliarobjectscalledseries—andcertainlynotconvergentseries.Indeed,manyasymptoticexpansions,iftreatedconventionallyi.e.selectavalueandcomputethetermsintheseries,turnouttobedivergent(although,exceptionally,someareconvergent).Ofcourse,numericalestimatesaresometimesrelevant,eithertogainaninsightintothenatureofthesolutionor,moreoften,toprovideastartingpointforaniterativesolutionoftheproblem.Becausetheseissuesmaybeofsomeinterest,wewill(in§1.4)deviatefromourmaindevelopmentandofferafewcommentsandobservations.Wemustemphasise,however,thatthethrust 161.Mathematicalpreliminariesofthistextistowardstheintroductionofmethodswhichaidthedescriptionofthestructureofasolution(inthelimitunderconsideration).Finally,beforewemoveon,webrieflycommentonfunctionsofacomplexvariable.(Wewillpresentnoproblemsthatsitinthecomplexplane,butitisquitenaturaltoaskifourdefinitionsofanasymptoticexpansionremainunaffectedinthissituation.)Givenandthelimitweareabletoconstructasymptoticexpansionsexactlyasdescribedabove,butwithoneimportantnewingredient.Becauseisapointinthecomplexplane,itispossibletoapproachi.e.takethelimit,fromanydirectionwhatsoever.(Forrealfunctions,thelimitcanonlybealongtherealline,eitherorHowever,ingeneral,theasymptoticcorrectnesswillholdonlyforcertaindirectionsandnotforeverydirectione.g.for(forsomeandforotherargstheasymptoticexpansion(withthesameasymptoticsequence,failsbecauseforsomen.1.4CONVERGENTSERIESVERSUSDIVERGENTSERIESSupposethatwehaveafunctionf(x)andaseriesthenisaconvergentseriesifasforallxsatisfying(forsomeR>0,theradiusofconvergence).Thisisastatementofthefamiliarpropertyofthetypeofseriesthatisusuallyencountered;sowehave,forexample,asthatandOneimportantconsequenceisthatwemayapproximateafunction,whichhasaconvergent-seriesrepresentation,toanydesiredaccuracy,byretainingasufficientnum-beroftermsintheseries.Forexamplewherethelimitasis2.Withtheseideasinmind,weturntothechallengeofworkingwithdivergentseries.Inthiscase,hasnolimitasforanyx(except,perhaps,attheonevaluex=a,whichaloneisnotuseful).Usuallydiverges—thesituationthatistypicalofasymptoticexpansions—butitmayremainfiniteandoscillate.Ineithercase,thissuggeststhatanyattempttouseadivergentseriesasthebasisfornumericalestimatesisdoomedtofailure;thisisnottrue.Adivergentseriescanbeusedtoestimatef(x) 17foragivenx,buttheerrorinthiscasecannotbemadeassmallaswewish.However,weareabletominimisetheerror,foragivenx,byretainingaprecisenumberoftermsintheseries–onetermmoreoronelesswillincreasetheerror.Thenumberoftermsretainedwilldependonthevalueofxatwhichf(x)istobeestimated.Thisimportantpropertycanbeseeninthecaseofa(divergent)serieswhichhasalternatingsigns—aquitecommonoccurrence—viaageneralargument.ConsidertheidentitywhereNisfinite;istheremainder.Supposethatandwith(and,correspondingly,areversalofallthesignsifthisdescribesthealternating-signpropertyoftheseries.LetuswritethenButtheremaindersareofoppositesign,sotheyalwaysadd(notcancel,approximately),whichwemayexpressassimilarlyHencethemagnitudeoftheremainder—theerrorinusingtheseries—islessthanthemagnitudeofthelasttermretainedandalsolessthanthatofthefirsttermomitted.Itisimportanttoobservethat,providedNremainsfinite,itisimmaterialtothisargumentwhethertheseriesisconvergentordivergent.Thus,foragivenx,westoptheseriesatthetermwiththesmallestvalueof(which,iftheseriesisconvergent,arisesatinfinityandiszero);thesumofthetermsselectedwillthenprovidethebestestimateforthefunctionvalue.Letusinvestigatehowthisideacanbeimplementedinaclassicalexample.E1.5TheexponentialintegralAproblemwhichexhibitsthebehaviourthatwehavejustdescribed,andforwhichthecalculationsareparticularlystraightforward,istheexponentialintegral:Weareinterested,here,inevaluatingEi(x)forlargex(andweobservethatasseeQ1.13);ofcourse,wecannotperformtheintegration,butwecan 181.Mathematicalpreliminariesgenerateasuitableapproximationviathefamiliartechniqueofintegrationbyparts.Inparticularweobtainandsoon,togiveNotethatwehaveusedastandardmathematicalprocedure,whichhasautomaticallygeneratedasequenceofterms—indeed,ithasgeneratedanasymptoticsequence,definedasThisisanotherimportantobservation:ourdefinitionshaveimpliedaselectionoftheasymptoticsequence,butinpracticeaparticularchoiceeitherappearsnaturally(ashere)oristhrustuponusbyvirtueofthestructureoftheproblem;wewillwritemoreofthislatterpointinduecourse.Here,fortheexpansionof(1.37)intheform(1.38),wemightregardasthenaturalasymptoticsequence.Itisclearthatwemaywrite,forexample,butwhatoftheconvergence,orotherwise,ofthisseries?Inordertoanswerthis,wewillusethestandardratiotest.Weconstruct(becausex>0andandifthisexpressionislessthanunityasforsomex,thentheseriesconverges(absolutely).Buttheexpressionin(1.39)tendstoinfinityasforallfinitex;hencetheseriesin(1.38)diverges.Toexaminethisseriesinmoredetail,letuswrite(1.38)intheformwheretheseriescanbeinterpretedasanasymptoticexpansionforistheremainder,givenby 19Itisconvenient,becauseitsimplifiesthedetails,ifweelecttoworkwithandthenwehaveandsoon.Thus,using(1.36a,b),weobtainsothat,ingeneral,Thebestestimate,foragivenx,isobtainedbychoosingthatnwhichminimisesthesmallerofthesetwobounds;inthisexample,thisisclearlyn=[x](where[]denotes‘theintegralpartof’).Infact,whenxisitselfaninteger,thesetwoboundsforareidentical.Asanumericalexample,weseekanestimateforEi(5)—andsinceourasymptoticexpansionisvalidasx=5appearstobearatherboldchoice.Theremainderthensatisfiesandi.e.0.1660(because(1.56)stillholds).Ofcourse,ifwepermitthelimitthenconditions(1.56)willbeviolated,althoughwemayallowptobeascloseaswedesiretozero.Thisobviouslypromptsthequestion:whatdoeshappentoourprocedure—theexpansionofexpansions—ifwedoselectp=0?Afterall—beingnaïve—itwouldseembutasmallstepfrompnearlyzero(whichispermitted)top=0(althoughweareallawarethattherecanbebigdifferencesbetweenandx=ainsomecontexts!).Infact,thissituationhereisnotunfamiliar;itisanalogoustothediscussionthatmustbeundertakenwhentheconvergenceofaseriesisinvestigated.Giventhataseriesisconvergentforanddivergentforitsstatusfor(i.e.thetwocasesx=a±R)mustbeinvestigatedviaindividualandspecialcalculations.Here,wewillemploythesamephilosophy,namely,toapplyourprocedureinthecasep=0,andnotetheresults;theymay,ormaynot,proveuseful.Intheevent,itwilltranspirethattheresultsarefundamentallyimportant,andleadtoaverysignificantpropertyofasymptoticexpansions.1.8THEMATCHINGPRINCIPLEAgain,wesupposethatwehavetwoasymptoticexpansions,onevalidforx=O(1)andoneforexactlyasdescribedintheprevioussection.Thistime,however,weexpandthefirstexpansionforandthesecondforx=O(1),i.e.theoverlapregionisthemaximumthatwecanenvisage(andonestepbeyondanythingpermittedsofar).Weknowthatthisprocedureisacceptableforthepairwith01andshowthat00withineachcase,findthefirsttwotermsineachoftheasymptoticexpan-sionsvalidforx=O(1)andforasShowthat,withtheinterpretationyourexpansionssatisfythematchingprinciple.(a)(thisexamplewasintroducedbyEckhaus);(b)(c)Q1.22CompositeexpansionsI.Forthesefunctions,giventhatfindthefirsttwotermsinasymptoticexpansionsvalidforx=O(1)andfor 461.MathematicalpreliminariesasHenceconstructadditivecompositeexpansions.(a)(b)Whatdoyouobserveaboutyourcompositeexpansionobtainedin(b)?Q1.23CompositeexpansionsII.SeeQ1.22;now,whentheyaredefined,findthecorre-spondingmultiplicativeexpansions.(Youmaywishtocompare,byplottingtheappropriatefunctions,theoriginalfunctionandthetwocompositeexpansions.)Q1.24Estimateoferror.SeeE1.16;findanestimateoftheerrorinusingthisformofthecompositeexpansion. 2.INTRODUCTORYAPPLICATIONSInthepreviouschapter,welaidthefoundationsofsingularperturbationtheoryand,althoughwewillneedtoaddsomespecifictechniquesforsolvingcertaintypesofdifferentialequations,wecanalreadytacklesimpleexamples.Inaddition,wewillseethatwecanapplytheseideasdirectlytoother,moreroutineproblems—andthisiswhereweshallbegin.Here,wewilldescribehowtoapproachtheproblemoffindingrootsofequations(whichcontainasmallparameter),andhowtoevaluateintegralsoffunctionswhicharerepresentedbyasymptoticexpansionswithrespecttoaparameter.Finally,webeginourstudyofdifferentialequationsbyexaminingafewimportant,fairlystraightforwardexampleswhichare,nonetheless,nottrivial.2.1ROOTSOFEQUATIONSAtsomestageinmanymathematicalproblems,itisnotunusualtobefacedwiththeneedtosolveanequationforspecificvaluesofanunknown.Suchaproblemmightbeassimpleassolvingaquadraticequation:orfindingthesolutionofmorecomplicatedequationssuchas 482.IntroductoryapplicationsInthissection,wewilldescribeatechnique(forequationswhichcontainasmallparameter,asinthoseabove)whichisanaturalextensionofsimplyobtaininganasymptoticexpansionofafunction,examiningitsbreakdown,rescaling,andsoon.WewillbeginbyexaminingthesimplequadraticequationandseekthesolutionsforTheessentialideaistoobtaindifferentasymptoticapproximationsforvalidfordifferentsizesofx,andseeiftheseadmit(ap-proximate)roots.Giventhatwecouldhaverootsanywhereontherealline,andsoallsizesofxmustbeexamined.(Wewillconsider,first,onlytherealrootsofequations;theextensiontocomplexrootswillbediscussedinduecourse.)Onefurthercommentisrequiredatthisstage:wedescribehereatechniqueforfind-ingrootsthatbuildsontheideasofsingularperturbationtheory.Inpractice,otherapproachesarelikelytobeusedinconjunctionwithourstosolveparticularequationse.g.sketchingorplottingthefunction,orusingastandardnumericalprocedure(suchasNewton-Raphson).Thereisnosuggestionthatthisexpansiontechniqueshouldbeusedinisolation—itissimplyoneofanumberoftoolsavailable.Returningto(2.1),ifx=O(1),thenandsowehavearootx=–1(approximately).Inordertogenerateabetterap-proximation,wemayuseanyappropriatemethod.Forexample,wecouldinvokethefamiliarprocedureofiteration,sowemaywritewithThenweobtainandsoon(butnotethatiterationmaynotgenerateacorrectasymptoticexpansionatagivenorderinItisclearfromthisapproachthatacompleterepresentationoftherootwillbeobtainedifweusetheasymptoticsequenceandsoanalternativeistoseekthisformdirectly—andthisismoreinkeepingwiththeideasofperturbationtheory.Thuswemightseekarootintheformsothat(2.1)canbewrittenas 49(wheretheuseof‘=0’hereistoimply‘equaltozerotoallordersinthusandsoon.WehaveonerootItisclearthat(2.2)admitsonlyoneroot;theother—itisaquadraticequationthatwearesolving—mustappearforadifferentsizeofx.The‘asymptoticexpansion’(wetreatthefunctionassuch)remainsvalidforandsothereisnonewrootforx=o(1);however,thisexpansiondoesbreakdownwhereWedefineandwriteorThisapproximationadmitstherootsX=0andX=–1,sonowthequadraticequa-tionhasatotalofthreeroots!Ofcourse,thiscannotbethecase;indeed,itisclearthattherootX=0isinadmissible,becausethe‘asymptoticexpansion’breaksdownwhere(whichisx=O(1)andsoreturnsusto(2.5)).TheonlyavailablerootisX=–1,andthisisthesecond(approximate)rootoftheequation(leavingX=0asnomorethana‘ghost’oftherootx~–1).TheexpansionforFdoesnotfurtherbreakdown(asandsotherearenootherroots—notthatweexpectedanymore!Wemayseekabetterapproximation,aswedidbefore,intheformwhichgivesi.e.thusorThetworootsofthequadraticequation,(2.1),aretherefore 502.Introductoryapplicationsitisleftasanexercisetoconfirmthattheseresultscanbeobtaineddirectlyfromthefamiliarsolutionofthequadraticequation,suitablyapproximated(byusingthebinomialexpansion)for(SimilarproblemsbasedonquadraticequationscanbefoundinexerciseQ2.1.)Thissimpleintroductoryexamplecoverstheessentialsofthetechnique:findallthedifferentasymptoticformsofandinvestigateifrootsexistforeach(dominant)asymptoticrepresentation.Letusnowapplythistoaslightlymoredifficultequationwhich,nevertheless,hasasimilarstructure.E2.1AcubicequationWearetofindapproximationstoalltherealrootsofthecubicequationforFirst,forx=O(1),wehaveandthisapproximationadmitstherootsx=±1;abetterapproximationisthenob-tainedbywritingsothatweobtainThisequationrequiresthatandsoon;tworootsarethereforeNowthe‘asymptoticexpansion’remainsvalidasbutnotasitbreaksdownwhereorWewriteandthenor 51TheonlyrelevantrootisX=1(becausetheothertwoaretheghostsoftherootsthatappearforx=O(1)).Toimprovethisapproximation,wesettogivei.e.sothatathirdrootisasThusthethree(real)rootsareOurintroductoryexample,andtheoneabove,havebeenratherconventionalpoly-nomialequations,butthetechniqueisparticularlypowerfulwhenwehavetosolve,forexample,transcendentalequations(whichcontainasmallparameter).Wewillnowseehowtheapproachworksinaproblemofthistype.E2.2AtranscendentalequationWerequiretheapproximate(real)roots,asoftheequationForx=O(1),wenowhavetwopossibilities:butonlythefirstoptionadmitsanyrootsforreal,finitex.Thusx=±1(approxi-mately)andthenonlythechoicex=+1isacceptable(becausewerequirex>0);abetterapproximationfollowsdirectly:The‘expansion’iswrittenwherethetermmustbeexponentiallysmallforx=O(1)orlarger(becausenorootsexistifthistermdominates).Now,forx>0,thereisnobreakdownas 522.IntroductoryapplicationsthusanyotherrootsthatmightexistmustariseasIndeed,asweseethattheexpansion(2.8)breaksdownwhereandsowesettogiveThisapproximationhasonerootatX=0,butthiscannotbeuseddirectlybecausetheexpansion,(2.9),itselfbreaksdownasThisoccurswherei.e.soafurtherscalingmustbeintroduced:toproduceSincewehavewehavearootneartoobtainanimprovedapproximation,wewriteandsoobtainAsthereisnofurtherbreakdown,andsowehavefoundtworealrootsAnumberofotherequations,bothpolynomialandtranscendental,arediscussedintheexercisesQ2.2,2.3and2.4.However,alltheseinvolvethesearchforrealroots;wenowturn,therefore,toabriefdiscussionofthecorrespondingproblemoffindingallroots,whetherrealorcomplex.Itwillsoonbecomeclearthatwemayoftenadoptpreciselythesameapproachwhenanyrootsarebeingsought(although,sometimes,theremaybeanadvantageinwritingandworkingwithtwo,coupled,realequations).Theonlysmallwordofwarningisthatthesizeoftherealandimaginaryparts,measuredintermsofmaybedifferente.g.x=O(1)nowimpliesthatwhich 53canbesatisfiedifeither,butnotnecessarilyboth,andareO(1).Letusseehowthisarisesinanexample.E2.3AnequationwithcomplexrootsHere,usingthemoreusualnotationforacomplexnumber,weconsiderandimmediatelyweobtainandsowehaverootsapproximately.ThuswewriteandsotheequationbecomesThisissatisfiedifetc.,andthuswehavetwocomplexrootsandweobservethattheimaginarypartisO(1),butthattherealpartisThefull‘expansion’isclearlynotuniformlyvalidasthereisabreakdownwhereorWeintroduceandwriteandsowhichproducesthesingle,availablerootnearZ=–1andthen,moreaccurately,wehaveTheequationhasthreeroots,twoofwhicharecomplex:Finally,aclassofequationsforwhichthisdirectapproach(forcomplexroots)isnotuse-fulischaracterisedbytheappearanceoftermssuchas(oranythingequivalent). 542.IntroductoryapplicationsInthiscase,itisalmostalwaysconvenienttoformulatetheprobleminrealandimag-inaryparts,andtheappearanceofasmallparameterdoesnotaffectthisapproachinanysignificantway;wepresentanexampleofthistype.E2.4Areal-imaginaryproblemWeseekalltherootsoftheequationasnotethatandsotheformofthisproblemdoesindeedexhibitthismorecomplicatedstructure.Letuswritez=x+iy,andthen(2.9)becomesorWeseeimmediatelythattheright-handsidesofthesetwoequationsdonotpresageabreakdownofthesecontributions,asxoryincreasesordecreases;thusweproceedwithx=O(1)andy=O(1).Nowequation(2.10b)possessesthesolutionsandthisistherelevantchoice(ratherthany=0)becausewerequiresinxcoshy>1(from(2.10a)).Thenequation(2.10a)givesandthisisconsistentonlyifn=2m(m=0,±1,±2,...)becausecoshy>0.Finally,thesolutionarisesonlyforsincecoshasweseethatasolutionexistswhereandsoweintroduceThusweobtain 55andsowehavethesetof(approximate)rootsAfewexamplesofotherequationswithcomplexroots(someofwhichmaybereal,ofcourse)aresetasexercisesinQ2.5.2.2INTEGRATIONOFFUNCTIONSREPRESENTEDBYASYMPTOTICEXPANSIONSOurseconddirect,andratherroutineapplicationoftheseideasistotheevaluationofintegrals.Inparticular,weconsiderintegralsoffunctionsthatarerepresentedbyasymptoticexpansionsinasmallparameter;thismayinvolveoneormoreexpansions,butifitisthelatter—anditoftenis—thentheexpansionswillsatisfythematchingprinciple.Theprocedurethatweadoptcallsupontwogeneralproperties:thefirstistheexistenceofanintermediatevariable(validintheoverlapregion;see§1.7),andthesecondisthefamiliardeviceofsplittingtherangeofintegration,asappropriate.Wethenexpresstheintegralasasumofintegralsovereachoftheasymptoticexpansionsoftheintegrand,theswitchfromonetothenextbeingatapointwhichisintheoverlapregion.Theexpansionsarethenvalidforeachintegrationrangeselectedand,furthermore,thevalueoftheoriginalintegral(assumingthatitexists)isindependentofhowwesplittheintegral.Thustheparticularchoiceofintermediatevariableisunimportant;indeed,itmaybequitegeneral,satisfyingonlythenecessaryconditionsforsuchavariable;see(1.56),forexample.Letusapplythistechniquetoasimpleexample.E2.5AnelementaryintegralWearegivenandwerequirethevalue,asoftheintegral(Notethattheintegralhereiselementary,totheextentthatitmaybeevaluateddirectly,althoughwewillintegrateonlytherelevantasymptoticexpansions;thisexamplehasbeenselectedsothattheinterestedreadermaychecktheresultsagainsttheexpansionoftheexactvalue.) 562.IntroductoryapplicationsFirst,weexpandforx=O(1)andfortogiveandbothforwehaveretainedtermsasfarasineachexpansion.(Youshouldconfirmthatthesetwoexpansionssatisfythematchingprinciple.)Nowthesetwoexpansionsarevalidintheoverlapregion,representedbydefinedbythusweexpresstheintegralasTheonlyrequirement,atthisstage,isthatweareabletoperformtheintegrationofthevariousfunctionsthatappearintheasymptoticexpansions.NotethatthefirstintegralhasbeenexpressedasanintegrationinX—themostnaturalchoiceofintegrationvariableinthiscontext.Toproceed,weobtainandthisistobeexpandedforand(note!).Thusweobtain 57wheretheellipsis(···)indicatesfurthertermsinthevariousbinomialexpansions;wekeepasmanyasrequiredinordertodemonstratethatvanishesidentically(atthisorder),toleaveThuswehavefoundthatasasfarastermsathereweseethattheintegrationoverx=O(l)providesthedominantcontributiontothisvalue.Thisexamplehaspresented,viaafairlyroutinecalculation,theessentialideathatunderpinsthismethodforevaluatingintegrals.Ofcourse,thereisnoneedtoexploitthistechniqueiftheintegralcanbeevaluateddirectly(aswasthecasehere);letusthereforeexamineanotherproblemwhichislesselementary.E2.6AnotherintegralWewishtoevaluatetheintegralashere,theexpansionoftheintegrandrequiresthreedifferentasymptoticexpansions(validforx=O(l),Thusweobtain 582.IntroductoryapplicationsInthisproblem,werequiretwointermediatevariables;thesearedefinedbyallasTheintegralisthenwrittenasandwewillnowretaintermsthatwillenableustofindanexpressionforcorrectatThuswefindthat 59(Youshouldconfirmthat,intheabove,bothandcancelidentically,tothisorder.)FurtherexamplesthatmakeuseoftheseideascanbefoundinexercisesQ2.6,2.7and2.8.Withalittleexperience,itshouldnotbetoodifficulttorecognisehowmanytermsneedtoberetainedineachexpansioninordertoproducetoadesiredaccuracy.Theregionthatgivesthedominantcontributionisusuallyself-evident,andquiteoftenthisalonewillprovideanacceptableapproximationtothevalueoftheintegral.Furthermore,termsthatcontaintheoverlapvariablescanbeignoredaltogether,becausetheymustcancel(althoughthereisacasefortheirretention—whichwasourapproachabove—asacheckonthecorrectnessofthedetails).2.3ORDINARYDIFFERENTIALEQUATIONS:REGULARPROBLEMSWenowturntoaninitialdiscussionofhowthetechniquesofsingularperturbationtheorycanbeappliedtotheproblemoffindingsolutionsofdifferentialequations—unquestionablythemostsignificantandfar-reachingapplicationthatweencounter.Therelevantideaswillbedeveloped,first,forproblemsthatturnouttoberegular(butwewillindicatehowsingularversionsoftheseproblemsmightarise,andwewilldiscusssomesimpleexamplesoftheselaterinthischapter).Clearly,weneedtolaydownthebasicprocedurethatmustbefollowedwhenweseeksolutionsofdifferentialequations.However,thesetechniquesaremanyandvaried,andsowecannothopetopresent,atthisstage,anall-encompassingrecipe.Nevertheless,thefundamentalprinciplescanbedevelopedquitereadily;toaidusinthis,weconsiderthedifferentialequationforThisproblem,weobserve,isnottrivial;itisanequationwhich,althoughfirstorder,isnonlinearandwithaforcingtermontheright-handside.ThefirststageistodecideonasuitableasymptoticsequencefortherepresentationofHere,wenotethattheprocessofiterationontheequation,whichcanbe 602.Introductoryapplicationswrittenforthispurpose(withaprimeforthederivative)asgivesandsoon,sothattakestheform(forappropriatefunctionsWhenthissolutionisusedtogenerateitisclearthatwewillproducetermsinandandsothispatternwillcontinue:theequationimpliesthe‘natural’asymptoticsequencesothisiswhatwewillassumetoinitiatethesolutionmethod.(Itshouldbenotedthattheboundaryconditionisconsistentwiththisassumption,asisthealternativeconditionOntheotherhand,aboundaryvaluewouldforcetheasymptoticsequencetobeadjustedtoaccommodatethisi.e.Thusweseekasolutionoftheproblem(2.11)intheformforsome(andwedonotknowwhichxswillbeallowed,atthisstage).Theexpansion(2.12)isusedinthedifferentialequationtogivewhere‘=0’meanszerotoallordersinthuswerequireandsoon.Similarly,theboundaryconditiongivessothatofcourse,toevaluateonx=1impliesthattheasymptoticexpansion,(2.12),isvalidhere—butwedonotknowthisyet.Thisiswrittendownbecause,iftheproblemturnsouttobewell-behavedi.e.regular,thenwewillhavethisreadyforuse;essentially,allwearedoingisnoting(2.14)—wecanrejectitiftheexpansionwillnotpermitevaluationonx=1.Thenextstepissimplytosolveeachequation(forinturn;weseedirectly(from(2.13a))thatthegeneralsolutionforis 61whereisanarbitraryconstant.Then,from(2.13b),wehavewhichcanbewritten(onusingtheintegratingfactorasThisproducesthegeneralsolutionwhereisasecondarbitraryconstant.Itisimmediatelyclearthatthesefirsttwotermsintheexpansionaredefined(andwell-behavedi.e.nohintofanon-uniformity)forsowemayimposetheboundaryconditions,(2.14a,b);theseproduceThusourasymptoticexpansion,sofar,isandthisiscertainlyuniformlyvalidforwehavea2-termexpansionofthesolution.(Notethatthespecificationofthedomainiscriticalhere;if,forexample,wewereseekingthesolutionwiththesameboundarycondition,butinthen(2.17)wouldnotbeuniformlyvalid:thereisabreakdownwherei.e.seetheproblemin(2.34),below.)Theevidencein(2.17)suggeststhatwehavethebeginningofauniformlyvalidasymptoticexpansioni.e.(2.12)isvalidforandfor(anditisleftasanexercisetofindandtocheckthattheinclusionofthistermdoesnotalterthisproposition).Inordertoinvestigatetheuniformvalidity,orotherwise,of(2.12),oneapproachistoexaminethegeneraltermintheexpansion;thisisthesolutionofwherewithThesolutionto(2.18)isbutandareboundedfunctionsforandhencesoisandthensoisandhencealltheInparticular,asandconstants)asthereisnobreakdownoftheasymptotic 622.Introductoryapplicationsexpansion.Theproblemposedin(2.11)isthereforeregular,resultinginauniformlyvalidasymptoticexpansion.Morecomplete,formalandrigorousdiscussionsofuniformvalidity,inthecontextofdifferentialequations,canbefoundinothertexts,suchasSmith(1985),O’Malley(1991)andEckhaus(1979).Typically,theseargumentsinvolvewritingwhereandthenshowingthatremainsboundedforandforWewilloutlinehowthiscanbeappliedtoourproblem,(2.11);first,weobtainwithforSinceeachsatisfiesanappropriatedifferentialequationandboundarycondition,thisgiveswherecomprisestheterms,andsmaller,fromtheexpansionof(afterdivisionbyAuniformasymptoticexpansionrequiresthatisboundedasforandToprovesucharesultisrarelyanelementaryexerciseingeneral,anditisnottrivialhere,althoughanumberofapproachesarepossible.OnemethodisbasedonPicard’siterativescheme(whichisastandardtechniqueforprovingtheexistenceofsolutionsoffirstorderordinarydifferentialequationsinsomeappropriateregionof(x,y)-space);thiswillbedescribedinanygoodbasictextonordinarydifferentialequations(e.g.Boyce&DiPrima,2001).Anotherpossibility,closelyrelatedtoPicardsmethod,isformallytointegratetheequationfortherebyobtaininganintegralequation,andthentoderiveestimatesfortheintegralterm(andhenceforWewilloutlineathirdtechnique,whichinvolvestheconstructionofestimatesdirectlyforthedifferentialequation,andthenintegratingareducedversionoftheequationforAtthisstagewedonotknowifisofonesign,fororifitchangessignonthisinterval;however,wemayproceedwithoutspecifyingorassumingthenatureofthisproperty,butitwillaffectthedetails;firstwewriteButwedoknowthateachisbounded(forandhencesoiswhichwewillexpressintheform(aconstantindependentofandso 63Thissamepropertyofthefunctionsleadstoacorrespondingstatementfor(again,independentofwhichnowgiveswheretheuppersignappliesifandthelowerif(Ofteninargumentsofthistype,wecannotincorporatebothsigns,andwearereducedtoworkingwiththemodulusofthefunction;wewillseeherethatwecanallowtheformgivenin(2.20).)Betweenthetwoinequalities,wehaveanexpressionassociatedwithaconstantcoefficientRiccatiequation;letusthereforeconsiderwhereandforarbitrary(bounded)functionsandIfanappropriateuniquesolutionof(2.21),satisfyingexistsforallandallasspecified,thenwewillcertainlyhavesatisfied(2.20).However,wewill,inthistext,giveonlytheflavourofhowthedevelopmentproceeds,byconsideringarestrictedversionoftheproblemwiththespecialchoice:andconstant(butsatisfyingthegivenbounds).Tosolve(2.21),weintroducetoobtainwhichhas,inourspecialcase,thegeneralsolutionwherethearbitraryconstantsareAandB,andtheauxiliaryequationfortheexponentsisTherootsofthisequationhavebeenwrittenaswherefori=1,2,asFinally,thesolutionwhichsatisfiestheconditiononx=1iswhichisboundedforasThustheerrorisforasrequired.(Acomprehensivediscussion 642.Introductoryapplicationsrequirestheanalysisoftheequationforwithgeneral,boundedandwhichispossible,butbeyondtheaimsofthistext.)Asshouldbeclearfromthiscalculation,itistobeanticipatedthatspecialprop-erties,relevanttoaparticularproblem,mayhavetobeinvoked.Here,forexample,wetookadvantageoftheunderlyingRiccatiequation;otherproblemsmayrequirequitedifferentapproaches.However,wemustalsoemphasisethat,formanypracticalandimportantproblemsencounteredinappliedmathematics,thesecalculationsareoftentoodifficulttosuccumbtosuchageneralanalysis.Indeed,theconventionalwisdomisthat,ifbreakdownshavebeenidentified,rescalingemployedandasymp-toticsolutionsfound(andmatched,asrequired),thenwehaveproducedasufficientlyrobustdescription.Itshouldbenotedthattheprocessofrescalingmightinvolveaconsiderationofallpossiblescalingsallowedbythegoverningequation,whichwillthengreatlystrengthenourtrustintheresultsobtained.Thosereaderswhopreferthemorerigorousapproachthatsuchdiscussionsaffordareencouragedtostudythetextspreviouslymentioned.Inthistext,however,weshallproceedwithoutmuchfurtherconsiderationofthesemoreformalaspectsoftheasymptoticsolutionofdifferentialequations.Nowthatwehavepresentedthesalientfeaturesofthemethodofconstructingsolutions,weapplyittoanotherexample.E2.7Aregularsecond-orderproblemWeseekanasymptoticsolution,asofwithandtheprimesheredenotederivatives.First,weassumethatthereisasolution,forsomeoftheformThusweobtainandsoon,with(iftheexpansionisvalidattheend-points).Thegeneralsolutionof(2.23a)is 65andthen(2.23b)becomeswhich,inturn,hasthegeneralsolutionwherearethenewarbitraryconstants.Thefunctionsandareclearlydefinedforandthereisnosuggestionofabreakdown,soweimposetheboundaryconditions(2.24)togiveandthenourasymptoticexpansion(tothisorder)isNowthatwehaveobtainedtheexpansion,(2.25),weareabletoconfirmthatwehavea2-termuniformlyvalidrepresentationofthesolution.Inordertoexaminethegeneralterminthisasymptoticexpansion,ifthisisdeemednecessary,wecanfollowthemethoddescribedearlier.Thuswemaywritewhere,inparticular,wehaveandforthegeneralsolutionforiswhereandaredeterminedtosatisfythetwoboundaryconditions.Theessentialsoftheargumentarethenaswehavealreadyoutlinedinourfirst,simple,presentation:isbounded(on[0,1]),soisandhencesoisetc.,forallFurther,asandasfortheasymptoticexpansionisuniformlyvalid.SomefurtherexamplesofregularexpansionscanbefoundinQ2.9and2.10,andaninterestingvariantofE2.7isdiscussedinQ2.15. 662.Introductoryapplications2.4ORDINARYDIFFERENTIALEQUATIONS:SIMPLESINGULARPROBLEMSNowthatwehaveintroducedthesimplestideasthatenablesolutionsofdifferentialequationstobeconstructed,wemustextendourhorizons.Thefirstpointtorecordisthat,onlyquiterarely,doweencounterproblemsthatcanberepresentedbyuniformlyvalidexpansions(although,somewhataftertheevent,wecanoftenconstructsuchexpansions—intheformofacompositeexpansion,forexample;see§1.10).Themorecommonequationsexhibitsingularbehaviour,inoneformoranother;thesimplestsituation,wesuggest,iswhenthetechniquesusedabove(§2.3)produceasymptoticexpansionsthatbreakdown,resultingintheneedtorescale,expandagainand(probably)invokethematchingprinciple.(Othertypesofsingularitycanarise,andthesewillbedescribedinduecourse.)Toseehowthisapproachisanaturalextensionofwhatwehavedonethusfar,wewillpresentaproblembasedontheequationgivenin(2.11).Weconsiderfortheimportantnewingredienthereisthevariablecoefficient(which,wenote,isforx=O(1)).Weseekasolutionintheformandwewillneedtofindthetermsand(atleast)inordertoincludeacontributionfromthenewpartofthecoefficient.Theequationsfortheareandsoon;theboundaryconditionrequiresthatInthisproblem,weshouldexpectthatevaluationoftheexpansiononx=1isallowed—alltermsaredefinedforx=O(1)—butwemustanticipatedifficultiesasThesolutionsforthefunctionsandfollowfromtheresultsgivenin(2.15)and(2.16),respectively,butwiththeparticularintegralomitted;thus 67The2-termasymptoticexpansion,isuniformlyvalidforLetusfindthenexttermintheexpansion;thisisthesolutionofThus,introducingtheintegratingfactorwehaveandsowherethearbitraryconstantmustbeA=0(tosatisfyThisthirdtermintheasymptoticexpansionisverydifferentfromthefirsttwo:itisnotdefinedonx=0,sowemustexpectabreakdown.Theexpansion,tothisorder,isnowasforx=O(1);asweclearlyhaveabreakdownwherethesecondandthirdtermsintheexpansionbecomethesamesizei.e.orNotethatthisbreakdownoccursforalargersizeofx(asxisdecreasedfromO(1))thanthebreakdownassociatedwiththefirstandthirdterms,sowemustconsiderTheproblemforisformulatedbywritingwheretherelabellingofyisanobviousconvenience(andwenotethaty=O(1)forTheoriginalequation,in(2.26),expressedintermsofXandY,requirestheidentityandthenweobtain 682.Introductoryapplicationsbuttheboundaryconditionisnotavailable,becausethisisspecifiedwherex=O(1).Equation(2.28)suggeststhatweseekasolutionintheformwhichgivesImmediatelyweobtain(anarbitraryconstant),andthenequation(2.29b)becomeswhichintegratestogivewhereisasecondarbitraryconstant.Theresulting2-termexpansionisthereforethetwoarbitraryconstantsaredeterminedbyinvokingthematchingprinciple:(2.30)and(2.27)aretomatch.Thuswewritethetermsin(2.27)asfunctionsofX,let(forX=O(1))andretaintermsO(1)and(whichareusedin(2.30));conversely,write(2.30)asafunctionofx,expandandretaintermsO(1),andFrom(2.27)weconstructandfrom(2.30)wewrite(Thisexpansionrequiresthestandardresult: 69whichtheinterestedreadermaywishtoderive.)Thetwo‘expansionsofexpan-sions’,(2.31)and(2.32),matchwhenwechooseandtheasymptoticexpansionforX=O(1)isthereforeWenowobservethat,althoughtheexpansion(2.27)isnotdefinedonx=0,theexpansionvalidfordoesallowevaluationonx=0i.e.X=0;indeed,from(2.33),weseethatInsummary,theprocedureinvolvestheconstructionofanasymptoticexpansionvalidforx=O(1)andapplyingtheboundarycondition(s)iftheexpansionremainsvalidhere.Theexpansionisthenexaminedforseekinganybreakdowns,rescalingandhencerewritingtheequationintermsofthenew,scaledvariable;thisproblemisthensolvedasanotherasymptoticexpansion,matchingasnecessary.Acoupleofgeneralobservationsarepromptedbythisexample.First,thematchingprinciplehasbeenusedtodeterminethearbitraryconstantsofintegrationbecausetheboundaryconditiondoesnotsitwherethustheprocessofmatchingisequivalent,here,toimposingboundaryconditions(andtherebyobtaininguniquesolutionsforInthecontextofdifferentialequations,thisistheusualroleofthematchingprinciple,anditisfundamentalinseekingcompletesolutions.Thesecondissueisrathermoregeneral.Inthisexample,theexpansionforx=O(1),(2.27),hadtobetakentothetermatbeforethenon-uniformity(asbecameevident.Thispromptstheobviousquestion:howmanytermsshouldbedeterminedsothatwecanbe(reasonably)surethatallpossiblecontributionstoabreakdownhavebeenidentified?Averygoodruleofthumbistoensurethattheasymptoticexpansioncontainsinformationgeneratedbyeveryterminthedifferentialequation.Thusourrecentexample,(2.26),requirestermstoincludethenonlinearityandtermsforthedominantrepresentationofthevaryingpartofthevariablecoefficient.Inaphysicallybasedproblem,theinterpretationofthisruleissimplytoensurethateverydifferentphysicaleffectisincludedatsomestageintheexpansion.Asanexampleofthisidea,considerthenonlinear,dampedoscillatorwithvariablefrequencydescribedbytheequation: 702.IntroductoryapplicationsAtO(1)wehavethebasicoscillator;atthevariablefrequency;atthenon-linearity;atthedamping.Thus,inordertoinvestigatetheleadingcontributions(atleast)toeachofthesepropertiesoftheoscillation,theasymptoticexpansionmustbetakenasfarastheinclusionofterms(Thereisnosuggestionthateachwillnecessarilyleadtoabreakdown,andanassociatedscaling,buteachneedstobeex-amined.)Onefurtherimportantobservationwillbediscussedinthenextsection;weconcludethissectionwithtwoexamplesthatexploitalltheseideas.E2.8Problem(2.11)extendedWeconsidertheproblemwhichisthesameequationandboundaryconditionasweintroducedin(2.11),butnowthedomainisTheasymptoticsolutionforx=O(1),whichsatisfiestheboundarycondition,is(2.17)i.e.andthisbreaksdownwhereorThusweintroducebutforthissizeofx,weobservethatandsothisalsomustbescaled:wewriteEquation(2.34)becomesandweseekasolutionwhichgivesandsoon.Thisresultmaycausesomesurprise:thissequenceofproblemsispurelyalgebraic—thereisnointegrationofdifferentialequationsrequiredatanystage.Equation(2.36a)hasthesolution 71andwewillneedtoinvokethematchingprincipletodecidewhichsignisappropriate.Thusfromthefirsttermin(2.35),weseethatfrom(2.37)weobtainandthen(2.38)and(2.39)matchonlyforthepositivesign.(Notethat,becauseyhasalsobeenscaled,thismustbeincludedintheconstructionwhichenablesthematchingtobecompleted.)Thesolutionforthefirsttermisthereforeandthenthesecondtermisobtaineddirectlyastheresulting2-termasymptoticexpansionisWehavefoundthatthisproblem,(2.34),requiresanasymptoticexpansionforx=O(1)andanotherforInaddition,itisclearthat(2.40)doesnotfurtherbreakdownas(anditisfairlyeasytoseethatnolatertermsintheexpansionwillalterthisobservation):twoasymptoticexpansionsaresufficient.Theappearanceofanalgebraicproblemimpliesthatallsolutionsarethesame—anyvariationbyvirtueofdifferentboundaryvaluesislostforhowisthispossible?Theexplanationbecomesclearwhen(2.35)isexaminedmoreclosely;thetermsassociatedwiththearbitraryconstants(ateachorder)areexponentialfunctions,andfortheseareallproportionaltotheyareexponentiallysmall.Suchtermshavebeenomittedfromtheasymptoticexpansionforiftheyhadbeenincluded,thenthematchingofthesetermswouldhaveensuredthatinformationabouttheboundaryvalueswouldhavebeentransmittedtothesolutionvalidforalbeitinexponentiallysmallterms. 722.IntroductoryapplicationsE2.9Anequationwithaninterestingbehaviournearx=0Weconsidertheproblemasweassumethatforsomex=O(1).Weobtainthesequenceofequationsandsoon;theboundarycondition(ifavailablehere)givesThegeneralsolutionof(2.42a)issimply(whichisnotdefinedonx=0,soweanticipatetheneedforascalingasifaboundedsolutionexists),andthen(2.42b)givesorTheasymptoticexpansionisthereforeandthisisdefinedforx=O(1),includingx=1,butnotasSotheboundarycondition,(2.43),canbeapplied,requiringthearbitraryconstantstobeandi.e. 73Asexpansion(2.44)breaksdownwhereorandthenweintroducethescaledvariablesandthentheequationin(2.41)becomesForthisequation,itisclearthatwemustseekasolutionintheformandthen(2.45)yieldsandsoon.Thefirstequationherecanbewrittenaswhereisanarbitraryconstant;thusandbothandthechoiceofsignaretobedeterminedbymatching.Fromthefirsttermin(2.44)weobtainandfrom(2.47)wehavewhichmatcheswith(2.48)onlyforthepositivesignandthenwithThus(2.47)becomes 742.Introductoryapplicationsandthenequation(2.46b)isorThegeneralexpressionforisthereforeandsowehavetheexpansionandthisistobematchedwith(2.44).From(2.44)weobtainand,correspondinglyfrom(2.49),wehaveorwhichmatchesonlyif(becausetheterminmustbeeliminated).ThesolutionvalidforisthereforeandthisexpansionisdefinedonX=0,yieldingWeobserve,inthisexample,thatthevalueofthefunctiononx=0iswell-definedfrom(2.51),butthatitdivergesasThisdemonstratestheimportantpropertythatwerequire,forasolutiontoexist,thattheasymptoticrepresentationbedefined 75forbutthatsolutionsobtainedfromtheseexpansionsmaydivergeaswemayhavex=0inthedomain,butTheexamplesthatwehavepresentedthusfar(andotherscanbefoundinQ2.11-2.15),andparticularlythosethatinvolvearescalingafterabreakdown,possessanimportantbutratherlessobviousproperty.Thisrelatestotheexistenceofgeneralscalingsofthedifferentialequation,andtheresulting‘balance’of(dominant)termsintheequation;thisleadsustotheintroductionofanadditionalfundamentaltool.Thisideawillnowbeexploredinsomedetail,andusemadeofitinsomefurtherexamples.2.5SCALINGOFDIFFERENTIALEQUATIONSLetusfirstreturntoourmostrecentexamplegivenin(2.41).Wemay,ifitisconvenientorexpedient,choosetousenewvariablesdefinedbywhereandarearbitrarypositiveconstants;XandYarenowscaledversionsofxandy,respectively.Thus,withequation(2.52)becomesandthenachoiceforandmightbedrivenbytherequirementtofindanewasymptoticexpansionvalidinanappropriateregionofthedomain.Inthisexample,thefirsttermoftheasymptoticexpansionvalidforx=O(1)isy~1/x(see(2.44))andsoanyscalingthatistoproduceasolutionwhichmatchestothismustsatisfyY~1/Xi.e.Withthischoice,equation(2.53)becomesandfromourpreviousanalysisofthisproblem,weknowthatthebreakdownoftheasymptoticexpansionvalidforx=O(1)occursfori.e.theissuehereiswhetherthiscanbededuceddirectlyfromthe(scaled)equation.Thecluetothewayforwardcanbefoundwhenweexaminetheterms,inthedifferentialequation,thatproducetheleadingtermsinthetwoasymptoticexpan-sions,onevalidforx=O(1)andtheotherforFrom(2.41)and(2.45), 762.Introductoryapplicationsthesearewhere‘–’denotestermsused,inthefirstapproximation,withx=O(1),andde-notes,correspondingly,thetermsusedwhere(Thederivativehasnotbeenlabelled,butitwillautomaticallyberetained,byvirtueofthemultiplicationofterms,whenlabelledtermsareusedinanapproximation.)Theimportantinter-pretationisthatsometerms—hereall—usedwherex=O(1)arebalancedagainstsometermsnotusedpreviously(inthefirstapproximation),butnowrequiredwhereWhenweimposethisrequirementon(2.54),andnotethatthebreakdownisasi.e.thentheonlybalanceoccurswhenwechooseor,becausewemaydefineinanyappropriateway,simplyItisimpossi-bletobalancetheterminagainsttheO(1)termshere(togivedifferentleadingterms)because,whenwedothis,thedominanttermthenbecomeswhichisplainlyinconsistent.Notethatandascanneverbebalanced.Thus,armedonlywiththegeneralscalingproperty,thebehavioury~1/xasandtherequirementtobalanceterms,weareledtothechoicethisdoesnotinvolveanydiscussionofthenatureofthebreakdownoftheasymptoticexpansion.Thisnewprocedureisveryeasilyapplied,isverypowerfulandisthemostimmediateandnaturalmethodforfindingtherelevantscaledregionsforthesolutionofadifferentialequation.Weusethistechniquetoexploretwoexamplesthatwehavepreviouslydiscussed,andthenweapplyittoanewproblem.E2.10Scalingforproblem(2.11)(seealso(2.34))ConsidertheequationwithaboundaryconditiongivenatwhereasthedomainiseitherorThesolutionofthefirsttermintheasymptoticexpansionvalidforx=O(1)is(see(2.15))Thegeneralscaling,X,in(2.55)gives 77ifthedomainisthenfrom(2.56)weseethaty=O(1)as(unless;seebelow).Soweselect(2.57)becomesandthereisnochoiceofscaling,aswhichbalancesthetermagainstWeconcludethatasecondasymptoticregiondoesnotexist,andhencethattheexpansionforx=O(1)isuniformlyvalidonthis(bounded)domain,whichagreeswiththediscussionfollowing(2.17).(Inthespecialcaseasandsoamatchedsolutionnowrequiresproducingagain,thereisnochoiceofaswhichbalancesagainstOntheotherhand,ifthedomainisthenasandwerequireforamatchedsolutiontoexist;equation(2.57)nowbecomesThistime,with(becausetheO(1)termsbalanceife.g.orwhichrecoversthescalingusedtogive(2.40)inE2.8.E2.11Scalingforproblem(2.22)Considertheequationwithandsuitableboundaryconditions(whichmaybothbeatoneend,oroneateachend).Thegeneralsolutionofthedominanttermsfrom(2.58),withx=O(1)asis(asusedtogenerate(2.25)).Inthisexample,theasymptoticexpansion,ofwhich(2.59)isthefirstterm,maybreakdownasorasor,justpossibly,aswithAllthesemaybesubsumedintoonecalculationbyintroducingasimpleextensionofourmethodofscaling:letwherewemayallowinthisformulation.Thenwiththeusual(2.58)becomes 782.Introductoryapplications(Notethatwehaveusedtheidentityandsimilarlyforthesecondderivative;thisisvalidforForgeneralAandB,(2.59)impliesthaty=O(1)assoweselectandnobalanceexists(thatis,betweenandasAgainwededuce,onthebasisofthisscalingargument,thattheasymptoticexpansionisuniformlyvalidfor(exactlyaswefoundwasthecaseforexpansion(2.25)).Thespecialcaseinwhichproducesasandsowenowrequirebutanybalanceisstillimpossible.E2.12ScalingprocedureappliedtoanewequationForourfinalexample,weconsidertheequationwherer(x)iseitherzeroorr(x)=x;thetwoboundaryconditionsareeitheroneateachendofthedomain,orbothatoneend—itisimmaterialinthisdiscussion.Thegeneralsolutionofthedominanttermsin(2.60)asforx=O(1),isthelatterapplyingwhenr(x)=x.ThegeneralscalingintheneighbourhoodofanyisandwhichgivesfortheequationwithBecausethisequationislinearandhomogeneous,thescalinginyisredundant:itcancelsidentically.(Wemaystillrequiretomeasurethesizeofy,butitcanplaynorôleinthedetermination,fromtheequation,ofanyappropriatescalingnearThebalanceofterms,asrequireseitherthatorthatandonlythelatterisconsistent,sotheformerbalancesthetermsandY,butthenisthedominantcontributor!Thusanyscaledregionmustbedescribedbyalthoughthisanalysiscannothelpusdecideifanexists,orwhatmightbe;wewillexaminethisissueinthenextsection. 79Thecaseofr(x)=xisslightlydifferent,becausetheequationisnolongerhomo-geneous:thereisaright-handside.ThesamescalingnowproducesbutcanbefoundbytheconditionthatanysolutionweseekforYmustmatchto(2.61b).Ify=O(1)asthenandtheresultisasbefore:theonlybalanceisprovidedbyOntheotherhand,ifthenassoandthenwehaveButthenewtermisthesamesizeasanexistingterm,andsothesameresultfollowsyetagain:istheonlyavailablechoice.(Takinggivesthesameresult.)Thetechniqueofscalingdifferentialequations,coupledwiththerequiredbehaviournecessaryifmatchingistobepossible,issimplebutpowerful(astheaboveexamplesdemonstrate).Itisoftenincorporatedatanearlystageinmostcalculations,andthatishowwewillviewitinthefinalintroductoryexamplesthatwepresent;additionalexamplesareavailableinQ2.16.Wewillshortlyturntoadiscussionofaclassicaltypeofproblem:thosethatexhibitaboundary-layerbehaviour(aphenomenonthatwehavealreadymetinE1.4;see(1.16)).However,beforewestartthis,afewcommentsofarathermoreformalmathematicalnatureareinorder,andmaybeofinteresttosomereaders.Letussupposethatwehavescaledanequationaccordingtoandandthatwehavechosentosatisfymatchingrequirements;wewillexpressthisasforsomeknownn.Thescaling,ortransformation,willberepresentedbythistransformationofvariablesbelongstoacontinuousgrouporLiegroup.(Notethatthisdiscussionhasnotinvokedthebalanceofterms,whichthenleadstoachoicethiswouldconstituteaselectionofonememberofthegroup.)Wenowexplorethepropertiesofthistransformation.Firstweapply,successively,thetransformationandthenthisisequivalenttothesingletransformationandsowehavethemultiplicationrule: 802.IntroductoryapplicationsButwealsohavesothislawiscommutative.Furthermore,theassociativelawissatisfied:inaddition,wehavetheidentitytransformationi.e.forallFinally,weformandsoisboththeleftandrightinverseofThustheelementsofforallrealformaninfinitegroup,whereistheparameterofthiscontinuousgroup.(Ifnisfractional,thenwemayhavetorestricttheparametertoAlthoughwehavenotusedthefullpowerofthiscontinuousgroup—weeventuallyselectonlyonememberforagiven—thereareothersignificantapplicationsofthisfundamentalpropertyinthetheoryofdifferentialequations.Forexample,ifaparticularscalingtransformationleavestheequationunchanged(exceptforachangeofthesymbols!)i.e.theequationisinvariant,thenwemayseeksolutionswhichsatisfythesameinvariance.Suchsolutionsare,typically,similaritysolutions(iftheyexist)oftheequation;thisaspectofdifferentialequationsisgenerallyoutsidetheconsiderationsofsingularperturbationtheory(althoughthesesolutionsmaybetherelevantonesincertainregionsofthedomain,inparticularproblems).2.6EQUATIONSWHICHEXHIBITABOUNDARY-LAYERBEHAVIOURTherearemanyproblems,posedintermsofeitherordinaryorpartialdifferentialequations,thathavesolutionswhichincludeathinregionnearaboundaryofthedomainwhichisrequiredtoaccommodatetheboundaryvaluethere.Suchregionsarethinbyvirtueofascalingofthevariablesintheappropriateparameterand,typically,thisinvolveslargevaluesofthederivativesneartheboundary.Theterminology—boundarylayer—isratherself-evident,althoughitwasfirstassociatedwiththeviscousboundarylayerinfluidmechanics(whichwewilldescribeinChapter5).Here,wewillintroducetheessentialideaviasomeappropriateordinarydifferentialequations,andmakeuseoftherelevantscalingpropertyoftheequation.Thenatureofthisproblemisbestdescribed,first,byananalysisofequation(1.16):withand(andwewillassumethatandarenotfunctionsofandthatInthispresentationoftheconstructionoftheasymptoticsolution,validaswewillworkdirectlyfrom(2.63)(althoughtheexactsolutionisavailablein(1.22),whichmaybeusedasacheck,ifsodesired).Becausewewishtoincorporateanapplicationofthescalingproperty,weneedtoknowwheretheboundarylayer(interpretedasascaledregion)issituated:isitnearx=0ornear 81x=1(or,possibly,both)?Here,wewillassumethatthereisasingleboundarylayernearx=0;theproblemoffindingthepositionofaboundarylayerwillbeaddressedinthenextsection,atleastforaparticularclassofordinarydifferentialequations.Letusnowreturntoequation(2.63).AsshouldbeevidentfromexampleE1.4,andwillbecomeveryclearinwhatfollows,itistheappearanceofthesmallparametermultiplyingthehighestderivativethatiscriticalhere.Thepresenceofinanothercoefficientisaltogetherirrelevanttothegeneraldevelopment;itisretainedonlytoallowdirectcomparisonwithE1.4.Weseekasolutionof(2.63)intheformandthenweobtainandsoon.Theonlyboundaryconditionavailabletous(becauseoftheassumptionaboutthepositionoftheboundarylayer)isThusandindeed,inthisproblem,wethenhavealthoughexponentiallysmalltermswouldberequiredforamorecompletedescriptionoftheasymptoticsolutionvalidforx=O(1);sowehaveThescaledversionof(2.63)isobtainedbywritingforand(becausey=O(1)asalthoughanyscalingonYwillvanishidenticallyfromtheequation);thusTherelevantbalance,aswehavealreadyseenin(2.62),isorgiving 822.IntroductoryapplicationsWeseekasolutionofthisequationintheusualform:whichgivesandsoon.Wehaveavailabletheoneboundaryconditionprescribedatx=0(whichisintheregionoftheboundarylayer)i.e.atX=0,soFrom(2.67a)and(2.68)weobtainwhereisanarbitraryconstant,andthen(2.67b)becomesThuswhereandarethearbitraryconstants,whichmustsatisfyfromthisgivesthesolutionThe2-termasymptoticexpansionvalidforisthereforewhichistobematchedto(2.65);thisshoulduniquelydetermineandWewrite(2.65)asretainingtermsO(1)and(asusedin(2.70));correspondingly,wewrite(2.70)as 83andthetermisincludedbecausethatwasobtainedfor(2.65)—althoughithadazerocoefficient.Inordertomatch(2.71)and(2.72),werequireandi.e.(Itisleftasanexercisetoshowthat(2.65)and(2.73)arerecoveredfromsuitableexpansionsof(1.22).)Weshouldnotethat(2.65)exhibitsnobreakdown—thereisonlyonetermhere,afterall—but(2.73)doesbreakdownwherei.e.x=O(1),aswewouldexpect.Anyindicationofabreakdownintheasymptoticexpansionvalidforx=O(1)willcomefromtheexponentiallysmallterms;letusbrieflyaddressthisaspectoftheproblem.Thefirstpointtonoteisthat,from(2.70)and(2.73),wewouldrequirenotonlyO(1)andbutalsoterms(andothers),inordertocompletethematchingprocedure;thisissimplybecauseweneed,atleastinprinciple,tomatchtoalltheterms(cf.(2.72))Thustheexpansionvalidforx=O(1)mustincludeatermtoallowmatchingtothisorder(andthenitisnottoodifficulttoseethatacompleteasymptoticexpansionrequiresallthetermsinthesequencen=0,1,2,…,m=0,1,2,…).Inpassing,weobservethatthisuseofthematchingprincipleisnewinthecontextofourpresentationhere.Weareusingit,first,inageneralsense,todeterminethetype(s)ofterm(s)requiredintheexpansionvalidinanadjacentregioninordertoallowmatching.Then,withthesetermsincluded,the‘full’matchingproceduremaybeemployedtocheckthedetailsandfixthevaluesofanyarbitraryconstantsleftundetermined.Wewillfindthefirstoftheexponentiallysmalltermshere.Forx=O(1),weseekasolutionwhereinthisexample);thusequation(2.63)be-comes(Again‘=0’meanszerotoallordersinWealreadyhavethatsatisfiestheoriginalequation,andsomustsatisfy 842.Introductoryapplicationswithweobtainor(constant).Thustheexpansion(2.74)becomesandthisistobematchedtotheasymptoticexpansion(2.73);from(2.75)weobtainandfrom(2.73)wehavewhichmatchifThusthesolutionvalidforx=O(1),i.e.awayfromtheboundarylayernearx=0,incorporatingthefirstexponentiallysmallterm,isOnefinalcomment:thissolution,(2.76),producesthevaluesonowtheboundaryvalueisinerrorbyTocorrectthis,afurthertermisrequired;wemustwrite(attheorderofallthefirstexponentiallysmallterms)whereItisleftasanexercisetoshowthatwiththesolutionensuresthattheboundaryconditiononx=1iscorrectatthisorder.Theinclusionofthetermintheexpansionvalidforx=O(1)forces,viathematchingprinciple,atermofthissameorderintheexpansionforandsothepatterncontinues.(Theappearanceofalltheseterms,inbothexpansions,canbeseenbyexpandingtheexactsolution,(1.22),appropriately.)E2.13Anonlinearboundary-layerproblemWeconsidertheproblem 85withforwearegiventhattheboundarylayerisnearx=1.Weseekasolution,for1–x=O(1),intheformandsoetc.,withforThusweobtainwhichleadstobutthissolutiondoesnotsatisfytheboundaryconditiononx=1.Forthesolutionnearx=1,weintroduce(withandsothati.e.Thesolutionintheformgivesandsoon;theavailableboundaryconditionrequiresthatThusandwhereandarethearbitraryconstantstobedeterminedbymatching;thatis,wemustmatch 862.Introductoryapplicationswith(2.78).From(2.79)weobtainfrom(2.78)wehaveandthesematchifweselectandthus(2.79)becomesThefundamentalissuerelatingtoboundary-layer-typeproblems,whichwehaveavoidedthusfar,addressesthequestionofwheretheboundarylayermightbelo-cated.Intheexamplesdiscussedabove,weallowedourselvestheadvantageofknowingwherethislayerwassituated—andtheconsistencyoftheresultingasymptoticsolutionconfirmedthatunique,well-definedsolutionsexisted,sopresumablywestartedwiththecorrectinformation.Wenowexaminethisimportantaspectofboundary-layerproblems.2.7WHEREISTHEBOUNDARYLAYER?Forthisdiscussion,weconsiderthegeneralsecond-orderordinarydifferentialequationintheformwithsuitableboundaryconditionsandfornotethatthecoefficientofmustbeThecoefficienta(x)willsatisfyeithera(x)>0ora(x)<0,fortheterm(orsmaller)asforandforallsolutionsthatmaybeofinterest.Ofcourse,thisdescribesonlyoneclassofsuchboundary-layerproblems,butthisdoescoverbyfarthemostcommononesencounteredinmathe-maticalmodelling.(Someoftheseconditions,bothexplicitlywrittenandimplied,canberelaxed;wewillofferafewgeneralisationslater.)Theguidingprinciplethatwewilladoptistoseeksolutionsof(2.80)whichremainboundedasforThestartingpointistheconstructionoftheasymptoticsolutionvalidforsuitablex=O(1),directlyfromtheequationaswrittenin(2.80)—butthiswillnecessarilygenerateasequenceoffirst-orderequations.Itisthereforeimpossibletoimposethetwoboundaryconditions(aswehavealreadydemonstratedinourexamplesabove);theinclusionofaboundarylayerremediesthisdeficiencyinthesolution.Weintroducetheboundarylayerinthemostgeneralwaypossible:defineforsomeg(x) 87andsomeandwriteThenwehaveandsoequation(2.80)becomeswhereWewillfurtherassume,whateverthechoiceof(andinalmostallproblemsthatweencounter,thatthetermsinanddominateasThusthe‘classical’choiceforthebalanceofterms,applieshereforgeneralg(x)(whichwehaveyettodetermine).Thedifferentialequationvalidintheboundarylayercannowbewrittenwhichhasthefirstterminanasymptoticexpansion,satisfyingAtthisstage,hasyettobedetermined;letuschoose(wemaynotchoosethenweareleftwiththesimple,genericproblemforwhichalsosolvesthedifficultyoverthemixingofthexandXnotationsin(2.83).Thusallboundary-layerproblemsinthisclasshavethesamegeneralsolution,from(2.84),However,wearenonearerfindingthepositionoftheboundarylayeritself;thiswenowdobyexaminingtheavailablesolutionsforg(x).Thegeneralformforg(x)iswhereCisanarbitraryconstant(andthisprovestobethemostconvenientwayofincludingtheconstantofintegration).First,wesupposethata(x)>0,andexamine 882.Introductoryapplications(2.85)whenexpressedintermsofx(aswillbenecessaryforanymatching);thisgivesButweareseekingsolutionsthatremainboundedasandthisispossibleonlyiftheexponentin(2.87)isnon-positiveforThuswerequireandthismeansthatforthesamexsi.e.(forotherwisex0,thebound-arylayermustsitneartheleft-handedgeofthedomain.Conversely,thesameargumentinthecasea(x)<0requiresthattheboundarylayerbesituatednear(theright-handedgeofthedomain).Whenweapplythisruletoequation(2.63):weseethatandsotheboundarylayerisintheneighbourhoodofx=0(andweintroducedforthisexample).Similarly,equation(2.77):hasa(x)=–1<0,andsotheboundarylayerisnownearx=1(andweusedItisrarelynecessarytoincorporatetheformaldefinitionofg(x)togeneratetheappropriatevariablethatistobeusedtorepresenttheboundarylayer(althoughitwillalwaysproducethesimplestformofthesolution).Forexample,theequationofthisclass:hasaboundarylayernearx=2(becausea(x)=–1/(2+x)<0forNowanappropriatescaledvariableissimplygivingandthischoicewillsuffice,eventhoughhigher-ordertermswillrequiretheexpansionof(butthisisusuallyasmallpricetopay—andwealreadyknowthatthisasymptoticexpansionwillbreakdownforsoretaininginthecoefficienthasnounforeseencomplications).Itshouldalsobenotedthat,exceptionally,aboundary-layer-typeproblemmaynotrequireaboundarylayeratall,inordertoaccommodatethegivenboundaryvalue(toleadingorderor,possibly,toallorders).Thisisevidentfortheequationgivenin(2.63)(andseealsotheexactsolution,(1.22));inthisexample,iftheboundaryvaluessatisfythespecialconditionthennoboundarylayerwhatsoeverisrequired.Note,however,thatifthen 89theboundarylayerispresent,butonlytocorrecttheboundaryvalueat—theleadingterm(forx=O(1))isuniformlyvalid.Wecalluponalltheseideasinthenextexample.E2.14Anonlinear,variablecoefficientboundary-layerproblemWeconsiderwithforBecausethecoefficientofisnegativefortheboundarylayerwillbesituatedattheright-handedgeofthedomaini.e.nearx=1.Awayfromx=1,weseekasolutionandsoetc.,andwemayusetheboundaryconditiononx=0:Thusweobtainandthentheapplicationoftheboundaryconditionyieldsthesolutionwenotethatremainsrealandpositiveas(Thesecondtermcanalsobefound,butitisaslightlytiresomeexerciseanditsinclusionteachesuslittleaboutthesolution.)Clearly,astheright-handboundaryisapproached,whichdoesnotsatisfythegivenboundaryconditionandsoaboundarylayerisrequired.(Ofcourse,ifthen(2.90)wouldbeauniformlyvalid1-termasymptoticsolution.)Weintroduce(sothatandwriteequation(2.88)thenbecomes 902.Introductoryapplicationswhichgives,withThesolutionof(2.92),whichsatisfiestheboundaryconditionisandistobedeterminedbymatching.From(2.90)wehavedirectlythatandfrom(2.93)weseethatthusmatchingrequiresandthefirsttermintheboundary-layersolutionisandthenacompositeexpansioncanbewrittendown,ifthatisrequired.Thefundamentalideasthatunderpinthenotionofboundary-layer-typesolutions,insecond-orderordinarydifferentialequations,havebeendeveloped,butmanyvariantsofthissimpleideaexist;seealsoQ2.17–2.20.Theseleadtoadjustmentsinthefor-mulation,ortogeneralisations,ortoaratherdifferentstructure(withcorrespondinginterpretation).Wenowdescribeafewofthesepossibilities,butwhatwepresentisfarfromprovidingacomprehensivelist;rather,wepresentsomeexampleswhichem-phasisetheapplicationofthebasictechniqueofscalingtofindthinlayerswhererapidchangesoccur.Inthenextsection,webrieflydescribeanumberofdifferentscenarios,andpresentanexampleofeachtype.2.8BOUNDARYLAYERSANDTRANSITIONLAYERSOurfirstdevelopmentfromthesimplenotionofaboundarylayerisaffordedbyanextensionofourdiscussionofthepositionofthislayer,viaequation(2.80);here,weconsiderthecasewhereSuchapointisanalogoustoaturningpoint(seeQ2.24)andthesolutionvalidneartakestheformofatransitionlayer.(Theterminology‘turningpoint’isusedtodenotewherethecharacterofthesolutionchangesor‘turns’,typicallyfromoscillatorytoexponential,inequationssuchasThegeneralapproachistoseekascaling–justasforaboundarylayer–butnowatthisinteriorpoint.Letussupposethatforgivenconstantsandn,thenequation(2.80)becomesandweintroduce 91togivewherewehaveusedthesamenotationasin(2.81).Thebalancethatweseekisgivenbythechoiceorprovidedthatn<1(inorderthatthisbalancedoesindeedproducethedominantterms,withF=O(1)orsmaller).Theprocedurethenunfoldsasfortheboundary-layerproblems,althoughwewillnowhavesolutionsinandinwheretogetherwithamatchedsolutionwhereIfthenthebalanceoftermsrequiresandtheneither(n=1)allthreetermsintheequationcontributetoleadingorder,or(n>1)thebalanceisbetweenandF.(ThisdescriptionassumesthatF=O(1);notealsothatthechosenbehaviourofa(x)usedhereneedonlyapplynearforthisapproachtoberelevant.)E2.15AnequationwithatransitionlayerConsidertheequationforwhere,forrealsolutions,weinterprettheboundaryconditionsareWewillfindthefirsttermsonlyintheasymptoticexpansionsvalidawayfromx=0(wherethecoefficientiszero),inandtheninandfinallyvalidnearx=0.Forx=O(l),wewriteandso,from(2.94),weobtainwedeterminethearbitraryconstantbyimposingtheboundaryconditions(2.95a,b),therebyproducingsolutionsvalideitherononeside,ortheother,ofx=0:Notethatthesesolutionsdonotholdintheneighbourhoodofx=0i.e.wemayallowbutwiththissolutionwouldbevalidfortoleadingorder,ifthefunctiongivenin(2.96)werecontinuousatx=0,andthennotransitionlayerwouldbeneeded(tothisorder,atleast). 922.IntroductoryapplicationsNearx=0,wewriteand,inordertomatch,werequirey=O(1),sowewriteequation(2.94)thengivesandhenceweselectThusinthetransitionlayerwehavetheequationandweseekasolutionwhere(Thelowerlimitintheintegralhereissimplyaconvenience;withthischoice,thearbitraryconstantfollowingthesecondintegrationisNotethatthisintegralexistsforThesearbitraryconstantsaredeterminedbymatching;from(2.96a,b)weobtaindirectlythatFrom(2.97),wefirstwritenowletusintroducetheconstant(whereisthegammafunction),thenandmatching(2.98)and(2.100)requiresandThefirsttermoftheasymptoticexpansionvalidinthetransitionlayer(aroundx=0)istherefore 93Thisexamplehasdemonstratedhowboundary-layertechniquesareequallyapplicabletointerior(transition)layersand,further,theyneednotberestrictedtosinglelayers.Someproblemsexhibitmultiplelayers;forexamplehastransitionlayersbothnearx=1/2andnearx=–1/2,andthesolutionawayfromtheselayersisnowinthreeparts.Atypeofproblemwhichcontainselementsofbothboundaryandtransitionlayersoccursifthecoefficient,a(x),ofiszeroatone(orboth)boundaries.Becauseataninternalpoint,itisnotatransitionlayer,butthefactthatattheend-pointaffectsthescaling—itisnolongeringeneral—andthismustbedetermineddirectly(aswedidforthetransitionlayer).E2.16Aboundary-layerproblemwithanewscalingWeconsidertheequationforwithandnotethatandsowemustexpectaboundarylayernearx=0.Awayfromx=0,weseekasolutionwhichgivesandsoon,withandforThusweobtaininordertosatisfytheboundarycondition(andasthisdoesnotapproachandsoaboundarylayeriscertainlyrequired).Theequationforisthereforeor 942.Introductoryapplicationswhichhasthegeneralsolutionandrequires(Itshouldbenotedthataswhich,alone,indicatesabreakdownwherethis,asweshallseebelow,isirrelevant).Fortheboundarylayer,wescalewiththentheequation,(2.101),becomesandsowemustselectwhichleadsto(Theapparentscaling,isthereforeredundant—itissmallerthanthatrequiredintheboundarylayer.)Weseekasolutionwithandtheavailableboundarycondition,thengivesFinally,wedeterminebyinvokingthematchingprinciple;from(2.102)weobtainandfrom(2.103)weseethatwhichmatchiftheboundary-layersolutionisthereforewherekisgivenin(2.104). 95Boundarylayersalsoariseevenintheabsenceofthefirst-derivativeterm;indeed,equationsoftheformwithforcanhaveaboundarylayerateachendofthedomain.(Ifthentherelevantpartofthesolutionisoscillatoryandboundarylayersarenotpresent.Ifa(x)=0ataninteriorpoint,thenwehaveaclassicalturning-pointproblemandnearthispointwewillrequireatransitionlayer.)Thesolutionawayfromtheselayersissimplygiven,toleadingorder,byy(x)=–f(x)/a(x).Toseethenatureofthisproblem,considerthecasea(x)=1.Theequationthatcontrolsthesolutionintheboundarylayersisthenandsoandexponentiallydecayingsolutions—ensuringboundedsolutionsas—ariseforA=0orB=0,appropriatelychosen,eitherontheleftboundaryorontherightboundary.E2.17TwoboundarylayersConsidertheequationforwiththesolutionforsuitablex=O(1)iswrittenastoleadingorder,whereThissolution,(2.106),clearlydoesnotsatisfytheboundaryconditionsasnorasTheboundarylayernearx=1isexpressedintermsofwithequation(2.105)thenbecomeswhichleadstothechoiceSeekingasolutionthenandforbounded(i.e.matchable)solutionsaswemusthaveTheboundarycondition,givesandso 962.Introductoryapplicationsandthisiscompletelydetermined,asis(2.106),somatchingisusedonlytoconfirmthecorrectnessoftheseresults(andthisisleftasanexercise).Theboundarylayernearx=0iswrittenintermsofwithagain);equation(2.105)nowbecomesTheboundarylayeris,notsurprisingly,thesamesizeatthisendofthedomain:andthenwithweobtainAboundedsolution,asrequiresthechoiceandwillensurethattheboundaryconditionissatisfiedi.e.Thenexttermintheasymptoticexpansionsatisfieswith(forboundedness)and(becausei.e.itisleftasanotherexercisetoconfirmthatthismatcheswith(2.106).Inourexamplessofar(andseealsoQ2.21,2.22),thecharacterandpositionoftheboundarylayer(oritsinteriorcounterpart,thetransitionlayer)havebeencontrolledbytheknownfunctiona(x),asin(ora(x)yinourmostrecentexample).Wewillnowinvestigatehowthesameapproachcanbeadoptedwhentherelevantcoefficientisafunctionofy.Inthissituation,wedonotknow,apriori,thesignofy—andthisisusuallycritical.Typically,wemakeanappropriateassumption,seekasolutionandthentesttheassumption.Forexample,ifthetwoboundaryvaluesfory—wearethinkinghereoftwo-pointboundaryvalueproblems—havethesamesign,thenwemayreasonablysupposethatyretainsthissignthroughoutthedomain.Ontheotherhand,iftheboundaryvaluesareofoppositesign,thenthesolutionmusthaveatleastonezerosomewhereonthedomain(andthisindicatestheexistenceofatransitionlayer).E2.18Aproblemwhichexhibitseitheraboundarylayeroratransitionlayer:I(AnexamplesimilartothisoneisdiscussedingreatdetailinKevorkian&Cole,1981&1996;seeQ2.23.) 97Weconsidertheequationforwithand(whereandareindependentofforsuitablex=O(1),weseekasolutionintheusualformwhichgivesandsoon.Thegeneralsolutiontoequation(2.108a)isweexcludethesolutionbecausewewillconsiderproblemsforwhichand(Anyspecialsolutionswhichmayneedtomakeuseofthezerosolutionareeasilyincorporatedifrequired.)Thenextterminthisasymptoticexpansionisobtainedfrom(2.108b)i.e.whichyieldswhereisthesecondarbitraryconstant(andwehavetakenItisclear,however,thatitisimpossibletoproceedwithoutmoreinformationabouttheboundaryvalues,andLetusexamine,first,theproblemforwhichbothvaluesarepositive;wethereforeassumethatasolution,y>0,existsandhencethatanyboundarylayermustbesituatedintheneighbourhoodofx=0(indicatedbythetermwithy>0).Withthisinmind,wemayusetheoneavailableboundaryconditionawayfromx=0,i.e.thusweobtainCorrespondingly,withweseethatwewillassumehereafterthat(butweclearlyhaveaninterestingcaseifforthenthesolutiondoeshaveazeronearevenwithpossibilitynotpursuedhere). 982.IntroductoryapplicationsNow,asweseethatandifthisdoesnotequalthenaboundarylayerisrequirednearx=0.Forthislayer,wewritewith(=O(1)),andthen(2.107)becomeswiththechoiceWeseekasolutionwhereandwehavechosentowritethearbitraryconstantofintegrationasanyotherchoiceproducesasolutionwhichcannotbematched—aninvestigationthatisleftasanexercise.Thenextintegralof(2.113)givesthegeneralsolutionandtosatisfythiscanbewrittenasThevalueoftheremainingarbitraryconstant,isnowdeterminedbymatching(2.114)and(2.111);from(2.111)weobtainandfrom(2.114)weseethatwhichrequiresthatThuswehavesuccessfullycompletedtheinitialcal-culationsintheconstructionofasymptoticexpansionsvalidforx=O(1)andforthesedemonstratethat,inthecaseandwehaveasolutionwhichsatisfiesy>0forE2.19Aproblemwhichexhibitseitheraboundarylayeroratransitionlayer:IIWerepeatexampleE2.18,butnowwiththeboundaryvalues(chosentoavoidthedifficultiesalreadynoted)andandthusthesolutionmustchangesign(atleastonce)inordertoaccommodatetheseboundaryvalues.Thisindicatestheneed 99foratransitionlayeratsomeanddeterminingbecomesanessentialelementintheconstructionofthesolution.Forx=O(1)wehave,asbefore(see(2.111)),butthiscanholdonlyforbutwhereisintroducedbelow);forwehavethecorrespondingsolutionNearwewritewithwhichessentiallyrepeats(2.112)i.e.andsoandthisgivesthesamegeneralsolution,toleadingorder,asbefore(see(2.113),etseq.):However,foratransitionlayer,wedonothaveanyboundaryconditions;here,wemustmatch(2.117)toboth(2.115)and(2.116).From(2.115)and(2.116)weobtainrespectively,bothforX=O(1);from(2.117),withwehaveasWeobserve,immediately,thatapropertyofthistransitionlayeristoadmitonlyachangeinvalueacrossitfromto(whichwillfixthevalueofandthatthematchingexcludes(sothiscannotbedeterminedatthisstage).Now(2.119)doesmatchwith(2.118)whenwechoosewhichrequiresthat 1002.IntroductoryapplicationsandhenceatransitionlayerexistsatprovidedIfthisconditionisnotsatisfied,forgivenandthentheadjustmenttothegivenboundaryvaluemustbethroughaboundarylayernearx=0.Thus,forexample,withandthereisatransitionlayeratandthejumpacrossthelayerisbetween±3/2(toleadingorder).Ontheotherhand,theproblemwithanddoesnotadmitatransitionlayer;theboundarylayernearx=0isusedtoaccommodatethechangeinvaluefrom(toleadingorder)toThedominantsolutioninthetransitionlayerisgivenby(2.117)withalthoughisunknownatthisstage.(Theroleofistodeterminethepositionofthetransitionlayer,correctatThesetwoexamples,E2.18andE2.19(andseealsoQ2.23),demonstratethecomplex-ityandrichnessofsolutionsthatareavailableforthistypeofproblem,dependingontheparticularboundaryvaluesthatareprescribed.Allthiscanbetracedtothenonlin-earityassociatedwiththeterm;ifthistermweresimplythenwewouldhaveafixedboundarylayer,orfixedtransitionlayers,independentofthespecificboundaryvalues(aswehaveseeninourearlierexamples).Weconcludethissectionwithanexamplewhichshowshowtheseideascanbeextended,fairlystraightforwardly,tohigher-orderequations.(Thefollowingexampleisbasedonthetypeofproblemthatcanarisewhenexaminingthedisplacementofaloadedbeam.)E2.20AproblemwithtwoboundarylayersWeconsiderfor(andtheuseofhereismerelyanalgebraicconvenience),withBeforewebeginthedetailedanalysisofthisproblem,acoupleofpointsshouldbenoted.First,thevariablecoefficient,in(1.121),ispositiveforandso,second,thisimpliesthatwehaveavailable(locally)twoexponentialsolutions.Thesearisefrom,approximately,andsowemayselectoneexponentialnearx=0andtheothernearx=1,ensuringexponentialdecayaswemoveawayfromtheboundaries.Thuswemustanticipate 101boundarylayersnearbothx=0andx=1;wewillassumethattheyexist—wecanalwaysignorethemiftheyarenotrequired(becausetheboundaryvaluesareautomaticallysatisfied).Hence,forx=O(1)awayfromx=0andx=1,weseekasolutionwithwherebutnoboundaryconditionsareavailable(byvirtueoftheassumedexistenceofbound-arylayers).Fortheboundarylayernearx=0,itisclearthatwerequire(butwedonotknowatthisstage;sincepresumablyasalthoughthesizeofinthislimitisunknown).Thusequation(2.121)becomesWeseekasolutionwheresatisfiesprovidedaswewillassumethatthisconditionapplies,andwewillcheckitshortly.TheboundaryconditionsareandwemustnotallowthetermwhichgrowsexponentiallyawayfromX=0,soandthenImmediatelyweobservethatthetermisunmatchabletounlessweselect(andthenthiswedo,sothatCorrespondingly,fortheboundarylayernearx=1,wewriteandnowchoosetogiveThusthefirsttermintheexpansion,satisfies 1022.Introductoryapplicationsandthen(exactlyasdescribedforweobtainWenowdeterminetheconstantsandbymatching(2.123)with,inturn,(2.124)and(2.125).First,from(2.123)withandweobtainfrom(2.124)wehavewhichrequiresandthenAgain,from(2.123),butnowwithandweobtainand,finally,(2.125)givesNowwerequireandthus,collectingalltheseresults,weseethatandhence,toleadingorder,wehavewithSo,indeed,boundarylayersarerequiredateachendinordertoaccommodatetheboundaryconditionsthere(althoughwemaynotethatthesolutionfordoessatisfybutnotthederivativeconditions).Somefurtherexamplesofhigher-orderequationsthatexhibitboundary-layerbe-haviourareofferedinQ2.26.Thischapterhasbeendevotedtoapresentationofsomeofthefairlyroutineap-plicationsofsingularperturbationtheorytovarioustypesofmathematicalproblem. 103Althoughwehavetouchedonmethodsforfindingrootsofequations,andoninte-gration,themainthrusthasbeentodevelopbasictechniquesforsolvingdifferentialequations—themostimportantuse,byfar,ofthesemethods.Weshalldevotetherestofthetexttoextendinganddevelopingthemethodsforsolvingdifferentialequations,bothordinaryandpartial,andtheirapplicationstomanypracticalproblemsthatareencounteredinvariousbranchesofmathematicalmodelling.Manyoftheexamplesandexercisesinthischapterare,perforce,inventedtomakeapointortotestideas;however,afewofthelaterexercisesthatareincludedattheendofthischapter(seeQ2.27–2.35)begintoemploythetechniquesinphysicallyrelevantproblems.Inthenextchapter,wewillshowhowtheseideascanbeappliedtoabroaderclassofprob-lemsand,inparticular,beginourdiscussionofpartialdifferentialequations.Thiswillallowus,inturn,tobegintoextendtheapplicationsofsingularperturbationtheorytomoreproblemswhicharisewithinaphysicallyrelevantcontext.FURTHERREADINGAfewoftheexistingtextsincludeadiscussionofthemethodsforfindingrootsofequations,andforevaluatingintegralsoffunctionswhichcontainasmallparameter;inparticular,theinterestedreaderisdirectedtoHolmes(1995)andHinch(1991).Differentialequationsthatgiverisetoregularproblemsaregivenlittleconsideration—theyarequiterare,afterall—butsomecanbefoundinHolmes(1995)andinGeorgescu(1995).Wehavealreadymentionedthosetextsthatpresentamoreformalapproachtoperturbationtheory(Eckhaus,1979;Smith,1985;O’Malley,1991),butsomefurtherdevelopmentsalongtheselinesarealsogiveninChang&Howes(1984).Thewholearenaofscalingwithrespecttoaparameter,andweshouldincludeheretheconstructionofnon-dimensionalvariables,isfairlyroutinebutverypowerful.Theseideasplayarôle,notonlyintheidentificationofasymptoticregions(aswehaveseen),butalsoinprovidingmoregeneralpointerstotheconstructionofsolutions.Averythoroughintroductiontotheseideas,andtheirconnectionwithasymptotics,canbefoundinBarenblatt(1996).Adiscussionoftheapplicationsofgrouptheorytothestudyofdifferentialequationsislikelytobeavailableinanygood,relevanttext;onesuch,whichemphasisespreciselytheapplicationtodifferentialequations,isDresner(1999).Thenatureofaboundarylayer(whichis,forourcurrentinterest,limitedtoapropertyofcertainordinarydifferentialequations)isdescribedatlength,andcarefully,inmostavailabletextsonsingularperturbationtheory.Wecanmention,asexamplesoftheextentanddepthofwhatisdiscussed,theexcellentpresentationsonthissubjectgivenbySmith(1985)andHolmes(1995).Thedeterminationofthepositionofaboundarylayerisalsocoveredinmostexistingtexts,althoughO’Malley(1991)probablyprovidesthemostdetailedanalysis.(Thisworkalsoincludesanumberofrelevantreferenceswhichtheinterestedreadermaywishtoinvestigate.)Anexcellentdiscussionoftheinterplaybetweenboundarylayersandtransitionlayers(fornonlinearequations)isgiveninKevorkian&Cole(1981,1996).(Thosereaderswhowishtoexaminetechniquesapplicabletoturningpoints,atthisstage,areencouragedtostudyWasow(1965)andHolmes(1995);wewilltouchontheseideasinChapter4.) 1042.IntroductoryapplicationsFinally,someexamplesofhigher-orderequations,whichexhibitboundary-layer-typesolutions,arediscussedinSmith(1985)andO’Malley(1991).EXERCISESQ2.1Quadraticequations.Writedowntheexactrootsofthesequadraticequations,whereisapositiveparameter.(a)(b)(c)Now,ineachcase,usethebinomialexpansiontoobtainpower-seriesrepre-sentationsoftheseroots,validforwritingdownthefirstthreetermsforeachroot.(Youmaywishtoinvestigatehowthesesameexpansionscanbederiveddirectlyfromtheoriginalquadraticequations.)Finally,obtainthecorrespondingpowerserieswhicharevalidforQ2.2EquationsI.Findthefirsttwotermsintheasymptoticexpansionsofalltherealrootsoftheseequations,for(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Q2.3EquationsII.RepeatQ2.2fortheseslightlymoreinvolvedequations.(a)(b)(c)(d)(e)Q2.4Kepler’sequation.Aroutineproblemincelestialmechanicsistofindtheeccentricanomaly,u,givenboththeeccentricityandthemeananomalynt(wheretistimemeasuredfromwhereu=0,andwherePistheperiod);uisthenthesolutionofKepler’sequation(seee.g.Boccaletti&Pucacco,1996).Formanyorbits(forexample,mostplanetsinoursolarsystem),theeccentricityisverysmall;findthefirstthreetermsintheasymptoticsolutionforuasConfirmthatyour3-termexpansionisuniformlyvalidforallnt.Q2.5Complexroots.Findthefirsttwotermsintheasymptoticexpansionsofalltherootsoftheseequations,for(a)(b)(c)(d)(e)(f) 105Q2.6Simpleintegrals.Obtainestimatesfortheseintegrals,forbyfirstfindingasymptoticexpansionsoftheintegrandforeachrelevantsizeofx,retainingthefirsttwotermsineachcase.(Theseintegralscanbeevaluatedexactly,soyoumaywishtocheckyourresultsagainsttheexpansionsoftheexactvalues.)(a)(b)(c)(d)Q2.7Moreintegrals.SeeQ2.6;repeatfortheseintegrals(buthereyouarenotexpectedtohaveavailabletheexactvalues).(a)(b)(c)for(d)for(e)(f)Q2.8Anintegralfromthinaerofoiltheory.Anintegralofthetypethatcanappearinthestudyofthinaerofoiltheory(forthevelocitycomponentsintheflowfield)isobtainthefirsttermsintheasymptoticexpansionsoftheintegrand(forwithxawayfromtheend-points,foreachof:(a)awayfromxandawayfrom(b)(c)HencefindanestimateforRepeatthecalculationswithandthenwithforawayfromtheend-points,andthenwithrespectively.Again,findestimatesforandforshowthatyourasymptoticapproximationsforsatisfythematchingprinciple.Q2.9Regularexpansionsfordifferentialequations.Findthefirsttwotermsintheasymptoticexpansionsofthesolutionsoftheseequations,satisfyingthegivenboundaryconditions.Ineachcaseyoushouldusetheasymptoticsequenceandyoushouldconfirmthatyour2-termexpansionsareuniformlyvalid.(Youmaywishtoexaminethenatureofthegeneralterm,andhenceproduceanargumentthatshowstheuniformvalidityoftheexpansiontoallordersin(a)(b)(c)(d)(e) 1062.IntroductoryapplicationsQ2.10Eigenvalueproblems.Astandardprobleminmanybranchesofappliedmathe-maticsandphysicsistofindtheeigenvalues(andeigenfunctions)ofappropriateproblemsbasedonordinarydifferentialequations.Intheseexamples,findthefirsttwotermsintheasymptoticexpansionsofboththeeigenvaluesandtheeigenfunctions;foreachusetheasymptoticsequence(a)(b)(c)Q2.11Breakdownofasymptoticsolutionsofdifferentialequations.Theseordinarydiffer-entialequationsdefinesolutionsonthedomainwithconditionsgivenonx=1.Ineachcase,findthefirsttwotermsinanasymptoticsolutionvalidforx=O(1)aswhichallowstheapplicationofthegivenboundarycondition(s).Show,ineachcase,thattheresultingexpansionisnotuniformlyvalidasfindthebreakdown,rescaleandhencefindthefirstterminanasymptoticexpansionvalidnearx=0,matchingasnecessary.Finallyfind,foreachproblem,thedominantasymptoticbehaviourofas(a)(b)(c)(d)(e)Q2.12Anotherbreakdownproblem.SeeQ2.11;repeatfortheproblembutshowthat,forarealsolutiontoexist,thedomainiswhereandthenfindthedominantasymptoticbehaviourofasQ2.13BreakdownasFindthefirsttwotermsinanasymptoticsolution,validforx=O(1)asofwithNowshowthatthisexpansionisnotuniformlyvalidasfindthebreakdown,rescaleandfindthefirsttwotermsinanexpansionvalidforlargex,matchingasnecessary.Show,also,thatthis2-termexpansionbreaksdownforevenlargerx,butdonottaketheanalysisfurther. 107Q2.14BreakdownasII.SeeQ2.11(a)and(c);fortheseequationsandbound-aryconditions,andtheasymptoticsolutionsalreadyfoundforx=O(l),takethedomainnowtobeHenceshowthattheexpansionsarenotuni-formlyvalidasfindthebreakdown,rescaleandthenfindthefirsttermsintheexpansionsvalidforlargex,matchingasnecessary.Q2.15ProblemE2.7reconsidered.Findthefirsttwotermsinanasymptoticexpansion,validforx=O(1)asofwithShowthat,formally,thisrequirestwomatchedexpansions,butthattheasymptoticsolutionobtainedforx=O(1)correctlyrecoversthesolutionforDi.e.itisuniformlyvalid.(Notethebalanceofterms,whenscalednearx=0!)Q2.16Scalingofequations.SeeQ2.11andQ2.14;usethedominanttermsonly,validforx=O(1),togetherwithappropriatescalingsassociatedwiththerelevantbalanceofterms,toanalysetheseequations.Compareyourresultswiththescalingsobtainedfromthebreakdownoftheasymptoticexpansions.Q2.17Boundary-layerproblemsI.Findthefirsttwotermsinasymptoticexpansions,validforx=O(1)(awayfromtheboundarylayer)asforeachoftheseequations,withthegivenboundaryconditions.Then,foreach,findthefirsttermintheboundary-layersolution,matchingasnecessary.(YoumaywishtouseyourexpansionstoconstructcompositeexpansionsvalidforD,tothisorder.)(a)(b)(c)(d)(e)Q2.18Boundary-layerproblemsII.SeeQ2.17;repeatforthesemoreinvolvedequations.(a)(b)(c)(d)(e)(f) 1082.IntroductoryapplicationsQ2.19Twoboundarylayers.ThefunctionisdefinedbytheproblemwithFindthefirsttwotermsinanasymptoticexpan-sionvalidforx=O(1),asawayfromx=0andx=1.Henceshowthat,inthisproblem,boundarylayersexistbothnearx=0andnearx=1,andfindthefirsttermineachboundary-layersolution,matchingasnecessary.Q2.20Aboundarylayerwithinathinlayer.ConsidertheequationforwithFindthefirsttermsineachofthreeregions,twoofwhicharenearx=0,matchingasnecessary.(Here,onlytheinner-mostregionisatrueboundarylayer;theotherissimplyascaled-thin-connectingregion.)Q2.21Boundarylayersortransitionlayers?Decideiftheseequations,onthegivendomains,possesssolutionswhichmayincludeboundarylayersortransitionlayers;givereasonsforyourconclusions.(a)(b)(c)(d)(e)Q2.22Transitionlayernearafixedpoint.Intheseproblems,atransitionlayerexistsatafixedpoint,independentoftheboundaryvalues.Find,for(a),thefirsttwoterms,andfor(b)thefirsttermonly,inanasymptoticexpansion(asvalidawayfromthetransitionlayerandthefirsttermonlyofanexpansionvalidinthislayer;matchyourexpansionsasnecessary.(a)(b)Q2.23Boundarylayerortransitionlayer?(ThisexampleisbasedontheonewhichisdiscussedcarefullyandextensivelyinKevorkian&Cole,1981&1996.)TheequationisforandgivenSupposethatatransitionlayerexistsnearfindtheleadingtermsintheasymptoticexpansionsvalidoutsidethetransitionlayer,andinthetransitionlayer.Hencededucethatatransitionlayerisrequiredifandareofopposite 109signandandthenfindtheleadingtermsinalltherelevantregionsfor:(a)(b)(c)Q2.24Transitionlayersandturningpoints.ConsidertheequationintroduceandfindachoiceofwhichproducesanequationforintheformandidentifyF.IfFchangessignonthedomainofthesolution,thenthepointwherethisoccursiscalledaturningpoint;findtheequationthatdefinestheturningpointsinthecaseQ2.25Transitionlayerataturningpoint.Considertheequationfindthepositionoftheturningpointandscaleintheneighbourhoodofthetransitionlayer.Writedownthegeneralsolution,toleadingorder,validinthetransitionlayer,as(ThissolutionisbestwrittenintermsofAiryfunc-tions.AuniformlyvalidsolutionisusuallyexpressedusingtheWKBmethod;seeChapter4.)Q2.26Higher-orderequations.Fortheseproblems,findthefirsttermsonlyinasymptoticexpansionsvalidineachregionofthesolution,for(a)(b)(c)Q2.27Verticalmotionundergravity.Consideranobjectthatisprojectedverticallyup-wardsfromthesurfaceofaplanetarybody(or,rather,forexample,fromourmoon,becausewewillassumenoatmosphereinthismodel).Theheightabovethesurfaceisz(t),wheretistime,andthisfunctionisasolutionofwhereRisthedistancefromthecentreofmassofthebodytothepointofprojection,andgistheappropriate(constant)accelerationofgravity.(Forourmoon,Theinitialconditionsarefindtherelevantsolutionintheformt=t(z)(andyoumayassumethat 1102.Introductoryapplications(a)Writeyoursolution,andthedifferentialequation,intermsofthenon-dimensionalvariablesandintroducetheparameterSupposethatthelimitofinterestis(whichyoumaycaretointerpret);findthefirsttwotermsofanasymptoticex-pansion,validfor(intheformdirectlyfromthegoverningequation.(Youshouldcomparethiswiththeexpansionoftheexactsolution.)Fromyourresults,findapproximationstothetimetoreachthemaximumheight,thevalueofthisheightandthetimetoreturntothepointofprojection.(b)Abettermodel,formotionthroughanatmosphere,isrepresentedbytheequationwhereisaconstant.(Thisisonlyarathercrudemodelforairresistance,butithastheconsiderableadvantagethatitisvalidforbothandNon-dimensionalisethisequationasin(a),andthenwritewhereRepeatallthecalculationsin(a).(c)Finally,inthespecialcase(i.e.theescapespeed),findthefirstterminanasymptoticexpansionvalidasNowfindtheequationforthesecondtermandaparticularintegralofit.Ontheassump-tionthattherestofthesolutioncontributesonlyanexponentiallydecayingsolution,showthatyourexpansionbreaksdownatlargedistances;rescaleandwritedown—donotsolve—theequationvalidinthisnewregion.Q2.28Earth-moon-spaceship(1D).InthissimplemodelforthepassageofaspaceshipmovingfromtheEarthtoourmoon,weassumethatboththeseobjectsarefixedinourchosencoordinatesystem,andthatthetrajectoryisalongthestraightlinejoiningthetwocentresofmass.(Morecompleteandaccuratemodelswillbediscussedinlaterexercises.)Wetakex(t)tobethedistancemeasuredalongthislinefromtheEarth,andthenNewton’sLawofGravitationgivestheequationofmotionaswheremisthemassofthespaceship,andthemassesoftheEarthandMoon,respectively,Gistheuniversalgravitationalconstantanddthedistancebetweenthemasscentres.Non-dimensionalisethisequation,usingdasthedistancescaleandasthetimescale,togivethenon-dimensionalversionoftheequation(xandtnownon-dimensional)as 111whereWewillconstructanasymptoticsolution,foras(Theactualvalues,fortheEarthandMoon,giveandatrajectoryfromsurfacetosurfacerequiresapproximately.)Writedownafirstintegraloftheequation.(a)Findthefirsttwotermsinanasymptoticexpansionvalidforx=O(1),byseeking(cf.Q2.27),andusethedataasandwrite(Here,isthenon-dimensionalinitialspeedawayfromtheEarth,issmallandtheconditiononkensuresthatthespaceshipreachestheMoon,butnotatsuchahighspeedthatitcanescapetoinfinity.)Showthatthisexpansionbreaksdownas(b)Seekascalingofthegoverningequationintheneighbourhoodofx=1bywriting(whichisconsistentwiththesolutionobtainedin(a),wherethefirstterm,providesthedominantcontributionatx=1).Findthefirstterminanasymptoticex-pansionofmatchtoyoursolutionfrom(a)andhencedetermine(Bewarnedthatlntermsappearinthisproblem.)Q2.29Eigenvaluesforavibratingbeam.The(linearised)problemofanelasticbeamclampedateachendisforwithwhereistheeigenvalue(whicharisesfromthetime-dependence),andYoung’smod-ulus.Findthefirstterminanasymptoticexpansionoftheeigenvalues.(Thisproblemcanbesolvedexactly,andthentheexponentsexpandedforthisisanalternativethatcouldbeexplored.)Q2.30Heattransferin1D.Anequationwhichdescribesheattransferinthepresenceofaone-dimensional,steadyflow(Hanks,1971)iswithtemperatureconditionsFindthefirsttwotermsinanasymptoticexpansion,validforx=O(1)asandtheleadingtermvalidintheboundarylayer,matchingasnecessary.Q2.31Self-gravitatingannulus.Aparticularmodelforthestudyofplanetaryringsisrepresentedbytheequation 1122.Introductoryapplicationswhereandareconstants,withthedensity,satisfy-ing(ThisexampleisbasedonthemoregeneralequationgiveninChristodoulou&Narayan,1992.)For(thenarrowannulusapproximation),introduceandthenwritethedensityasfindthefirstthreetermsinanasymptoticex-pansionforP.Onthebasisofthisinformation,deducethatyourexpansionwouldappeartobeuniformlyvalidforQ2.32Anelasticdisplacementproblem.Asimplifiedversionofanequationwhichde-scribesthedisplacementofa(weakly)nonlinearstring,inthepresenceofforcing,whichrestsonanelasticbed,iswhereisaconstant,withFindthefirsttwotermsinanasymptoticexpansion,forandusethisevidencetodeducethatthisexpansionwouldappeartobeuniformlyvalidforQ2.33Laminarflowthroughachannel.Amodelforlaminarflowthroughachannelwhichhasporouswalls,throughwhichsuctionoccurs,canbereducedtowhereisanarbitraryconstantofintegration,with(ThisistakenfromProudman,1960;seealsoTerrill&Shrestha,1965,andMcLeodinSegur,etal.,1991;here,thestreamfunctionisproportionaltothefunctionand1/(Reynolds’Number).)AssumethatA(0)existsandisnon-zero,andthenfindthefirstterminanasymptoticexpansionforandforvalidforx=O(1),andthenthefirsttwotermsvalidintheboundarylayer(thefirstbeingsimplytheboundaryvaluethere).Q2.34Sliderbearing.Thepressure,p,withinthefluidfilmofasliderbearing,basedonReynolds’equation,canbereducedtotheequationwritteninnon-dimensionalform;here,isaconstantandisthegiven(smooth)filmthickness,with(andFindthefirsttwotermsinanasymptoticexpansion,forvalidforx=O(1),andthenthefirsttermonlyintheboundarylayer,matchingasnecessary.(Thefirsttermintheboundarylayercanbewrittenonlyinimplicitform,butthisissufficienttoallowmatching.)Q2.35Anenzymereaction.Theconcentration,ofoxygeninanenzymereactioncanbemodelledbytheequation 113withwhereisaconstant.Theboundaryconditionsspecifytheconcentrationonr=1andthatthefluxofoxygenmustbezeroatr=0;theseareexpressedas(a)Findthesizeoftheboundarylayernearr=1(intheformforsuitableandhenceshowthatsatisfieswherewehaveassumedthatandas(whichisconsistentwiththeequation).(b)Fromtheresultin(a),deducethatisexponentiallysmallasfor1–r=O(1),seekasolution(whichisexponentiallysmall)intermsofthescaledvariableandshowthatwheresatisfiesSolvethisequation,applytherelevantboundarycondition,matchandhenceshowthatwhereisaconstant(independentofwhichshouldbeidentified. Thispageintentionallyleftblank 3.FURTHERAPPLICATIONSTheideasandtechniquesdevelopedinChapter2havetakenusbeyondelementaryapplications,andasfarassomemethodsthatenableustoconstruct(asymptotic)solutionsofafewtypesofordinarydifferentialequation.Theaimnowistoextendthesemethods,inparticular,topartialdifferentialequations.Thefirstreactiontothisproposalmightbethatthemovefromordinarytopartialdifferentialequationsisaverybigstep—anditcancertainlybearguedthusifwecomparethesolutions,andmethodsofsolution,forthesetwocategoriesofequation.However,inthecontextofsingularperturbationtheory,thisisamisleadingpositiontoadopt.Withoutdoubtwemusthavesomeskillsinthemethodsofsolutionofpartialdifferentialequations(albeitusuallyinareduced,simplifiedform),butthefundamentalideasofsingularperturbationtheoryareessentiallythesameasthosedevelopedforordinarydifferentialequations.Theonlyadjustment,becausethesolutionwillnowsitinadomainoftwoormoredimensions,isthatanappropriatescalingmayapply,forexample,inonlyonedirectionandnotintheothers,orintimeandnotinspace.Inthischapterwewillexaminesomefairlystraightforwardproblemsthatarerepre-sentedbypartialdifferentialequations,startingwithanexampleofaregularproblem.Theapproachthatweadoptwillemphasisehowthemethodsforordinarydifferentialequationscarryoverdirectlytopartialdifferentialequations.Inaddition,wewilltaketheopportunitytowritealittlemoreaboutmoreadvancedaspectsofthesolutionofordinarydifferentialequations,inpartasapreparationfortheverypowerfulandgeneralmethodsintroducedinChapter4. 1163.Furtherapplications3.1AREGULARPROBLEMAsimple,classicalprobleminelementaryfluidmechanicsisthatofuniformflowofanincompressible,inviscidfluidpastacircle.(Thisistakenasatwo-dimensionalmodelforacircularcylinderplacedintheuniformflow.)Representedasacomplexpotential(w),thesolutionofthisproblemcanbewrittenaswhereisthevelocitypotential,thestreamfunction,Utheconstantspeedoftheuniformflow(movingparalleltothex-axis)andaistheradiusofthecircle,centredattheorigin.Intermsofcomplexpotentials,thissolutionisconstructedfromthepotentialfortheuniformflow(Uz)andthatforadipoleattheoriginthecomplexvariableisBothandsatisfyLaplace’sequationintwodimensions:and,ifweelecttoworkwiththestreamfunction(asisusual),thenwehaveexpressedinplanepolarcoordinates.Wenowformulateavariantofthisproblem:uniformflowpastaslightlydistortedcircle.Letthedistortedcircleberepresentedbywherewillbeoursmallparameter;intermsoftheproblemistosolve(wheresubscriptsdenotepartialderivatives)withandThecondition(3.4)ensuresthatthereistheprescribeduniformflowatinfinityi.e.as(see(3.1)),and(3.5)statesthatthesurfaceofthedistortedcircleisastreamline(designatedoftheflow.Wesee,immediately,thatthere 117isasmallcomplicationhere:isembeddedinthesecondboundarycondition,(3.5).Inordertouseourfamiliarmethods,wemustfirstreformulatethiscondition.Weassume(andthismustbecheckedattheconclusionofthecalculation)thatoncanbeexpandedasaTaylorseriesaboutr=ai.e.Nowtheproblem—albeitviaaboundarycondition—containstheparameterinaformwhichsuggeststhatwemayseekasolutionforbasedontheasymptoticsequenceThuswewriteandthenwithandfrom(3.6):andsoon.Theproblemforispreciselythatfortheundistortedcircle,soasgivenin(3.1).TheproblemfornowbecomesthatoffindingasolutionofLaplace’sequa-tionwhichsatisfiesandasThemostnaturalwaytoproceedistorepresentasaFourierSeries,andthenasolutionforfollowsdirectlybyemployingthemethodofseparationofvariables.Asaparticularlysimpleexampleofthis,letussupposethatandsotherelevantsolutionisthen 1183.FurtherapplicationsandsowehaveThistwo-termasymptoticexpansionisclearlyuniformlyvalidforandforanditisalsoanalyticinthisdomain,sotheuseoftheTaylorexpansiontogenerate(3.6)isjustified.Itisleftasanexercise(Q3.1)tofindthenextterminthisasymp-toticexpansionandthentodiscussfurtherthevalidityoftheexpansion;itisindeeduniformlyvalid.AfewotherexamplesofregularexpansionsobtainedfromproblemsposedusingpartialdifferentialequationscanbefoundinQ3.2andQ3.3.Thisexercisehasdemonstratedthat,aswithanalogousproblemsbasedonordinarydifferentialequations,wemayencounterasymptoticexpansionsthatareessentiallyuniformlyvalidi.e.theproblemisregular.However,thisisverymuchararity:mostproblemsthatwemeet,andthatareofinterest,turnouttobesingularperturbationproblems.Wenowdiscussthisaspect,inthecontextofpartialdifferentialequations,andhighlightthetwomaintypesofnon-uniformitythatcanarise.3.2SINGULARPROBLEMSIThemoststraightforwardtypeofnon-uniformity,aswehaveseenforordinarydiffer-entialequations,ariseswhentheasymptoticexpansionthathasbeenobtainedbreaksdownandtherebyleadstotheintroductionofanew,scaledvariable.Thissituationistypicalofsomewavepropagationproblems,forwhichanasymptoticexpansionvalidneartheinitialdatabecomesnon-uniformforlatertimes/largedistances.Indeed,thegeneralstructureofsuchproblemsisreadilycharacterisedbyanexpansionoftheformwherecisthespeedofthewaveandthefunctionsf,gandharebounded(andtypicallywell-behaved,oftendecayingforHowever,forasolutiondefinedin(whichisexpectedinwaveproblems),weclearlyhaveabreakdownwhenirrespectiveofthevalue(size)of(x–ct).Thuswewouldneedtoexaminetheprobleminthefar-field,definedbythenewvariables(Inthisexample,wehaveusedx–ct(c>0)forright-runningwaves;correspondingly,forleft-runningwaves,wewouldworkwithx+ct.)Wepresentanexampleofthistypeofsingularperturbationproblem.E3.1Nonlinear,dispersivewavepropagationAmodelequationwhichdescribessmall-amplitude,weaklydispersivewaterwavescanbewrittenas 119wherewehaveagainusedsubscriptstodenotepartialderivatives,andherethespeed(associatedwiththeleftsideoftheequation)isone.(ThisequationisusuallycalledtheBoussinesqequationandithappenstobeoneoftheequationsthatiscompletelyintegrable,inthesenseofsolitontheory,forallseejohnson,1997,andDrazin&Johnson,1992.)Ourintentionistofindanasymptoticsolutionof(3.9),validassubjecttotheinitialdatawhereas(and,further,allrelevantderivativesoff(x)satisfythissamerequirement).Theequationfor(whereuistheamplitudeofthewave),onusingourfamiliar‘iteration’argument,suggeststhatwemayseekasolutionintheformforsomexs(distance)andsomets(time).Thusweobtainfrom(3.9)(where‘=0’meanszerotoallordersinwhichgivesandsoon.Theinitialdata,(3.10),thenrequiresandEquation(3.12a)istheclassicalwaveequationwiththegeneralsolution(d’Alembert’ssolution)forarbitraryfunctionsFandG;applicationoftheinitialdatafor(givenin(3.13))thenproduces(Wenote,inthisexample,thattheparticularinitialdatawhichwehavebeengivenproducesawavemovingonlytotheright(withspeed1),atthisorder.) 1203.FurtherapplicationsTheequationfor(3.12b),ismostconvenientlywrittenintermsofthecharacteristicvariablessothatwehavetheoperatoridentitiesThus(3.12b)becomeswhichcanbeintegrateddirectlytogivewhereJandKarearbitraryfunctions.Theinitialdata,(3.14),requiresandorwhereAisanarbitraryconstant.ThesetwoequationsenableustofindJandKashencethesolutionforbecomeswherewehavewrittenForconvenience,letussetthenwehavetheasymptoticsolution(tothisorder):Thistwo-termasymptoticexpansion,(3.17),containstermsf,Handallofwhichareboundedanddecayastheirargumentsapproachinfinity(becauseofourgivenf(x)),andsothereisnonon-uniformityassociatedwiththese.However,ifwe 121followtheright-goingwave(byselectinganyx–t=constant)then,astincreasesindefinitely,wewillencounterabreakdownwhenThisleadsustointro-duceanewvariableandotherwisewemayuse(becausewearefollowingtheright-runningwave)andweobservethatnoscalingisassociatedwiththisvariable.Thuswetransformfrom(x,t)variables(thenear-field)tovariables(thefar-field),i.e.andsoouroriginalequation,(3.9),becomeswhereThenatureoftheappearanceofinthisequationisidenticaltotheoriginalequation,suggestingthatagainwemayseekasolutionintheformwhichgivesandsoon.Thisequationcanbeintegratedonceinand,whenweinvokedecayconditionsatinfinity(i.e.allaswhichmirrorsourgivenconditionsonf(x)),weobtainThisequationisverydifferentfromourpreviousleading-orderequation(inthenear-field):thatwassimplytheclassicalwaveequation.Ourdominantequationinthefar-field,(3.21),describesthetimeevolutionofthewaveintermsofthewave’snonlinearityanditsdispersivecharacterEquation(3.21)isavariantofthefamousKorteweg-deVries(KdV)equation;itssolutions,andmethodofsolution,initiated(fromthelate1960s)theimportantstudiesinsolitontheory.Forsolutionsthatdecayasitcanbedemonstratedthatthelatertermsintheasymptoticexpansioncontributeuniformlysmallcorrectionstoas(Thisisafar-from-trivialexerciseandisnotundertakeninthistext;theinterestedreaderwhowishestoexplorethisfurthershouldconsultanygoodtextonwavepropagatione.g.Whitham,1974.)Insummary,wehaveseenthatthisexample(whichwehaveworkedthroughrathercarefully)describesapredominantlylinearwaveinthenear-field(wheret=O(1)orsmaller)but,for(thefar-field),thewaveis,toleadingorder,describedbyanonlinearequation.Equation(3.21)canbesolvedexactlyand,further,wemay 1223.Furtherapplicationsimposeinitialdataatintermsofmatching(firsttermtofirstterm),thisisequivalenttosolving(3.21)withas(whereFisasuitablearbitraryfunction)andhence,fortheright-goingwave,weselectandthesolutionisuniformlyvalidfori.e.Thisuniformvaliditygeneratedbythefar-fieldsolutionisanunlooked-forbonus,butitshouldberememberedthatthebasicexpansion(3.11)isnotuniformlyvalid,andsowecertainlyhaveasingularperturbationproblem.Otherexamplesofwavepropagationproblems,whichexhibitabreakdownandcon-sequentrescaling,aresetasexercisesinQ3.4–3.8.Inaddition,wepresentonefurtherexamplewhichembodiesthissamemathematicalstructure,butwhichisalittlemoreinvolved.However,thisisaclassicalproblemwhichshouldappearinanystandardtextand,further,ithasvariousdifferentlimitsthatareofpracticalinterest(andsomeofthesewillbediscussedlater;seealsoQ3.10andQ3.11).E3.2Supersonic,thin-aerofoiltheoryWeconsiderirrotational,steady,supersonicflowofacompressiblefluidpasta(two-dimensional)thinaerofoil.Theequationsofmassandmomentumconservationcanbereducedtothesingleequationwhereaisthelocalsoundspeedinthegas,andthevelocityisThecorrespondingenergyequation(Bernoulli’sequation)iswhereandasandisaconstantdescribingthena-tureofthegas:pressure(anisentropicgas).Theaerofoilisdescribedbywithwheretheupper/lowersurfacesaredenotedby+/–.UsingUandtonon-dimensionalisethevariables,eliminatingandthenwritingtheresultingnon-dimensionalvelocitypotentialasweobtainwhereistheMachnumberoftheoncomingflowfrominfinity.Theaerofoilisnowwrittenas(whichdefinesforthisproblem)andsoensuresthatwehaveathinaerofoil.(Thusimpliesthatthereisnoaerofoilpresent,andthenthenon-dimensionalvelocitypotentialissimplyxi.e. 123Figure4.Sketchoftheuniformflow(speed=1,Machnumber>1)pasttheaerofoileverywhere.)Thenon-dimensionalversionofthephysicalconfigurationisshowninfigure4.Finally,theboundaryconditionsareandwherethefirst,(3.23),ensuresthat,upstream,wehaveonlythegivenuniformflowwithandthesecond,(3.24),statesthatthesurfaceoftheaerofoilisastreamlineoftheflow.Equation(3.22),withandispredominantlyawaveequation(buthereexpressedinspatialvariables,xandy);theformofthisequationsuggeststhatweseekanasymptoticsolutionandthenweobtainandsoon.Correspondingly,theboundaryconditionsgiveand 1243.FurtherapplicationsThissecondconditionisrewrittenasanevaluationony=0,byassumingthatpossessesaTaylorexpansion(cf.§3.1),sowenowobtainorandsoon.Itisnotpossibletosolve,inanysimplecompactway,forboththeupperandlowersurfacestogether(aswillbecomeclear),sowewillconsideronlytheuppersurface,(Theproblemforthelowersurfacefollowsasimilar,butdifferent,construction.)Thegeneralsolutionofequation(3.26),forthecaseofsupersonicflowisgivenbyd’Alembert’ssolution:whereandFandGarearbitraryfunctions.ThecontributiontothesolutionfromG,intheupperhalf-plane,extendsintox<0andso,inordertosatisfy(3.23),wemusthave(ThecontributionfromFextendsintox>0fory>0.)Condition(3.29a)(uppersurface)nowrequiresthatandsotowithinanarbitraryconstant,whichmaybeignoredinthedeterminationofavelocitypotential(becausesuchaconstantcannotcontributetothevelocityfield).Thuswehaveiny>0.(Itwillbenotedthatthecorrespondingprobleminthelowerhalf-planerequirestheretentionofwithThesolutionofequation(3.27),forisapproachedinthesamewaythatweadoptedforthesimilarexerciseinE3.1:weintroducecharacteristicvariables,whichhereareWiththesevariables,equation(3.27)becomes 125for(wherewehaveusedoursolutionforandotherwisewhichgeneratesonlythezerosolutionwhentheboundaryconditionsareapplied.Theboundaryconditionsrelevanttothesolutionof(3.31)are(from(3.28)and(3.29b))andbecauseoftheformofthislatterboundarycondition,itisalittlemoreconvenienttofindtheasymptoticsolutionfor(Ofcourse,fromthisitisthenpossibletodeducebothandifthesearerequired;seelater.)From(3.31)weobtaindirectlythatwhereJandKarearbitraryfunctions,andcondition(3.32)thenrequiresthatcf.thesolutionforTheboundarycondition(3.33)issatisfiedifandthenweobtaindirectly(becausewemayuse(forwherewehavewrittenThetwo-termasymptoticexpansionforisthereforeasforandItisclear,forandbounded,thattheexpansion(3.35)breaksdownwhere—thefar-field.(ThissamepropertyisexhibitedbytheexpansionsforandTheassumptionthatwehaveandbounded(andcorrespondingly,andforthelowersurface),forimpliesthattheseaerofoilsaresharpatboththeleadingandtrailingedges—whichiscertainlywhatisaimedforintheirdesignandconstruction.However,iftheseedgesaresuitablymagnifiedthanitwillbecomeevidentthatarealaerofoilmusthaveroundededgesonsomescale.Thisinturnimpliesthatstagnationpointsexist,whereandthentheasymptoticexpansion,(3.35),certainlycannotbeuniformlyvalidneartoandevenfory=O(1):aboundary-layer-typestructureisrequirednearandnear 1263.FurtherapplicationsThis,andotheraspectsofcompressibleflowpastaerofoils,willbeputtoonesideinthecurrentdiscussion(butsomeadditionalideasareaddressedinQ3.10and3.11).Here,wewillworkwiththemathematicalmodelfortheflowinwhichandarebounded,sothattheonlynon-uniformityarisesasThesolutioninthefar-fieldiswrittenintermsofthevariablesandwithEquation(3.22)nowbecomeswhere,forsimplicity,wehavewrittendownonlytheleadingtermsontheright-handsideoftheequation.WeseekasolutionandthensatisfiesthenonlinearequationorandnotethatThegeneralsolutionforiswhereHisanarbitraryfunction;solution(3.39)provides,forgeneralH,anim-plicitrepresentationonlyfor(whichhasfar-reachingconsequences,asweshallseeshortly).WedetermineHbymatching,andthisismosteasilydonebymatchingand—butforthiswerequiretheexpansionofItisleftasanexercisetoshowthat,from(3.35),weobtainwhichgives 127whenweretaintheO(1)termonly.From(3.39),wefindthatwherewehaveretainedtheO(1)andterms(andwehaveassumedthatHpossessesasuitableTaylorexpansion).Thesetwoexpansions,(3.41)and(3.42),matchpreciselywhenweselectandthenWeconcludethisimportantexamplebymakingafewobservations.First,thebehaviourofthefar-fieldsolution,(3.43),asrecoversthenear-fieldsolution,andso(tothisorder,atleast)thefar-fieldsolutionisuniformlyvalidin(althoughouroriginalexpansion,(3.25),exhibitsasingularbehaviour).Thesolution(3.43),asYincreases,canbecompletelydescribedbythecharacteristiclines,intheformalongwhichTheselinesarethereforestraight,butnotpar-allel;theyfirstintersectforsome(whichdependsonthedetailsofthefunctionandthenthesolutionbecomesmulti-valuedin—whichisunacceptable,unlesswereverttoanintegralformofequation(3.38)andthenadmitadiscontinuoussolution.Thisdiscontinuity,atadistancefromthesurfaceoftheaerofoil,man-ifestsitselfinthephysicalworldasheraldingtheformationofashockwave.Itshouldbenosurprisethatthecharacteristicvariableplaysasignificantrôleinthesolutionofthiswave-type(hyperbolic)equation;indeed,theresultsdescribedherecanbeobtainedbyseekingasymptoticexpansionsforthese,ratherthanforthefunctionsthemselves(seeQ3.9).Afinalpoint,whichembodiesanimportantidea,istonotethatthenear-fieldsolutiontakesessentiallythecorrectform(toleadingorder)evenforthefar-field,inthesensethatthesolutionisreplacedbywhereistheappropriateapproximationtothecharacteristicvariable.Onewaytointerpretthisistoregardasthecorrectsolution,butthatitisinthe‘wrongplace’i.e.itisnotconstantalonglinesbutrather,alonglinesWehaveseen,inthesetwosomewhatroutineexamples,andthesimilarproblemsintheexercises,howthesimplesttypeofsingularperturbationproblemcanarise.Theotherfundamentallydifferentproblemthatwemayencounter,justasforordinarydifferentialequations,iswherethesmallparametermultipliesthehighestderivative: 1283.Furtherapplicationstheboundary-(ortransition-)layerproblem.Wenowturntoanexaminationofthisclassicalsingularperturbationprobleminthecontextofpartialdifferentialequations.3.3SINGULARPROBLEMSIITherearetwopartialdifferentialequation-typesthatareoftenencounteredwithsmallparametersmultiplyingthehighestderivative(s):theellipticequation(e.g.Laplace’sequation)andtheparabolicequation(e.g.theheatconduction,ordiffusion,equation).Thesetwo,togetherwiththewaveequation(i.e.ofhyperbolictype)discussedin§3.2,completethesetofthethreethatconstitutestheclassificationofsecond-orderpartialdifferentialequations.Thetwonewequationsareexemplifiedbyrespectively.Ofcourse,theuseofsingularperturbationmethodstosolvethesepartic-ularexamplesissomewhatredundant,becauseweareabletosolvethemexactly(forusingstandardtechniques.Thuswewilldiscusstwosimple—butnotcompletelytrivial—extensionsofthesebasicequations.E3.3Laplace’sequationwithnonlinearityWearegoingtofindanasymptoticsolution,foroftheequationwhereuisprescribedontheboundaryoftheregion.First,wewillmakeafewgeneralobservationsaboutthisproblemandthenobtainsomeofthedetailsinaparticularcase.Theappearanceofinequation(3.44)suggeststhatwemayseekasolutionwhichturnsouttobeconsistentwiththematchingrequirementstotheboundarylayers;thusweobtainandsoon.Immediatelyweseethatwecanfindwhichsatisfiesthegivendataonx=0andx=a,butonlyinveryspecialcircumstanceswillthisalsosatisfythedataony=0andy=b:thesolutionwill(ingeneral)requireboundarylayersneary=0andneary=b.Nosuchlayersexistnearx=0andx=a.Allthisfollowsfromthetermthetermsimplycontributesa(small)nonlinearadjustment 129tothesolution.Toproceed,letusbegiven,asanexample,thespecificdata:andwewillfindthefirsttwotermsinthesolutionvalidawayfromtheboundarylayer,andthenthefirsttermsineachboundary-layersolution.From(3.46a),weobtainwhereAandBarearbitraryfunctions;theavailabledata,(3.47),requiresthatandsowehaveThenextterm,satisfies(3.46b)withEquation(3.46b)canbewrittenandsothesolutionforisandwehavetheasymptoticsolutionThistwo-termasymptoticexpansiondoesnotsatisfythegivendataony=0orony=1;thuswerequire(thin)layersnearthesetwoboundariesofthedomain.Thefirststageinvolvesfindingthesizeoftheboundarylayers;letusintroducewithasfortheboundarylayerneary=0.Further,wenotethatu=O(1)hereand,ofcourse,thereisnoscalinginx.Thus,withequation(3.44)becomes 1303.FurtherapplicationsandwemustchooseorsimplyandthenseekasolutionwithThematchingcondition,from(3.98),givessoThesimplestsolutionavailableforinvolvesusingthemethodofseparationofvariablesandso,notingtheboundaryvalueony=0,i.e.Y=0,givenin(3.48),wewriteThusweobtainwhereCandDarearbitraryconstants;aboundedsolutionvalidawayfromthebound-arylayerrequiresC=0andthenasTheconditiononY=0isthensatisfiedifweselectD=1,andthereforethesolutionintheboundarylayerneary=0isNeary=1,theboundarylayerisclearlythesamethickness,soherewewriteandsettoobtaintheequationcf.equation(3.50)withThematchingconditionthistime,againfrom(3.98),isweseekasolutionwherewithThesolutionisobtainedaltogetherroutinelybywritingsothatwith 131whichproduceswheretheconstantsaredeterminedasthecoefficientsoftheFourier-seriesrep-resentationofthisisleftasanexercise.Thusthesolutionintheboundarylayerneary=1iswheretheareknown.Thiscompletes,forourpurposes,theanalysisofthisboundary-layerproblem.Thisexamplehasadmirablydemonstrated,wesubmit,howtheideasofsingularper-turbationtheory(hereexhibitedbytheexistenceofboundarylayers)developedforordinarydifferentialequations,carryoverdirectlytopartialdifferentialequations.Theboundarylayershavebeenrequiredinthey-direction,byvirtueofthepresenceoftheparameterbutnotinthex-direction.Ofcourse,themethodofsolutionhasrequiredsomeknowledgeofthemethodsforsolvingpartialdifferentialequations,butthatwastobeexpected;otherwisenoothercomplicationshaveariseninthecalculations.Foroursecondexample,weconsideraphysically-basedproblem:heatconduction,andsothegoverningequationwillnowbeparabolic.E3.4HeattransfertofluidflowingthroughapipeWeconsideracircularpipeofradiusr=1(wewilldescribethisproblem,fromtheoutset,intermsofnon-dimensionalvariables),extendinginastraightlineintheroleplayedbythelengthofthepipewillbediscussedlater.Throughthepipeflowsafluid,withaknownvelocityprofilerepresentedbyu=u(r),theequationforthetemperature,ofthefluidiswherewehaveassumednovariationintheangularvariablearoundthepipe.Thenon-dimensionalparameter,isproportionaltothethermalconductivityofthefluid.Weseekasolutionofequation(3.53),forsubjecttotheboundaryconditionsThefluidentersthepipe(atx=0)withaninitialtemperaturedistributionandthetemperatureofthepipewall(r=1)isprescribedalongthepipei.e.heat 1323.Furtherapplicationsistransmittedto(orpossiblylostby)thefluidasitflowsalongthepipe.(Notethat,inordertoavoidadiscontinuityintemperatureatthestartofthepipe—whichisnotanessentialrequirementintheformulationoftheproblem—thenwemusthaveFinally,observethatmultipliesthehighestderivativeterms,sowemustexpectaboundary-layerstructure.Wewillchoosethevelocityprofiletobethatassociatedwithalaminar,viscousflowi.e.andthenweseekasolutionwherewehavebeencarefulnottocommitourselvestothesecondterminthisasymptoticexpansion.Thuswehave,from(3.53),whenweinvoketheboundarycondition(3.54a);thissolutionis,apparently,validforallbut(ingeneral)itcannotpossiblyaccommodatetheboundaryconditiononr=1inx>0,(3.54b).Thisobservation,togetherwiththeformofthegoverningequation,(3.53),suggeststhatweneedaboundarylayernearr=1;letussetwithasandwriteThusequation(3.53)becomesand,aswemustusethebalance(usingthe‘old’term/‘new’termconcept):whichissatisfiedbythechoiceandsowehaveWeseekasolutionofthisequationintheformsothatwithandamatchingconditionforAlthoughitispossibletofindtheappropriatesolutionof(3.57),satisfyingthegivenboundaryconditions,itissomewhatinvolvedandwearelikelytolosemuchofthetransparencyoftheresults.Thuswewillcompletethesolutioninthespecialcase:constantwalltemperaturex>0,andwewillseekasolutioninx>0,therebyignoringthediscontinuitythatisevidentas 133(Thisdifficultyinthesolutioncanbediscussedseparately—butnothere—orthediscontinuitycanbereplacedbyasmoothfunctionthattakesfromatNow,sincewemustfindasolutionof(3.57),subjecttoToproceedwiththesolutionofthisproblem,wewriteandsothisissolvedverysimplybyintroducingthesimilaritysolution.Setforsomem,thenwefinddirectlythatm=–1/3andwhereAandaarearbitraryconstants,andthenalltheboundaryconditionsaresatisfiedbyThusthesolutionintheboundarylayer,R=O(1),isinthisspecialcase.Wehavethefirsttermsineachoftheouterregion(awayfromthepipewall)andtheregionclosetothepipewall(theboundarylayer):(3.55b)and(3.58),respectively.Wehavecompletedallthedetailedcalculationsthatwewillpresent,althoughwemakeafewconcludingcommentsthathighlightsomeintriguingissues.First,knowingthetemperatureinthefluidnearthepipewallenablesustofindtheheattransferredtoorfromthefluid,ifthatisapropertyofparticularinterest.Second,amoretechnicalmatter:whatistheformofthesolutionintheouterregion,i.e.whattermsintheasymptoticexpansionmustbepresentinordertomatchtotheboundary-layersolution?Toanswerthis,weneedtoknowthebehaviouroftheboundary-layersolutionas(becausewehavefor1–r=O(1)asForoursimilaritysolution,(3.58),thiscanbeobtainedbyusinga 1343.FurtherapplicationsFigure5.Schematicoftheboundary-layerregion,whichgrowsfromthepipewall,andthe‘fullydeveloped’regionforandlarger.suitableintegrationbyparts:Thuswewillneedtomatchtoasforx=O(l)and1–r=O(l).Thatis,theasymptoticexpansionvalidawayfromtheboundarylayermustincludetheexponentiallysmalltermoftheformforsuitablefunctionsandg(r,x).(Itturnsoutthatgsatisfiesanonlinear,firstorderpartialdifferentialequation,andthatisgovernedbyalinear,firstorderpartialdifferentialequation,withcoefficientsdependentong.)ThisresulthasparticularlydramaticconsequencesinthecaseforthenthesolutionawayfromthepipewallisapparentlyexactlyHowever,therequirementtomatchtotheboundary-layersolutionintroducesanexponentiallysmallcorrectiontotheoutersolution—andthisisthesoleeffectofthepresenceofthedifferenttemperatureatthepipewall,atleastforx=O(1).Finally,weaddresstheproblem-andthereisone-associatedwiththelengthofthepipe.Aswehavejustcommented(andwewillrelateallthisonlytothesimplecaseofthesimilaritysolutionwithifthetotallengthofthepipeis 135asmeasuredintermsofthentheflowawayfromthewallist=1,withonlyexponentiallysmallcorrections.Theboundarylayerremainsthinalongthefulllengthofthepipe,butnotethatlinesofconstanttemperature,emanatingfromtheneighbourhoodofthewall,arethelinesi.e.Thus,definingtheboundary-layerthicknessintermsofaparticulartemperature,thisthicknessincreasesasxincreases,althoughitremainsHowever,ifthepipeissolongthatthentheexponentialtermin(3.59)becomesO(1),andthetemperaturenowhasanO(1)correction.Inotherwords,theO(1)temperatureatthepipewallhascausedheattobeconductedthroughthefluidtoaffectthewholepipeflow,Indeed,weseethatthescalinginouroriginalequation,(3.53),balancesdominanttermsfrombothsidesoftheequationforthereisnolongeraboundarylayeratthewall;thisisshownschematicallyinfigure5.Thetwoexamplespresentedabovehaveshownhowthenotionofaboundarylayer,asdevelopedforordinarydifferentialequations,isrelevanttopartialdifferen-tialequations—andessentiallywithoutanyadjustmenttothemethod;somesimilarexamplescanbefoundinexercisesQ3.12–3.14.Wehavenowseenthetwobasictypesofproblem(breakdownandrescale;selectascalingrelevanttoalayer),althoughtheequationsthatwehaveintroducedasthevehiclesforthesedemonstrations—quitedeliberately—havebeenrelativelyuncomplicated.Weconcludethissectionwithanexamplethat,ultimately,possessesasimpleperturbationstructure(asin§3.2),butwhichinvolvesasetoffour,coupled,nonlinearpartialdifferentialequations.Asbe-fore,thepurposeoftheexampleistoexhibitthepower(andinherentsimplicity)ofthesingularperturbationapproach.E3.5Unsteady,one-dimensionalflowofaviscous,compressiblegasTheflowofacompressiblegas,withtemperaturevariationsandviscosity,isdescribedbytheequationswhicharetheequationsofmomentum,massconservation,energyandstate,respec-tively.Here,isthecoefficientof(Newtonian)viscosity,thethermalconductivity,thegasconstantandisassociatedwiththeisentropic-gasmodel(seeE3.2);weshalltakealltheseparameterstobeconstant.Anymovementofthegasisinthedirection—weuseprimesheretodenotephysicalvariables—withnovariationinother 1363.Furtherapplicationsdirections,soanydisturbancegeneratedinthegasisassumedtopropagate(andpossiblychange)onlyinistime.Thespeedofthegasisanditspressure,densityandtemperatureareandrespectively.First,wesupposethatthegasinitsstationaryundisturbedstateisdescribedbyallconstant.Thegasisnowdisturbed,therebyproducingaweakpressurewave(oftencalledanacousticwave,althoughthisisusuallytreatedwithtemperaturefixed);thesizeoftheinitiatingdisturbancewillbemeasuredbytheparameterWeintroducethesoundspeed,ofthegasinitsundisturbedstate,definedbyandthenwemovetonon-dimensionalvariables(withouttheprimes)bywritingweletatypicalorappropriatelengthscale(e.g.anaveragewavelength)beandalsodefineThegoverningequations,(3.60)–(3.63),thereforebecomewheretheReynoldsNumberisandthePrandtlNumberiswith(Notethatwehaveelectedtodefinethespeedinthedefi-nitionofaswhichisproportionaltothescaleofthespeedgeneratedbythedisturbance;asuitablechoiceofisacrucialstepinensuringthatweobtainthelimitofinterest.) 137Wenowseekasolutionofthesetofequations(3.65)–(3.68),forandfixedasbywritingwhereq(andcorrespondinglyrepresentseachofp,Tandu.Thefirsttermsineachoftheseasymptoticexpansionssatisfytheequationswhichfollowfrom(3.65)–(3.68),respectively.Theseequationsthengiveandthen,selectingtheright-goingwave(forsimplicity),wehaveforsomef(x)att=0.Forasolution-setwhichrecoverstheundisturbedstateforwealsohaveHowever,ourexperiencewithhyperbolicproblems(see§3.2)isthatasymptoticex-pansionslike(3.69)arenotuniformlyvalidast(orx)forx–t=O(1)i.e.inthefar-field.Letusinvestigatetheresultofthenon-uniformitydirectly,withoutexaminingthedetailsofthebreakdown(whichisleftasanexercise).Thevariablesthatwechoosetouseinthefar-fieldareforeachq,sowehavetheidentitiesequations(3.65)–(3.68)become(whenweretainonlythosetermsrelevanttothedeterminationofthedominantcontributionstoeachQ): 1383.FurtherapplicationswhereAgainweexpandforeachQ;immediatelyweseethattheleading-ordertermssatisfy(cf.(3.71)),butthesefunctionsareotherwiseunknown.Thetermsaredefinedbytheequationsfrom(3.72)–(3.75),respectively,andthislastequationisusedonlytodefineThefirstthreeequationsinvolvethecombinationsofterms:respectively,andsowemayeliminateallofandbetweenthem,whichwedo.Theresultingsingleequationinvolvingandiswrittenintermsofonefunction—say—byusing(3.77);thisgivestheleadingterm(forU)inthefar-fieldasthesolutionofThusisdescribedbyanequationinwhichthetimeofthesolutioniscontrolledbybothnonlinearityanddissipationcf.the 139KdVequation,(3.21).Equation(3.78),liketheKdVequation,isalsoanimportantequation;itistheBurgersequation(Burgers,1948)whichcanbesolvedexactlybyapplyingtheHopf-Coletransformation(Hopf,1950;Cole,1951)whichtransformstheequationintotheheatconduction(diffusion)equation.AswiththecorrespondingKdVproblem(inE3.1),thematchingconditionisasand,forsuitabletheasymptoticexpansions(3.76)areuniformlyvalidasThesolutionthatwehaveobtaineddescribesaweakpressurewavemovingthroughthenear-field(t=O(1))and,astincreasesintothefar-field,thewave-frontsteepens,butthiseffectiseventuallybalancedbythediffusion(whenFinallythewavewillsettletosomesteady-stateprofile—aprofilewhichisregardedasamodelforashockwaveinwhichthediscontinuityissmoothed;seeQ3.5.Thisconcludesallthatweshallpresenthere,asexamplesoffairlyroutinesingularper-turbationproblemsinthecontextofpartialdifferentialequations.Furtherideas—verypowerfulideas—whichareapplicabletobothordinaryandpartialdifferentialequa-tionswillbedevelopedinthenextchapter.Wecompletethischapteronsomefurtherapplicationsbyexaminingtworathermoresophisticatedproblemsthatinvolveordinarydifferentialequations.Thefirstemploystheasymptoticexpansioninaparameterinordertostudyanimportantequationinthetheoryofordinarydifferentialequations:Mathieu’sequation.Theseconddevelopsatechnique,whichisanextensionofoneofourearlierproblemsassociatedwithwavepropagation,thatenablestheasymptoticso-lutionofcertainordinarydifferentialequationstobewritteninaparticularlycompactandusefulform—indeed,onethatexhibitsuniformvaliditywhennoneappearstoexist.3.4FURTHERAPPLICATIONSTOORDINARYDIFFERENTIALEQUATIONSTheMathieuequation,forx(t),whereandareconstants,hasalongandexaltedhistory;itarosefirstintheworkofE-L.Mathieu(1835–1900)ontheproblemofvibrationsofanellipticalmembrane.Italsoappliestotheproblemoftheclassicalpenduluminwhichthepivotpointisoscil-latedalongaverticalline,oneresultbeingthat,forcertainamplitudesandfrequenciesofthisoscillation(whichcorrespondstocertainandthependulumbecomesstableintheupposition!Itisalsorelevanttosomeproblemsinelectromagnetic-wavepropagation(inamediumwithaperiodicstructure),someelectricalcircuitswithspe-cialoscillatorypropertiesandincelestialmechanics.TheequationisconventionallyanalysedusingFloquettheory(seee.g.Ince,1956)inwhichthesolutionforingeneral,acomplexconstant,hasy(t)periodic(withperiodorforcertainWewillshowhowsomeofthepropertiesofthisequationarereadilyaccessible,atleastinthecase 1403.FurtherapplicationsE3.6Mathieu’sequationforWeconsidertheequationwhereweusetheover-dottodenotethederivativewithrespecttot;specialcurvesintheseparatestable(oscillatory)fromunstable(exponentiallygrowing)solutions:onthesecurvesthereexistbothoscillatoryandlinearlygrowingsolutions.WewillseekthesecurvesinthecaseFirst,withweobtainwhichhasperiodicsolutions,withperiodoronlyif(n=0,1,2...)althoughn=0mightbethoughtanunimportantexceptionalcase.Theformoftheequationsuggeststhatweshouldseekasolutionintheformandinvoketherequirementthateachbeperiodic;eachisaconstantinde-pendentofWewillexplorethecasesn=0andn=1(andthecasen=2isleftasanexercise).(a)Casen=0Equation(3.79),with(3.80a,b),becomeswhere,asisourconvention,‘=0’meanszerotoallordersinThuswehavethesequenceofequationsandsoon.Theonlyperiodicsolutionfor—andtriviallyso—iswhichwewillnormaliseto=1.Thenequation(3.81b)becomesandthesolutionforwhichisperiodicrequiresthus 141whereAisanarbitraryconstant.Finally,fromequation(3.81c),weobtainwhichhasaperiodicsolutionforonlyifandsothemethodproceeds.Thusthetransitionalcurveinthealongwhichaorperiodicsolutionexists,is(b)Casen=1Thistime,althoughthegeneralprocedureisessentiallythesame,theappearanceofnon-trivialperiodicsolutionsatleadingordercomplicatesthingssomewhat.Equation(3.79)nowbecomesandthusweobtaintheequationsandsoon.Thegeneralsolutionofequation(3.83a)issimplyforarbitraryconstantsAandB;wemaynowproceed,collectingalltermspro-portionaltoA,andcorrespondinglytoB,butitisfareasier(andmoreusual)totreatthesetwosetsoftermsseparately.ThusweselectA=1,B=0,andA=0,B=1;thiswillgeneratetwotransitionalcurves:oneassociatedwithandonewithwhichistheusualpresentationadopted.Letuschoosethen(3.83b)canbewrittenandaperiodicsolutionforrequires(becauseotherwisetherewouldbeatermproportionaltotthereforeweobtain 1423.FurtherapplicationswhereCandDarearbitraryconstants.Finally,from(3.83c),wehavetheequationforwhichperiodicsolutions,require(andD=0).Thus,tothisorder,thetransitionalcurveforitisleftasanexercisetoshowthat,withthechoicethenInsummary,wehavethestabilityboundariesgivenbySetasanexercise,thecasen=2canbefoundinexerciseQ3.15.Acorrespondingcal-culationtothosedescribedhere,butnowformulatedinawayconsistentwithFloquettheory,issetinQ3.16.MoreinformationaboutMathieuequationsandfunctions,andtheirapplications,isavailableintheexcellenttext:McLachlan(1964).Forourfinaldiscussioninthischapter,wewillincorporatetheideaintroducedattheendofE3.2,namely,a‘correct’solutioninthe‘wrongplace’,butappliedheretoordinarydifferentialequations.Wewilldescribethemethodofstrainedcoordinates,whichhasalonghistory;itwasusedfirst,inanexplicitway,byPoincaré(1892),butotherauthorshadcertainlybeenawareiftheideaearlier,inoneformoranother.SomeauthorsrefertothisasthePLKmethodafterPoincaré,Lighthill(1949)andKuo(1953).Theideaisexactlyasmentionedabove:thesolution(say),whichisnotuniformlyvalid,ismadesobywritingwhereisasuitablychosenstrainedcoordinate.Ofcourse,onlyrelativelyspecialproblemshavesolutionsthatpossessthisstructure,butitisregardedasasignificantimprovement—overmatchedexpansions—whenitoccurs.Indeed,wehavemetaproblemofthistypeinChapter1:E1.1,ourveryfirstexample.Therewefoundthatastraightforwardasymptoticexpansionledto 143(seeequation(1.7)),buttheuseofthe‘strained’coordinate(see(1.8))immediatelyremovesthenon-uniformitythatisotherwisepresentasFurther,theleadingtermabove(~sint)isessentiallycorrect,but‘inthewrongplace’;iftisreplacedbythensinbecomesauniformlyvalidfirstapproximationasWewillshowhowthismethoddevelopsforaparticulartypeofequation(firstexaminedbyLighthill,1949&1961;similarexampleshavebeendiscussedbyCarrier,1953&1954).E3.7AnordinarydifferentialequationwithastrainedcoordinateasymptoticstructureWeconsidertheproblem(andpossessesthepropertythatitcanbeexpandeduniformlyforasthisisessentiallytheproblemdiscussedbyLighthill(1949).Theideasaresatisfactorilypresentedinaspecialcase(whichleadstoamoretransparentcalculation);wechoosetoexaminetheproblemwithwhereandareconstantsindependentofThus(3.84)and(3.85)becomeWewillstartbyseekingtheconventionaltypeofasymptoticsolution,asintheformandthensatisfieswhichproducesthegeneralsolutionwhereAisanarbitraryconstant.Itisimmediatelyclearthat,withthissolutionisnotdefinedonx=0,althoughwemayusetheboundaryconditiononx=1, 1443.Furtherapplicationswhichwechoosetodo,togive:ThebehaviourofisdiscussedinQ3.18,whereitisshownthattheasymptoticexpansionisnotuniformlyvalidasitbreaksdownwhereHere,wewillapproachtheproblemoffindingasolutionbyintroducingastrainedcoordinate.Thestrainedcoordinate,isdefinedbywhich,ifthisexpansionisuniformlyvalidonthedomainin(thatcorrespondstomaybeinvertedtofindnotethat,if(3.89)isuniformlyvalid,thenforallxonthedomain.Thesolutionweseekisnowwrittenintermsofthestrainedcoordinateasandthereasonforusing(3.89),ratherthanbecomesevidentwhenweseethatwetransformoftheoriginalproblemintofunctionsof,andderivativeswithrespectto,onlyThus,withtheequationin(3.86)canbewrittenwhere,asinourpreviousconvention,‘=0’meanszerotoallordersinFromequation(3.91)weobtainandsoon.Becausewehavedefined(3.89)witheachtheboundarycondi-tiononx=1becomessimply 145Thesolutionof(3.98),with(3.94),producesexactlythesolutionobtainedearlier,(3.88),butnowexpressedintermsofratherthanx:Wenowturntothevexingissueofsolving(3.93)—anditisvexingbecausethisisoneequationintwounknownfunctions:andHowdoweproceed?Theaimofthisnewtechniqueistoobtainauniformlyvalidasymptoticexpansion—ifthatisatallpossible—for1],althoughwehaveyettodetermine(whichcorrespondstox=0).Ifthereistobeanychanceofsuccess,thenwemustremoveanytermsthatgeneratenon-uniformitiesintheasymptoticexpansionforfromQ3.18,itisclearthattheonlysuchtermhereisi.e.Thuswedefinesoastoremovethistermfromtheequationforitissufficienttoremovesuchsingulartermsinanysuitablemanner,butifitispossibletochoosesoastoleaveanhomogeneousequationforthenthisistheusualmove.(Otherchoicesproducedifferentasymptoticrepresentationsofthesamesolution,butallequivalenttoagivenorderinHere,therefore,weelecttowritetheequationforasleavingAnimmediateresponsetothisprocedureistoobservethatthetermthatcausesallthedifficulty,hasnowappearedasaforcingtermintheequationfor—soallwehavesucceededindoingismovingthenon-uniformityfromoneasymptoticexpansiontoanother!Asweshallsee,thisisindeedthecase,butthenon-uniformityintheexpansionforthestrainedcoordinateisnotassevereasthatintheexpansionfory.Beforeweaddressthiscriticallyimportantissue,wemaynote,from(3.97),thatwhereBisanarbitraryconstant;butfrom(3.94)weseethatwerequireB=0andsoFurther,ifthissamestrategyforselectingtheequationsforeachisadopted,thenforeveryn=1,2,...,andtheexactsolution,intermsofbecomesItistypicaloftheseproblemsthatitisnotnecessarytosolvecompletelytheequationsforeachitissufficienttoexaminethenatureofthesolutionsasThuswewillemploythesameapproachasdescribedinQ3.18.First,from(3.92),wesubstitute 1463.Furtherapplicationsforinto(3.96)andthencanceleverywhere),togiveandthen,forthiscanbewrittenThisiseasilyintegratedtoproducethesolutionwheretheterminvolvingthearbitraryconstantofintegrationissuppressedbecauseitislesssingularthanthetermretained.Thustheasymptoticexpansion(3.89)becomeswhichexhibitsabreakdownwhereexactlyasfor(3.87)(seeQ3.18)—justwhatwemostfeared!Butthereisaveryimportantdifferencehere:theoriginalexpansionbrokedownwhereandwestillrequiredasolutionvalidonx=0;nowwehaveabreakdownatbut,becausex=0correspondstoobtaineddirectlyfrom(3.99),werequiretobenosmallerthanOfcourse,theburningquestionnowis:areweallowedtouse(3.99)withandhencedefineTheanswerisquitesurprising.WehaveseenthatwhereistheconstantitisfairlystraightforwardtoshowthatwheretheareconstantsboundedasThustheasymptoticexpansion(3.89),forthestrainedcoordinate,becomes 147andthisseriesconvergesforwhereissomeconstantindependentofHencetheexpansionconvergesfor—butistobenosmallerthanwhichislargerthanThustheasymptoticexpansionwith(whichmapstoisnotjustuniformlyvalid—itisconvergent!Thisisanaltogetherunlooked-forbonus;themethodofstrainedcoordinates,inthisexample,hasprovedtobeaverysignificantimprovementonourstandardmatched-expansionsapproach.SomefurtherexamplesthatmakeuseofastrainedcoordinatearesetasexercisesQ3.19–3.25.Althoughourexample,E3.7,hasdemonstrated,tothefull,theadvantagesofthestrainedcoordinatemethod,notallproblemsarequitethissuccessful.ManyordinarydifferentialequationsofthetypediscussedinE3.7doindeedpossessconvergentseriesforthecoordinate—sothecompletesolutionisnolongersimplyasymptotic—butotherproblemsmayproduceastrainedcoordinatethatisuniformlyvalidonly(withoutbeingconvergent).Intheexercises,thequestionofconvergenceisnotexplored(but,ofcourse,theinterestedandskilfulreadermaywishtoinvestigatethisaspect).Inthistext,atthisstage,wehaveintroducedmanyoftheideasandtechniquesofsingularperturbationtheory,andhaveappliedthem—inthemain—toordinaryandpartialdifferentialequationsofvarioustypes.Inthenextchapterwepresentonefurthertechniqueforsolvingsingularperturbationproblems.Thisisamethodwhichsubsumesmostofwhatwehavepresentedsofarandis,probably,thesinglemostpowerfulapproachthatwehaveavailable.Whenthishasbeencompleted,wewillemployallourmethodsintheexaminationofaselectionofexamplestakenfromanumberofdifferentscientificfields.FURTHERREADINGAfewregularperturbationproblemsthataredescribedbypartialdifferentialequationsarediscussedinvanDyke(1964,1975)andalsoinHinch(1991).Adiscussionofwavepropagationandbreakdown,andespeciallywithreferencetosupersonicflowpastthinaerofoils,canbefoundinvanDyke(above),Kevorkian&Cole(1981,1996)andinHolmes(1995).Anygoodtextoncompressiblefluidmechanicswillcovertheseideas,andmuchmore,fortheinterestedreader;wecanrecommendCourant&Friedrichs(1967),Ward(1955),Miles(1959),Hayes&Probstein(1966)andCox&Crabtree(1965),buttherearemanyotherstochoosefrom.Anicecollectionofpartialdifferentialequationsthatexhibitboundary-layerbehaviourarepresentedinHolmes(1995).MostofthestandardtextsthatdiscussmoregeneralaspectsofsingularperturbationtheoryincludeMathieu’sequation,andrelatedproblems,asexamples;good,dedicatedworksonordinarydifferentialequationswillgiveabroadandgeneralbackgroundtothe 1483.FurtherapplicationsMathieuequation(suchasInce,1956).Alotofdetail,withanalyticalandnumericalresults,includingmanyapplications,canbefoundinMcLachlan(1964).ThemethodofstrainedcoordinatesisdescribedquiteextensivelyinvanDyke(1964,1975),Nayfeh(1973),Hinch(1991)and,withonlyslightlylessemphasis,inKevorkian&Cole(1981,1996).EXERCISESQ3.1Flowpastadistortedcircle.Findthethirdtermintheasymptoticexpan-sion,foroftheproblemdescribedbywithandforHencewritedowntheasymptoticsolutiontothisorderandobservethat,formallyatleast,thereisabreakdownwhereDeducethatthesolutioninthenewscaledregionisidentical(totheappropriateorder)tothatobtainedforr=O(1),theonlyadjustmentbeingtheorderinwhichthetermsappearintheasymptoticexpansion(andsotheexpansioncanberegardedasregular).Useyourresultstofindanapproximationtothevelocitycomponentsonthesurfaceofthedistortedcircle.Q3.2Weakshearflowpastacircle.Cf.Q3.1;nowweconsideraflowwithconstant,smallvorticitypastacircle.Lettheflowatinfinitybewhichhastheconstantvorticity(inthetheproblemisthereforetosolvewithasandSeekasolutionfindthefirsttwotermsand,onthebasisofthisevidence,showthatthisconstitutesatwo-term,uniformlyvalidasymptoticexpansion.Indeed,showthatyourtwo-termexpansionistheexactsolutionoftheproblem.Q3.3Potentialfunctionoutsideadistortedcircle.(Thisisequivalenttofindingthepotentialoutsideanearlycircular,infinitecylinder.)Weseekasolution,oftheproblemwith 149whereWritefindandthendetermineinthecaseOnthebasisofthisevi-dence,confirmthatyouhaveatwo-term,uniformlyvalidasymptoticexpansioninQ3.4Theclassical(model)Boussinesqequationsforwaterwaves.Theseequationsarewrit-tenasthecoupledpairwhereisthehorizontalvelocitycomponentintheflow,thesurfacedisplacementi.e.thesurfacewave,andisaconstantindependentofFindthefirsttermsintheasymptoticexpansionsforx=O(1),t=O(1),as(thenear-field).Thenintroducethefar-fieldvariables:forright-runningwaves,andhencefindtheequationsdefiningtheleadingorder;showthattheequationforutakestheformtheKorteweg-deVriesequation.Showthatasolutionofthisequationisthesolitarywavewhereisafreeparameter.Q3.5Long,small-amplitudewaveswithdissipation.Amodelforthepropagationoflongwaves,withsomecontributionfromdissipation(damping),iswhereandarepositiveconstantsindependentofFollowthesameprocedureasinQ3.4(near-fieldthenfar-field,althoughheretheright-goingcharacteristicwillbeShowthat,inthefar-field,theleadingterm,foru(say),satisfiesanequationoftheformtheBurgersequation.Showthatthisequationhasasteady-stateshock-profilesolutionforsuitableconstantsCand(>0),whichshouldbeidentified.(ThissolutionisusuallycalledtheTaylorshockprofile;Taylor,1910.) 1503.FurtherapplicationsQ3.6Anonlinearwaveequation.AwaveisdescribedbytheequationwhereSeekasolution(forright-runningwavesonly)intheformfindthefirsttwotermsanddemonstratetheexistenceofabreakdownwhenNowintroduceandshowthattheleadingtermintheexpansionvalidforsatisfiestheequationThiscalculationisnowextended:defineandthendeterminefandgsothatwheresatisfies(*)andsatisfiesthecorrespondingequationforleft-goingwaves.(YoumayassumethatbothfandgpossessTaylorexpansionsaboutx+tandx–t,respectively.)Q3.7Amulti-wavespeedequation.Aparticularwaveprofile,u(x,t;withasisdescribedbytheequationwhereandcareconstants(independentofShowthat,ifthenonthetimescalethewavemovingatspeedandthewavemovingatspeedeachdecayexponentially(intime),toleadingorderasShow,also,thatonthetimescalethewavemovingatspeedchasdiffusedadistanceaboutthewavefrontand,toleadingorder,itsatisfiesaBurgersequation(seeQ3.5).Q3.8Waterwaveswithweaknonlinearity,dampinganddispersion.Thepropagationofaone-dimensionalwaveonthesurfaceofwatercanbemodelledbytheequations 151whereisthehorizontalvelocitycomponentintheflow,andisthesurfacewave;cf.Q3.4.Findthefirsttermsinthenear-fieldexpansionsofandasandthenobtaintheequationfortheleadingterminvalidinthefar-fieldYoushouldconsideronlyright-goingwaves.(TheequationthatyouobtainhereisaKorteweg-deVries-Burgers(KVB)equation;seeJohnson,1997.)Q3.9Supersonic,thin-aerofoiltheory:characteristicapproach.Thecharacteristicsforequa-tion(3.22)canbedefinedbytheequationwhereisthestreamlinedirection(sothatandistheinclinationofthecharacteristicrelativetothestreamline(sothattanwhereMisthelocalMachNumber).Showthatandhencededucethat,onthecharacteristics,Finally,sincetoleadingordershowthatandconfirmthatthisisrecoveredfromequation(3.39).Q3.10Thinaerofoilinatransonicflow.Showthattheasymptoticexpansion(3.35)isnotuniformlyvalidas(a)Setwriteandandhencededucethatascalingconsistentwithequations(3.22)and(3.24)isandthatthensatisfies,toleadingorder,(b)Giventhatusethescalingin(a)toshowthatthereisadistinguishedlimitinwhichwhatnowistheequationfortoleadingorder?Q3.11Thinaerofoilinahypersonicflow.See(3.35);showthatthisexpansionbreaksdownaswhenIntroduceleavexun-scaledandwriteshowthattermsfromboththeleft-handandright-handsidesofequation(3.22)areofthesameorderinthecaseforaparticularchoiceofWhatistheresulting 1523.Furtherapplicationsleading-orderequationfor[Thisresultcanonlygiveaflavourofhowthingsmightproceedbecause,withstrongshockwavesmostcer-tainlyappearandthenapotentialfunctiondoesnotexist.Toinvestigatethisproperly,weneedtoreturntotheoriginalgoverningequations,withouttheisentropicassumption.]Q3.12Asymmetricalbendingofapre-stressedannularplate.Thelateraldisplacement,ofaplateisdescribedbytheequation(writteninnon-dimensionalvariables)andcorrespondstoweakbendingrigidity.TheannularplateisdefinedbywiththeboundaryconditionswhereisaconstantindependentofSeekasolutionfindtheequationforuandthenfindthefirsttwotermsineachoftheasymptoticexpansionsofasvalidawayfromtheboundariesoftheregion,andinthetwoboundarylayers(nearMatchyourexpansionsasnecessary.[Formoredetails,seeNayfeh,1973.]Q3.13Anonlinearellipticequation.ThefunctionsatisfiestheequationanditisdefinedinTheboundaryconditionsareUsetheasymptoticsequenceandhenceobtainthefirsttwotermsintheasymptoticexpansionvalidawayfromtheboundariesx=0andy=0;thissolutionisvalidony=1.Nowfindthefirsttermonlyintheasymptoticexpansionvalidintheboundarylayerneary=0,havingfirstfoundthesizeofthelayer;matchasnecessary.Repeatthisprocedurefortheboundarylayernearx=0andshowthat,toleadingorder,nosuchlayerisrequired.However,deducethatoneisneededtoaccommodatetheboundaryconditionatWritethesolutioninthisboundarylayeras(writteninappropriatevariables)andformulatetheproblemfortheleadingtermintheasymptoticexpansionforV,butdonotsolveforV. 153Q3.14Thesteadytemperaturedistributioninasquareplate.Thetemperature,inaplateisdescribedbytheheatconductionequation(writteninnon-dimensionalvariables)wheretheplateiswiththetemperatureontheboundarygivenbyandSeekthefirsttermofacompositeexpansionbywritingwhereistherelevantboundary-layervariable.DeterminecompletelyandandthenshowthatwhereQ3.15Mathieu’sequationforn=2.SeeE3.6;findtheasymptoticexpansionofasfarastheterminthecasen=2(andtherewillbetwoversionsofthis,dependingonthechoiceofeithersinorcos).Q3.16Mathieu’sequationbasedonFloquettheory.WritetheMathieuequationasanequationinbysettingwhereisaconstant;isperiodicwithperiodor(NotethatthetransitionalcurvesarenowSeekasolutioninthecasen=1;becausewenowhavethesolutionwillbethatwhichisvalidnearthetransitionalcurves.Showthatwhereisafreeparameter. 1543.FurtherapplicationsQ3.17AparticularHillequation.(Hill’sequationisageneralisationoftheMathieuequation.)ConsidertheequationwhereisaconstantindependentofseekasolutionwheretheareindependentofImposetheconditionthatandaretobeperiodic;whatcondition(s)mustandsatisfy?Q3.18MatchedexpansionappliedtoE3.7.Considerwith(andconstantsindependentofFindtheequationforthesecondtermoftheexpansion,andfromthisdeducethatthiscanbedonebyfirstapproximatingtheequationforbeforeintegratingit—seethemethodleadingtoequation(3.96).Henceshowthattheasymptoticexpansionforbreaksdownwhererescalexandyintheneighbourhoodofx=0,andthenfindandsolvetheequationdescribingthedominantterminthisregion(matchingasnecessary).Whatisthebehaviourofasbasedonyoursolution?Q3.19Astrained-coordinateproblemI.(ThisisaproblemintroducedbyCarrier,1953.)FindanasymptoticsolutionofasintheformFindanduseyoursolutiontofindthedominantbehaviourofwhere(YoumayassumethattheasymptoticexpansionofthecoordinateisuniformlyvalidonQ3.20Astrained-coordinateproblemII.SeeQ3.19;followthesameprocedurefortheproblemasUseyourresultstoshowthat 155Q3.21Astrained-coordinateproblemIII.SeeQ3.20;followthissameprocedureforasShowthataswhereistheappro-priatesolutionoftheequation(whichdoespossessonerealroot).Q3.22Astrained-coordinateproblemIV.SeeQ3.19;followthissameprocedurefortheproblemas(Youmayobservethatthisproblemcanbesolvedexactly.)Q3.23AstrainedcoordinateproblemV.SeeQ3.19;followthissameprocedurefortheproblemasIftheboundaryconditionhadbeenwiththesamedomain,brieflyinvestigatethenatureofthisnewproblem.Q3.24Duffing’sequation.Theequationforthemotionofasimplependulum,withouttheapproximationforsmallanglesofswing,takestheformIfxissmall,andweretaintermsasfarasweobtainanequationlikethisisDuffing’sequation(Duffing,1918)whichwasintroducedtoimprovetheapproximationforthesimplependulum(withoutthecomplicationsofworkingwithsinx).Seekasolutionofthisequation,forandbyusingastrained-coordinateformulation:wheretheareconstants.Determinethesolutionasfarastermsinchoosingeachinordertoensurethatthesolutionisperiodic. 1563.FurtherapplicationsQ3.25Weaklynonlinearwavepropagation.Awavemotionisdescribedbytheequationwith(whichensures,certainlytoleadingorder,thatwehaveonlyaright-goingwave).Seekasolutionintheformandhencefindthesolutioncorrectat 4.THEMETHODOFMULTIPLESCALESThefinalstageinourpresentationoftheessentialtoolsthatconstitutesingularper-turbationtheoryistoprovideadescriptionofthemethodofmultiplescales,arguablythemostimportantandpowerfultechniqueatourdisposal.Theidea,asthetitleimplies,istointroduceanumberofdifferentscales,eachone(measuredintermsofthesmallparameter)associatedwithsomepropertyofthesolution.Forexample,onescalemightbethatwhichgovernsanunderlyingoscillationandanotherthescaleonwhichtheamplitudeevolves(asinamplitudemodulation).Indeed,thistypeofproblemisthemostnaturalonewithwhichtostart;wewillexploreaparticularlysimpleexampleandusethisasavehicletopresentthesalientfeaturesofthemethod.However,beforeweembarkonthis,onewordofwarning:thisprocessnecessarilytransformsalldifferen-tialequationsintopartialdifferentialequations—evenordinarydifferentialequations!Thiscouldwellcausesomeanxiety,butthecomfortingnewsisthattheunderlyingmathematicalproblemisnomoredifficulttosolve.So,forexample,anordinarydif-ferentialequation,subjectedtothisprocedure,involvesanintegrationmethodthatisessentiallyunaltered;theonlyadjustmentissimplythatarbitraryconstantsbecomearbitraryfunctionsofalltheothervariables.4.1NEARLYLINEAROSCILLATIONSWewillshowhowtheseideasemergeinthisclassofrelativelysimpleproblems;indeed,westartwithanexampleforwhichanexactsolutionexists.Letusconsiderthelinear, 1584.ThemethodofmultiplescalesFigure6.(a)Upperfigureisaplotofthefunctionforwith(b)LowerfigureisaplotofthefunctionforwithdampedoscillatorwhichisgovernedbywithwhereThisproblemcanbesolvedexactly(andhavingthesolutionavailablewillhelpinitiatethediscussion);thesolutioniswherethe‘e’subscriptdenotes‘exactsolution’.Thissolutionrepresentsanoscillation,withafixedperiod,andwithanamplitudewhichdecaysexponentially,albeitslowly.(Thistypeofsolutionisdepictedinfigure6a,andanotherfunctionwithadifferentmodulationisshowninfigure6b.)Nowthissolution,(4.2),hasthreeimportantcharacteristics:first,itisanoscillationcontrolledby(usuallycalledthefastscale);second,theamplitudedecaysslowlyaccordingto(usuallycalledtheslowscale);third,evenifweexpressthesolutionintermsofTanditwillstillrequireanasymptoticexpansion,asbyvirtueofthefactorinthedenominator.Anyconstructionofanasymptoticsolutiondirectlyfrom(4.1)mustaccommodatealltheseelements. 159Figure7.Sketchofthewiththelineincluded.Theappearanceoftwotimescalesin(4.2)isquiteclear:T(fast)and(slow),sowecouldwritethesolutionasTheunderlyingideainthemethodofmultiplescalesistoformulatetheoriginalproblemintermsofthesetwoscalesfromtheoutsetandthentotreatfunctionoftwovariables;thiswillleadtoapartialdifferentialequationforX.Clearly,Tandarenotindependentvariables—theyarebothproportionaltot—sowehave,apparently,asignificantmathematicalinconsistency.How,therefore,doweproceedwithanyconfidence?Themethodandphilosophyaresurprisinglystraightforward.Weseekanasymptoticsolutionforasasafunctionwithitsdomainin2-space;thisiscertainlymoregeneralthanintheoriginalformulation.TheaimistoobtainauniformlyvalidexpansioninandThiswill,typically,requireustoinvokeperiodicity(andboundedness)inT,andboundedness(anduniformity)inIfweareabletoconstructsuchanasymptoticsolution,itwillbevalidthroughoutthequadrantinBecausethesolutionisvalidinthisregion,itwillbevalidalonganyandeverypaththatwemaywishtofollowinthisregion;inparticular,itwillbevalidalongthelinewhichisthestatementthatTandaresuitablyrelatedtot.Thisimportantinterpretationisrepresentedinfigure7,andthisideaisattheheartofallmultiple-scaletechniques.Wewillnowapplythismethodto(4.1),presentedasaformalexample. 1604.ThemethodofmultiplescalesE4.1Alinear,dampedoscillatorGivenwithweintroducewheretheareconstants,andwritewhereeachistobeperiodicinTand,wehope,uniformlyvalidasandas(NotethatthechoiceofT,in(4.5),followsexactlythepatternofastrainedcoordinate;cf.Q3.24.)From(4.5)weobtaintheoperatoridentityandsoequation(4.3)becomeswhere,asusual,‘=0’meanszerotoallordersinWhenweinsert(4.6)forX,weobtainthesequenceofequations 161andsoon.Theinitialconditions,(4.4),becomewhichgiveetcetera.Itisnowaverystraightforwardexercisetosolveeachoftheseproblems,insequence.Thegeneralsolutionof(4.7a)isalthoughitisrathermoreconvenienttowritethisaswhereandarearbitraryfunctions;werequire,from(4.8),whichwesatisfybyselectingTheequationfor(4.7b),cannowbewrittenwhere,forsimplicity,wehavewrittenButfortobeperiodicinT,alltermsontheright-handside,inandmustvanish(forotherwisewewillgenerateparticularintegralsforlikethisremovalofsecular(i.e.non-periodic)termsrequiresThesolutionof(4.11a)andthen(4.11b),with(4.10),givesdirectly 1624.ThemethodofmultiplescalesleavingThus,atthisstage,wehavetheasymptoticsolutionbutandare,asyet,unknown.Beforeweproceedtoexaminetheequationforitisinstructivetonote,in(4.13),thattheoscillatorycomponentofthesolutiondependsonThus,atthisorder,vanishesidenticallyandsowemayjustaswellsetthisistheusualsimplificationthatisadoptedintheseproblems.Thereasonforthisredun-dancyreadilybecomesclear:inthedefinitionofthefastscale,thetermcouldbewrittenandthensubsumedintothegeneralofthesolution.Thustosetfromtheoutsetispermitted.Theequationforfrom(4.7c)andwithbecomeswheretomakeplainalltheterms(sinT,cosT)whichgeneratesecularbehaviour,wefurtherwritethisasThusisperiodicinTifwithinitialconditions(from(4.48))andso 163Equation(4.15)canbewrittenwhereisanarbitraryconstant;but(4.16)thenrequiresEquation(4.14)canbewritteninasimilarfashion:anditfollowsthatthesolutionforwillcontainatermunlessWemustmakethischoiceinordertoavoidanon-uniformityas(becausethissecondtermintheasymptoticexpansionwillgrowlikerelativetothefirst).Thusweareleftwithandhence,noting(4.17),wehave(andthearbitraryisnowunim-portant).Thuswehavethesolutionwherethisshouldbecomparedwiththeexpansionoftheexactsolution,(4.2).Wehaveusedthisexampletointroduceandillustratealltheessentialfeaturesofthetechnique.Itshouldbeclearthatthetransformationfromanordinarytoapartialdifferentialequationdoesnotintroduceanyunduecomplicationsinthemethodofsolution.Wedoseethatwemustimposeperiodicityanduniformityateachorder,andthatthisproducesconditionsthatuniquelydescribethesolutionatthepreviousorder.Indeed,theremovaloftermsthatgeneratesecularitiesisfundamentaltotheapproach.Further,aswehaveseen,suitablefreedominthechoiceofthefastscaleenablesnon-uniformitiesalsotoberemoved.Theonlyremainingquestion,atleastinthecontextofordinarydifferentialequations,ishowthemethodfareswhentheequationcannotbesolvedexactly(sowehavenosimpleguidetotheformofsolution,aswedidabove).Weexplorethisaspectviaanotherexample.E4.2ADuffingequationwithdampingWeconsidertheproblemwith 1644.Themethodofmultiplescaleswhereandisaconstantindependentofcf.Q3.24.Weseekanasymptoticsolutionintheformwhereandnotethatwehaveomittedtheterminaccordancewithourearlierobserva-tion.Equation(4.18)becomeswhichgivesthesequenceofequationsandsoon.Theinitialconditions,(4.19),becomeasfarastermsThesolutionto(4.20a),with(4.21)and(4.22),isTheproblemforcanbewrittenwherewith 165ThefunctionisperiodicinTonlyifthus,with(4.23),weobtain(Notethat,ifweallowthenandwhichcorrespondstotheresultsobtainedinQ3.24;thechangeofsigninisbecauseheretheleadingterminvolvescosratherthansin.)Thisleavesthesolutionof(4.24)aswithTheanalysisofequation(4.20c),forfollowsthesamepattern(andseealsoE4.1),butherethedetailsareconsiderablymoreinvolved;wewillnotpur-suethiscalculationanyfurther(forwelearnnothingofsignificance,otherthantoshowthatwhichisleftasanexercise).Thesolution,tothisorder,isthereforeOtherexamplesbasedonsmalladjustmentstotheequationforalinearoscillatorcanbefoundinexercisesQ4.1–4.9.Itmightbeanticipated,inthecontextofoscilla-torsgovernedbyordinarydifferentialequations,thatthemethodofmultiplescalesissuccessfulonlyiftheunderlyingproblemisalinearoscillationi.e.controlledbyanequationsuchasthiswouldbefalse.Inanimportantextensionofthesetechniques,Kuzmak(1959)showedthattheyworkequallywellwhentheoscillatorispredominantlynonlinear.Ofcourse,thefundamentaloscillationwillnolongerberepresentedbyfunctionslikesinorcos,butbyfunctionsthataresolutionsofnonlinearequationse.g.theJacobianellipticfunctions.4.2NONLINEAROSCILLATORSEquationssuchaspossesssolutionsthatcanbeexpressedintermsofsn,cnordn,forexample.(IntheAppendixwepresentallthebasicinformationaboutthesefunctionsthatisnecessaryfor 1664.Themethodofmultiplescalesthecalculationsthatwedescribe.)Wediscussanexample,takenfromKuzmak(1959),whichexemplifiesthistechnique.E4.3AnonlinearoscillatorwithslowlyvaryingcoefficientsWeconsidertheproblemfortogetherwithsomesuitableinitialdata;thisisaDuffingequationwithslowlyvaryingcoefficients.Theequationclearlyimpliesthattheslowscaleshouldbebutwhatdoweuseforthefastscale?Here,wedefineageneralformoffastscale,T,bywhereistobedetermined,andtheperiodoftheoscillationinTisdefinedtobeaconstant—anessentialrequirementintheapplicationofmultiplescalesinthisproblem.(Insomemoreinvolvedproblems,itmightbenecessarytointroduceEquation(4.27)istransformedaccordingtowhichgives,withandweseekasolutionwhereeachisperiodicinT.(Theboundednessanduniformvalidity,thatweaimforaswilldependontheparticularwewillassumethatthesefunctionsallowthis.)Now(4.30)in(4.29)yieldsthesequenceofequationsasfarasNotethat,althoughthefirstequation—for—isnecessarilynonlinear,allequationsthereafterarelinear. 167Anexactsolutionofequation(4.31)iswhere(Theconfirmationofthis,usingthepropertiesgivenintheAppendix,isleftasanexercise.)Here,isaconstant(toensurethattheperiodinTisaconstantwhich,withthechosenform(4.33),is1).Givenandequations(4.34a,b)providetwoequationsforthethreeunknownfunctions:andAthirdequationisobtainedbyimposingperiodicityonThetaskofsolving(4.32)isnotasdifficultasitmightappear;theimportantma-noeuvreistowriteandthentosolveforEquation(4.32),with(4.35),givesbutfrom(4.31)wealsohavethatandso(4.36)becomes,aftermultiplicationbyNow,fortobeperiodicwithperiod1,mustbesimilarlyperiodic(becauseis,in(4.35)).Thuswemusthave,foranyT,whichcanbewrittenwhichisourthirdrelation.When(4.33)isusedin(4.37),theintegrationispossible(butthisdoesrequiresomeadditionalskillswith,andknowledgeof,ellipticfunctionsandintegrals;seee.g.Byrd&Friedman,1971),togive 1684.ThemethodofmultiplescalesinitialdatawilldetermineboththisconstantandThen,forknownandsuitableandequations(4.34a,b)and(4.38)enablethecompletedescriptionofthefirsttermintheasymptoticexpansion(4.30).Thisexamplehasdemonstratedthatwearenotrestrictedtonearly-linearoscillations,althoughwemustacceptthatmathematicalintricacies,andtherequiredmathematicalskills,arerathermoreextensiveherethaninthetwopreviousexamples.Inaddition,problemsofthistype,becausetheyarestronglynonlinear,oftenforceustoaddressotherdifficulties:wehaveusedaperiodicitycondition,(4.37),butthisfailsifthesolutionisnotperiodic—andthiscanhappen.Ifthesolutionevolvessothatthentheperiodicityislostbecausetheperiodbecomesinfiniteinthislimit.Inthissituation,itisnecessarytomatchthesolutionform=1totheperiodicsolutionapproximatedasanexampleofthisprocedurecanbefoundinJohnson(1970).SomeadditionalmaterialrelatedtothistopicisavailableinQ4.10&4.11.4.3APPLICATIONSTOCLASSICALORDINARYDIFFERENTIALEQUATIONSThemethodofmultiplescalesisparticularlyusefulintheanalysisofcertaintypesofordinarydifferentialequationwhichincorporateasuitablesmallparameter.Wewilldis-cussthreesuchproblems,thefirstofwhichwehavealreadyencountered:theMathieuequation(§3.4andE3.6).Thenexttwoinvolveadiscussionofaparticularclassofproblems—associatedwiththepresenceorabsenceofturningpoints(see§2.8)—withasolution-techniqueusuallyreferredtoasWKB(or,sometimes,WKBJ);wewillwritemoreofthislater.TheMathieuequation,discussedin§3.4istowhichwewillapplythemethodofmultiplescales.However,beforeweundertakethis,weneedtoknowwhatthefastandslowscalesshouldbe;thisrequiresalittlecare.Letusconsidertheequationwith(fixedindependentofthenwecouldselectTheequationforbecomesIngeneral,wefindthateachhasaparticularintegralproportionaltounlessandthenwehaveparticularintegralsthatgrowint.Thisconditionwilloccurforandsothecriticalvaluesofare(n=0,1,2...).Fromthesepointsonthewillemanatethetransitionalcurves, 169inthewhichseparatepurelyoscillatoryfromexponentiallygrowingso-lutions;seeFurther,foranygivenn,theasymptoticexpansionwilltaketheformandsoweshouldthenusetheslowscaleWenotethatthefastscalecanbetakensimplyast.Withthesepointsinmind,weconsideranexampleinsomedetail.E4.4Mathieu’sequation(n=1)forWeusethemethodofmultiplescalesfortheequationwiththuswehavethecasen=1andsoweintroducethescalesEquation(4.39),withthenbecomeswherehasbeenexpandedintheusualway,andthenweseekasolutionthuswegeneratethesequenceofequationsandsoon.Equation(4.41a)hasthegeneralsolutionandthen(4.41b)canbewritten 1704.ThemethodofmultiplescalesThusasolutionforwhichisperiodicinTrequireswhichpossessthegeneralsolutionwhereandarearbitrary—possiblycomplexwhichthenrequiresthecomplexconjugatetobeincluded—constants.Weseethat(a)ifthenandareoscillatoryandsoisoscillatoryinbothTand(b)ifthenthereexistsasolutionwhichgrowsexponentially;(c)ifthen(see(4.42))oneoforisconstantandtheothergrowslinearly.Themethodofmultiplescaleshasenabledus,albeitinthelimittodescribealltheessentialfeaturesofsolutionsofMathieu’sequation,andhowthesechangeastheparametersselectdifferentpositionsandregionsinThecorrespondingproblemforn=2isdiscussedinQ4.12,andrelatedexercisesaregiveninQ4.13&4.14.Wenowturntoanimportantclassofproblemsthatareexemplifiedbytheequation(cf.Q2.24)whereisgiven.Inthesimplestproblemofthistype,takesonesignthroughoutthegivendomain(D)i.e.a>0(oscillatory)ora<0(exponential).Amoreinvolvedsituationarisesifachangessigninthedomain:aturning-pointproblem(seeQ2.24).Theintentionhereistoexaminethesolutionoftheequationinthecasesothat,forthecoefficientisslowlyvarying.Weusethemethodofmultiplescalestoanalysethisproblemandhencegiveapresentationofthetechniqueusuallyreferredtoas‘WKB’.(ThisisafterWentzel,1926;Kramers,1926;Brillouin,1926,althoughtheessentialideacanbetracedbacktoLiouvilleandGreen.SomeauthorsextendthelabeltoWKBJ,toincludeJeffreys,1924.)WewillformulateanoscillatoryproblemandusethisexampletodescribetheWKBapproach.E4.5WKBmethodforaslowly-varyingoscillationWeconsider 171withforandwithsuitableinitialconditions.Themostnaturalandconvenientformulationofthemultiple-scaleproblemfollowsE4.3:weintroduceandseekasolutionEquation(4.44)becomesand,inthisfirstexercise,wewillnotexpand(butseeQ4.15).Thus(4.45)leadstothesequenceofequationsfortheO(1)andproblems.Tosolve(4.46a),wefirstselectandthenobtainthegeneralsolutionwhereandarearbitraryfunctions.(Otherchoicesofleadtoaformulationintermsofbutthentheperiodwilldependonwhichleadstoadditionalnon-uniformities—seeE4.6—unlessisconstant;wehavemadethesimplestchoiceofthisconstant.)TheequationfornowbecomeswherewehavewrittenThesolutionforisperiodicinT,i.e.inifandsoSo,forexample,ifwearegiventheinitialdatathenwehave 1724.Themethodofmultiplescalesandasolution(withT=0atsay),expressedintermsofisThisdescribesafastoscillation(byvirtueofthefactorwithaslowevolutionoftheamplitude;thesearethesalientfeaturesofaWKB(J)solution.(Notethatapropertyofthissolutioniswhichisusuallycalled‘action’;typically,energyandsoisconserved—nottheenergyitself.)Theproblemoffindinghigher-ordertermsintheWKBsolutionisaddressedinQ4.15,andthecorrespondingproblemwithisdiscussedinQ4.16,andaninterestingassociatedproblemisdiscussedinQ4.17.Wenowconsiderthecaseofaturningpoint.Itisapparentthatthesolution(4.47)isnotvalidifwhichisthecaseataturningpoint.In(4.43),wewillwritewiththroughoutthedomainD,andanalytic(totheextentthatmaybewrittenasauniformlyvalidasymptoticexpansion,asforThischoiceofhasasingle(simple)turningpointatx=0;aturningpointelsewherecanalwaysbemovedtox=0byasuitableoriginshift.Theintentionistofindasolutionvalidneartheturningpointandthen,awayfromthisregion,usetheWKBmethodinx<0andinx>0.Thustheturning-pointsolutionistobeinsertedbetweenthetwoWKBsolutionsand,presumably,matchedappropriately.Wewillpresentalltheseideas,usingthemethodofmultiplescalesvalidasinthefollowingexample.E4.6Aturning-pointproblemWeconsiderwherebothandarepositive,O(1)constants,andanalyticforandTheturningpointisatx=0,andthefirstissueistodecidewhatscalestouseintheneighbourhoodofthispoint;thishasalreadybeenaddressedinLetuswrite(andanyscalingonyisredundant,insofarasthegoverningequationisconcerned,becausetheequation,(4.48),islinear)togiveandsoabalanceoftermsispossibleifi.e.fastscale.TheslowscaleissimplyHowever,amoreconvenientchoiceofthefastvariable 173(cf.§2.7)iswhereh(X)>0istobedetermined;thenwehavetheoperatoridentityFinally,beforeweusethistotransformequation(4.48),weneedtodecidehowtoreplacexinthecoefficientinequation(4.48):shouldweuseZorXorboth?Nowtheimportantpropertyofthefastscale,(4.49),isthatitiszeroattheturningpoint;thusweelecttowriteanypartofacoefficientwhichhasthissamepropertyintermsofZ,andotherwiseuseX.Withthisinmind,weuse(4.49)and(4.50)in(4.48)andthen,withweobtainwhere,forsimplicity,wehavewrittenWeseeka(bounded)solutionintheformandhenceweobtainthesequenceofequationsandsoon.Theboundedsolutionof(4.52a)canbeexpressedintermsoftheAiryfunction,Ai(seee.g.Abramowitz&Stegun,1964).Atthisstage,however,wehavenotyetmadeasuitablechoiceforh(X);letuschooseandthenwehave 1744.ThemethodofmultiplescalesThissolutionisoscillatoryforZ>0andexponentiallydecayingforZ<0;inpar-ticularandTheequationfornowbecomesandaparticularintegralofthisequationisnecessarilyproportionaltoZAi(Z),whichimmediatelyleadstoanon-uniformityinThuswemustselectitisleftasanexercisetoshowthat,ifwehadwrittenthenanothernon-uniformitywouldbepresentunlessk=constant,andwehavealreadysetk=1.WealludedtothisdifficultyattheendofE4.5.Finally,theleading-ordersolutionwillbecompletelydeterminedoncewehavefoundh(X)(introducedin(4.49)).From(4.53)and(4.51),wehavetheequationweconsiderthecaseX>0,thenitisconvenienttowriteThisisandsowhichgives(withtheappropriatechoiceofsign) 175withX>0andh(X)>0.(ItfollowsdirectlyfromthisresultthatThecaseX<0,wherethecorrespondingchoiceisG(X)=–Xh(X),isleftasanexercise;youshouldfindthatasacontinuoush(X)canbedefined.Inordertocompletethecalculation,werequirethesolutioninX>0andinX<0,awayfromtheneighbourhoodoftheturningpoint.FollowingE4.5wehave,forX>0andwritingwhereandandarearbitraryconstants(becausewewillnotimposeanyparticularconditionsatForX<0(seeQ4.16)weobtainforaboundedsolution(asandwith(Re-memberthat,sincetheoriginalequation,(4.48),issecondorder,onlytwoboundaryconditionsmaybeindependentlyassigned.)ThesolutionintheneighbourhoodoftheturningpointiswhereG(X)isgivenby(4.55)andisanarbitraryconstant.Thefinaltaskisthereforetomatch(4.57)with(4.56a,b).First,inZ<0,X<0,wehave(from(4.57)and(4.54b))From(4.56b)wehave 1764.Themethodofmultiplescaleswherewehaveused(4.55)(andnotethatthusmatchingoccursifwechooseForZ>0,X>0,from(4.57)and(4.54a),wehavefrom(4.56a)weobtainwhichalsomatchesifwechooseThematchingconditionsareusuallycalled,inthiscontext,connectionformulae:they‘connect’thesolutionsoneithersideoftheturningpointi.e.therelationbetweenandHere,wehavethreerelationsbetweenthefourconstantsandsoonlyoneisfree;thatonlyoneoccurshereisbecause,ofthetwo(independent)boundaryconditionsthatwemayprescribe,onehasbeenfixedbyseekingaboundedsolutioninX<0i.e.theexponentiallygrowingsolutionhasalreadybeenexcluded.Anumberofotherexamplesofturning-pointproblemsareofferedintheexercises;seeQ4.18–4.21.Thiscompletesallthatwewillwriteabouttheroutineapplicationstoordinarydifferentialequations;wenowtakeabrieflookathowthesesametechniquesarerelevanttothestudyofpartialdifferentialequations.4.4APPLICATIONSTOPARTIALDIFFERENTIALEQUATIONSItisnoaccidentthatwewilldiscusspartialdifferentialequationswhichareassociatedwithwavepropagation;thistypeofequationisanalogoustooscillatorysolutionsofordinarydifferentialequations.(Thesetwocategoriesofequationsarethemostnaturalvehiclesforthemethodofmultiplescales,althoughothersarecertainlypossible.)Inparticularwewillstartwithanequationthathasbecomeaclassicalexampleofitstype:Bretherton’smodelequationfortheweak,nonlinearinteractionofdispersivewaves(Bretherton,1964). 177E4.7Bretherton’sequationTheequationthatwewilldiscussiswhereisdefinedinandwewillfurtherassumethatwehavesuitableinitialdataforthetypeofsolutionthatweseek.Theaimistoproduceasolution,viathemethodofmultiplescales,forFirstweobservethat,withthereisasolutionwhere,givenk(thewavenumber),(thefrequency)isdefinedbythedispersionrelation:Thepresenceoftheparameter,togetherwithanaïveasymptoticsolution(generatingtermsproportionaltoorsuggeststhatwemustexpectchangesontheslowscalesandThefastscaleisdefinedinmuchthesamewaythatweadoptedforE4.3andE4.5;thuswewriteThesolutionnowsitsinadomainin3-space,definedbyAsolutiondescribedbythesevariableswillhavethepropertythatboththewavenumberandthefrequencyslowlyevolve.(Notethatthecorrectformofthesolutionwithisrecoveredifk=constantandTheretentionoftheparameterinthedefinitionsofkandallowsustotreatthesefunctionsasasymptoticexpansions,ifthatisusefulandrelevant;oftenthisisunnecessary.From(4.61)wehavetheoperatoridentitiesandalsoetc.,asfarasItissufficient,fortheresultsthatwepresenthere,totransformequation(4.58)butretaintermsnosmallerthanthus,with 1784.ThemethodofmultiplescalesweobtainBeforeweproceedwiththedetails,weshouldmakeanimportantobservation.Thetransform,(4.61),definingthefastscale(usuallycalledthephaseintheseproblems),impliesaconsistencyconditionthatmustexistifisatwice-differentiablefunction,namelythisadditionalequationiscalledtheconservationofwaves(orofwavecrests),anditarisesquitenaturallyfromanelementaryargument.Consider(one-dimensional)wavesenteringandleavingtheregionthenumberofwaves,perunittime,crossingintotheregionisgivenasandthenumberleaving,acrossx,wewillwriteasThetotalnumberofwaves(wavecrests)betweenandxisifthenumberofwavesdoesnotchange(whichiswhatistypicallyobserved,eveniftheychangeshape)thenor,uponallowingdifferentiationwithrespecttox,whichimmediatelyrecovers(4.63)ifthedependenceon(x,t)isvia(X,T).Returningtoequation(4.62),weseekasolutionwithandthenweobtainthesequenceofequations 179andsoon.Wewilltakethesolutionof(4.65)tobe(cf.(4.59))andthisrequiresTheequationfor(4.66),willincludetermsgeneratedbywhichinvolveor—andallthesemustberemovedifistobeperiodic.ThetermsingiveandthoseinleavingThetwoequations,(4.69)and(4.70),whichensuretheremovalofsecularterms,lookratherdaunting,butquitealotcanbedonewiththem.However,wefirstneedtointroducetwofamiliarpropertiesofapropagatingwave.Oneisthespeedatwhichtheunderlyingwave—thecarrierwave—travels,usuallycalledthephasespeed.Thisisdefinedasthespeedatwhichlinesmovei.e.linessuchthatsoTheother—andforus,thefarmoresignificant—propertyisthespeedatwhichtheenergypropagates(andtherefore,forexample,thespeedatwhichtheamplitudemodulationmoves);thisisdefinedasthegroupspeed.Fromourresultin(4.68),wehave 1804.ThemethodofmultiplescalesEquation(4.70),aftermultiplicationbycanbewrittenasorwhichdescribestheproperty(thewaveaction;seeE4.5)propagatingatthegroupspeed,andevolvingbyvirtueofthenon-zeroright-handsideofthisequation.Similarly,equation(4.69)canbewrittenfirstasandifweelecttowritethen(4.61)and(4.64)allowustointerpretandandthenweobtainThusthecorrectiontothephase,alsopropagatesatthegroupspeedandevolves.Finally,from(4.63),andnotingthat(4.68)(or(4.72))impliesweobtainthewavenumber(and,correspondingly,thefrequency)propagateunchangedatthegroupspeed.(TheinitialdatawillincludethespecificationofThisexamplehasdemonstratedthatthemethodofmultiplescalescanbeusedtoanalyseappropriatepartialdifferentialequations,evenifwehadtoteaseoutthedetailsbyintroducing,inparticular,thegroupspeed.ArelatedexercisecanbefoundinQ4.22.InouranalysisofBretherton’sequation,weworked—notsurprisingly—withreal-valuedfunctionsthroughouti.e.Thereis,sometimes,anadvantageinworkingwithinacomplex-valuedframeworke.g.complexconjugate.Wewillusethisapproachinthenextexample,butalsotakenoteofthegeneralstructurethatisevidentinE4.7.E4.8Anonlinearwaveequation:theNLSequationAwaveprofile,satisfiestheequation 181whereInthisproblemweseekasolutionwhichdependsonx–ct(here,cisthephasespeed),onisthegroupspeed)andont(time).Weexpectthatthedependenceonisslowi.e.intheformbutwewillallowanevenslowertimedependence;wedefinealthoughwehaveyettodeterminecandWithand(4.74),equation(4.73)becomesandweseekasolutionwith‘cc’denotesthecomplexconjugate.Thissolutionrepresentsaprimaryharmonicwavetogetherwithappropriatehigherharmonics(thenumberofwhichdependsoni.e.onthehierarchyofnonlinearinteractions).Thewavenumberoftheprimarywave,k,isgivenandreal;XandTarereal(ofcourse),buteachiscomplex-valued.WewillassumethatcandareindependentofFirstweuse(4.76a)in(4.75)togivethesequenceofequationsasfarastheterms.Thesolutionof(4.77a)isgivenintheform(see(4.76b)),andsowe-requirewhichdefinesthephasespeed;atthisstageisunknown.Equation(4.77b),fornowbecomes 1824.Themethodofmultiplescaleswiththegivenformofsolutionwhichrequireswith(andstillarbitrary.(Wenotethatandthat,withwehavesotheclassicalresultforgroupspeed.)Thefinalstageistofindthesolution—or,atleast,therelevantpartofthesolution—ofequation(4.77c).Theusualaiminsuchcalculationsistofind,completely,thefirsttermin(wehope)auniformlyvalidasymptoticexpansion.Inthiscasewehaveyettofind(althoughweknowbothc(k)andthedeterminationofarisesfromthetermsinequation(4.77c).Wewillfindjustthisoneterm;therestofthesolutionforisleftasanexercise,theessentialrequirementbeingtocheckthattherearenoinconsistenciesthatappearasisdetermined.Theseterms,in(4.77c)givetheequationwheretheover-bardenotesthecomplexconjugate.Whentheearlierresultsarein-corporatedhere,wefindtheequationforThisequation,(4.79),isaNonlinearSchrödinger(NLS)equation,anotheroftheex-tremelyimportantexactly-integrableequationswithintheframeworkofsolitontheory;seeDrazin&Johnson(1992).Thuswehaveacompletedescriptionofthefirsttermintheasymptoticexpansion(4.76a,b):whereisasolutionof(4.78)andbothc(k)andareknown.Weshouldcommentthat,becauseoftheparticularformofsolutionthatwehaveconstructedinthisexample,itisappropriateonlyforcertaintypesofinitialdata.Thus,from(4.80),weseethat,att=0,wemusthaveaninitialwave-profilethatispredominantlyaharmonicwave,butonethatadmitsaslowamplitudemodulationi.e. 183Anyinitialconditionsthatdonotconformtothispatternwouldrequireadifferentasymptotic,andmulti-scale,structure.OtherexamplesthatcorrespondtoE4.8aresetasexercisesQ4.23–4.26;seealsoQ4.27.Beforewedescribeonelastgeneralapplicationofthemethodofmultiplescales—perhapsarathersurprisingone—wenoteaparticularlimitationonthemethod.4.5ALIMITATIONONTHEUSEOFTHEMETHODOFMULTIPLESCALESTheforegoingexamplesthathaveshownhowtogenerateasymptoticsolutionsofpar-tialdifferentialequationsappearreasonablyroutineandhighlysuccessful.However,thereisanunderlyingproblemthatisnotimmediatelyevidentandwhichcannotbeignored.Inthecontextofwavepropagation,whichisthemostcommonapplicationofthistechniquetopartialdifferentialequations,weencounterdifficultiesifthepre-dominantsolutionisanon-dispersivewavei.e.waveswithdifferentwavenumber(k)alltravelatthesamespeed.Toseehowthisdifficultycanarise,wewillexamineanexamplewhichisclosetothatintroducedinE4.8.E4.9Dispersive/non-dispersivewavepropagationWeconsidertheequation(cf.(4.73))whereisagivenconstantandweseekasolutionintheformwhereTheequationthenbecomesandwelookforasolutionperiodicinintheformofaharmonicwave:whereallthisfollowstheproceduredescribedinE4.8.Here,wefindthatbutisyettobedetermined.Atthenextorder,theequationforcanbewritten 1844.ThemethodofmultiplescalesandsowerequireItisimmediatelyevidentthatwehaveanon-uniformityasFrom(4.82)weseethatcorrespondstoanon-dispersivewavei.e.c=±1forallwavenumbers,k.Whenwesetin(4.83),andusenosolutionexists,althoughforanynocomplicationsariseandwemayproceed.Thisfailureisnottoberegardedasfatal:fortheassumedinitialdata(theharmonicwave),oranyotherreasonableinitialconditions,anasymptoticsolution(withcanbefoundbythemethodofstrainedcoordinates.Forawaveproblem,aswehaveseen(e.g.Q3.25),thissimplyrequiresasuitablerepresentationofthecharacteristicvariables.Nevertheless,inthemostextremecases,whenthemethodofmultiplescalesisstilldeemedtobethebestapproach,theresultingasymptoticsolutionmaynotbeuniformlyvalidasT(orinourdiscussionofordinarydifferentialequations)Typically,themultiple-scalesolutionwillbevalidforT(ornolargerthanO(1),butthisisusuallyanimprovementonthevalidityofastraightforwardasymptoticexpansion.4.6BOUNDARY-LAYERPROBLEMSTheexamplesthatwehavediscussedsofarinvolve,usuallywitharatherstraightforwardphysicalinterpretation,theslowevolutionordevelopmentofanunderlyingsolution.Boundary-layerproblems(see§2.6–2.8),ontheotherhand,mightappearnottopossessthisstructure.Suchproblemshavedifferent—butmatched—solutionsawayfrom,andnearto,aboundary.However,thesolutionofsuchproblemsexpressedasacompositeexpansion(see§1.10)exhibitspreciselythemultiple-scalestructure:afastscalewhichdescribesthesolutionintheboundarylayer,andaslowscaledescribingthesolutionelsewhere.Wewilldemonstratethedetailsofthisprocedurebyconsideringagainourstandardboundary-layer-typeproblemgiveninequation(2.63)(andseealso(1.16)).E4.10Aboundary-layerproblemWeconsiderwithwhereandareconstantsindependentofandTheboundary-layervariableforthisproblem(see(2.66))isandsowewritewhichleadstoequation(4.84) 185writtenintheformwithNotetheappearanceoftheevaluationonwhichmaycausesomeanxiety;wewillnowaddressthisissue.Toseetheconsequencesofthis,andbeforewewritedownthecompleteasymptoticexpansion,itisinstructivefirsttosolvefori.e.withtheboundaryconditionsThissecondconditioninvolvesandtoaccommodatesuchtermstheasymptoticexpansionmustincludethem;thusweseekanasymptoticsolutionandevaluationonisnownolongeranembarrassment.Thesequenceofequations,generatedbyusing(4.87)in(4.85),startsThegeneralsolutionof(4.88a)isandthenwemaywrite(4.88b)aswhichitselfhasthegeneralsolutionWeseekasolutionwhichisuniformlyvalidforandbutthislatterimpliesthuswerequire,foruniformity, 1864.ThemethodofmultiplescalesTheboundaryconditions,atthisorder,giveandsowehavethesolutionswhichcompletelydeterminesthefirsttermoftheasymptoticexpansion:asgiveninequation(2.76).Theothertermsintheexpansionfollowdirectlybyim-posinguniformityateachorder;thetermsforareusedtoremovetheex-ponentiallysmallcontributionsthatappearintheboundaryconditiononx=1.Themethodofmultiplescalesisthereforeequallyvalid,andbeneficial,fortheanalysisofboundary-layerproblems.However,theexamplethatwehavepresentedisparticu-larlystraightforward(and,ofcourse,wehaveavailabletheexactsolution).Weconcludethischapterbyapplyingthemethodtoamoretestingexample(takenfromQ2.17(a)).E4.11Anonlinearboundary-layerproblemWeconsidertheproblemwithforTheslowscaleisclearlyx,butforthefastscalewewillusethemostgeneralformulationoftheboundary-layervariable;see§2.7.Thusweintroduceandthenwewritesothatequation(4.89)becomeswithThetermsthatareexponentiallysmallonx=1,asareandsoweseekasolution 187Thefirsttwoequationsinthesequence,obtainedbyusing(4.91)in(4.90),areand(4.92a)hasthegeneralsolutionwithEquation(4.92b)canthenbewrittenwhichhasthegeneralsolutionAsolutionthatisuniformlyvalidasrequiresandso,incorporatingtheboundaryconditions,(4.94),wereadilyobtainCombiningthiswith(4.93),wehavethefirsttermofauniformlyvalidrepresentationofthesolution:whichshouldbecomparedwiththesolutionobtainedinQ2.17(a). 1884.ThemethodofmultiplescalesNowthatwehaveseenthemethodofmultiplescalesappliedtoboundary-layerprob-lems,itshouldbeevidentthatthisprovidesthesimplestandmostdirectapproachtothesolutionofthistypeofproblem.Notonlydoweavoidtheneedtomatch—although,ofcourse,thecorrectselectionoffastandslowvariablesisessential—butwealsogenerateacompositeexpansiondirectly,whichmaybeusedasthebasisfornumericalorgraphicalrepresentationsofthesolution.FurtherexamplesaregiveninexercisesQ4.28–4.35.Thisconcludesourpresentationofthemethodofmultiplescales,andisthelasttechniquethatweshalldescribe.Inthefinalchapter,theplanistoworkthroughanumberofexamplestakenfromvariousbranchesofthemathematical,physicalandrelatedsciences,groupedbysubjectarea.These,wehope,willshowhowourvarioustechniquesarerelevantandimportant.Itistobehopedthatthosereaderswithinterestsinparticularfieldswillfindsomethingtoexcitetheircuriosityandtopointthewaytothesolutionofproblemsthatmightotherwiseappearintractable.FURTHERREADINGMostofthetextsthatwehavementionedearlierdiscussthemethodofmultiplescales,toagreateroralesserextent.Twotexts,inparticular,giveagoodoverviewofthesubject:Nayfeh(1973),inwhichanumberofproblemsareinvestigatedinmanydif-ferentways(includingvariantsofthemethodofmultiplescales),andKevorkian&Cole(1996)whichprovidesanup-to-dateandwide-rangingdiscussion.Theappli-cationstoordinarydifferentialequationsarenicelypresentedinbothSmith(1985)andO’Malley(1991),wherealotoftechnicaldetailisincluded,aswellasacarefuldiscussionofasymptoticcorrectness.Holmes(1995)providesanexcellentaccountofboththemethodofmultiplescalesandtheWKBmethod.ThislatterisalsodiscussedinWasow(1965)andinEckhaus(1979).Finally,anexcellentintroductiontotherôleofasymptoticmethodsintheanalysisofoscillations(mainlythosethatarenonlinear)canbefoundinBogoliubov&Mitropolsky(1961)—anoldertext,butaclassicthatcanbehighlyrecommended(eventhoughitdoesnotpossessanindex!).EXERCISESQ4.1NearlylinearoscillatorI.Aweaklynonlinearoscillatorisdescribedbytheequationwherea(>0)isaconstant(independentofandtheinitialconditionsareUsethemethodofmultiplescalestofind,completely,thefirstterminauniformlyvalidasymptoticexpansion.(ItissufficienttousethetimescalesT=tandExplainwhyyoursolutionfailsifa<0. 189Q4.2NearlylinearoscillatorII.SeeQ4.1;repeatthisfortheproblemwithQ4.3NearlylinearoscillatorIII.SeeQ4.1;repeatthisfortheproblemwhere(>0)isaconstant(independentofandisanintegrablefunction;theinitialconditionisWhatconditionmustsatisfyif,onthebasisoftheevidenceofyourtwo-termexpansion,theasymptoticexpansionistobeuniformlyvalid?[Hint:rememberthatthegeneralsolutionofwhereisanarbitraryconstant.]Q4.4Anearlylinearoscillatorwithforcing.SeeQ4.1;repeatthisfortheproblemwhereandareconstants,bothindependentofFindthegeneralformofthefirstterm,giventhatExplaintheconsequencesof(a)(b)Also,writedowntheequationsdefiningtheslowevolutionofthesolutioninthecases:(c)(d)(SeeQ4.7,4.8formoredetails.)Q4.5Aslowlyvaryinglinearoscillator.Adamped,linearoscillatorisdescribedbytheequationwithandObtainthecompletedescriptionofthe(general)firsttermofauniformlyvalidasymptoticexpansion,forwhichyoushouldusethefastscaledefinedby(andtheslowscaleisQ4.6Acoupledoscillatorysystem.AnoscillationisdescribedbythepairofequationswithandtheinitialconditionsareIntroduceandusethemethodofmultiplescalesandfind,completely,thefirsttermofauniformlyvalidasymptoticexpansions. 1904.ThemethodofmultiplescalesAlso,fromtheequationsthatdefinethethirdtermintheexpansions,showthatthesolutionisuniformlyvalid(asonlyifQ4.7ForcingnearresonanceI.ConsideranoscillationdescribedbyaDuffingequationwith(weak)forcing:withisthegivenfrequencyoftheforcingandtheinitialcondi-tionsareGiventhat(whereisaconstantindependentoffindtheequationwhichdescribescompletelythefirsttermofauniformlyvalidasymptoticexpansion.[Hint:writetheforcingtermaswherearethefast/slowscales,respectively.]Q4.8ForcingnearresonanceII.SeeQ4.7;repeatthisforintheequa-tion(seeQ4.4),whereisaconstantindependentof(andnotetheappearanceofsubharmonics).Q4.9Failureofthemethodofmultiplescales.AnoscillatorisdescribedbytheequationwithIntroduceand(withandanalyseasfarasthetermatwhichensuresthecompletedescriptionofthesolutionasfarasShowthatauniformlyvalidsolutioncannotbeobtainedusingthisapproach.(Youmaywishtoinvestigatewhythishappensbyexaminingtheenergyintegralforthemotion.)Q4.10NonlinearoscillationI.A(fully)nonlinearoscillationisdescribedbytheequationwithShowthatasolutionwithisx=acn[4K(m)t;m]forasuitablerelationbetweenaandm.Nowusethemethodofmultiplescales,withthescalesT(whereandtofindthefirsttermofanasymptoticexpansionwhichisperiodicinT.(Theperiodicityconditionshouldbewrittenasanintegral,butthisdoesnotneedtobeevaluated.)Q4.11NonlinearoscillationII.SeeQ4.10;followthissameprocedurefortheequationwhereasolutionwithcanbewrittenforsuitablea,bandm. 191Q4.12Mathieu’sequationforn=2.SeeE4.4;considerMathieu’sequationwith(whichisthecasen=2).IntroduceT=tandshowthatandfindtheequationforthetermintheasymptoticexpansionforx;fromthis,deducethattheexponentwhichdescribestheamplitudemodulation(cf.E4.4)isWhatisthenatureofthesolutionofthisMathieuequation,forvarious(TheseresultsshouldbecomparedwiththoseobtainedinQ3.15.)Q4.13AparticularHillequation.SeeQ4.12;followthissameprocedurefortheHillequationwhereisafixedconstant(independentofandforShowthatandthattheleadingtermisperiodicinTandonlyifwherearetobeidentified.(TheseresultsshouldbecomparedwiththoseobtainedinQ3.17.)Q4.14Mathieu’sequationawayfromcriticalSeeQ4.12;considerMathieu’sequa-tion,butnowwithawayfromthecriticalvalues:setn=0,1,2,..)andfixedindependentofIntroducewithandandfindthesolutioncorrectat(Youshouldnotethesingularities,forvariousthatareevidenthere.)Q4.15WKB:higher-orderterms.SeeE4.5;considertheequationwithintroduceandwriteShowthatandthenexpandbothDetermineandintermsofandtheconstant(whichisindependent 1924.Themethodofmultiplescalesof(Thisprocedureisaveryneatwaytoobtainhigher-ordertermsintheWKBapproach.)Q4.16WKB(exponentialcase).ConsidertheequationwhereforandIntroduceandandhencefind,completely,thefirsttermofauniformlyvalidasymp-toticexpansion(whichwillbethecounterpartofequation(4.47)).Q4.17Eigenvalues.Usethemethodofmultiplescales,intheWKBform,tofindtheleadingapproximationtotheeigenvaluesoftheproblemwithanda>0for(Youshouldin-troduceandEvaluateyourresults,explicitly,forthecases:(a)(b)a(X)=1+X.[Writtenintheformitisevidentthatthisistheproblemoffind-ingapproximationstothelargeeigenvalues.]Q4.18Aturning-pointproblemI.Considerwherewithf>0andanalyticthroughoutthegivendomain.Showthattherelevantscaling(cf.§2.7)intheneighbourhoodoftheturningpoint(atx=1)isandthenintroducethemoreusefulfastscaleandusexastheslowscale.Showthatwhere=constantandAi(X)isthe(bounded)Airyfunction(asolutionofDetermineh(x)andwritedownthefirsttermoftheasymptoticexpansionofQ4.19Aturning-pointproblemII.Showthattheequationwhereandhasturningpointsatx=0andatx=1.[Hint:writeUsetheWKBapproach(fortofindthefirsttermineachoftheasymptoticexpansionsvalidinx<0,01.Alsowritedowntheleadingtermintheasymptoticexpansionsvalidnearx=0andnearx=1.Q4.20Ahigher-orderturning-point.Showthattheequation 193withandf>0throughoutthegivendomain,hasaturningpointatx=0.Findtheequationthat,onanappropriatefastscale(X),describesthesolutionnearx=0(asShowthatthisequationhassolutionswhichcanbewrittenintermsofBesselfunctions:forsuitableandv.Q4.21Schrödinger’sequationforhighenergy.Thetime-independent,one-dimensionalSchrödingerequationforasimple-harmonic-oscillatorpotentialcanbewrittenwhereE(=constant)isthetotalenergy.Letuswriteanddefinetogivethisequationhasturningpointsatandwerequireexponentiallyde-cayingsolutionsas(andoscillatorysolutionsexistforFindtheleadingtermineachoftheregions,match(andthusdevelopap-propriateconnectionformulae)andshowthattheeigenvalues(E)satisfywherenisalargeinteger.(Thisproblemcanbesolvedexactly,usingHermitefunctions;itturnsoutthatourasymptoticevaluationofEisexactforalln.)Q4.22Aweaklynonlinearwave.AwaveisdescribedbytheequationwhereIntroduceandderivetheequationsthatcompletelydescribetheleading-ordersolutionwhichisuniformlyvalid.(DonotsolveyourequationforQ4.23Aweaklynonlinearwave:NLSI.AwaveisdescribedbytheequationwithIntroduce(wherec(k)andareindependentofandseekasolution 1944.ThemethodofmultiplescaleswhereDeterminec(k)and(andconfirmthatsatisfiestheusualconditionforthegroupspeed)andfindtheequationforQ4.24Aweaklynonlinearwave:NLSII.SeeQ4.23;repeatallthisfortheequationQ4.25Aweaklynonlinearwave:NLSIII.SeeQ4.23;repeatallthisfortheequationQ4.26KdVNLS.ConsidertheKorteweg-deVries(KdV)equationwhereisaparameter.Introduceand(andhereandwillbecorrectionstotheoriginalcandbecausethegivenKdVequationhasalreadybeenwritteninasuitablemovingframe).Seekasolutionwherefindc(k),andtheequationfor(ThisexampledemonstratesthatanunderlyingstructureoftheKdVequationisanNLS(NonlinearSchrödinger)equation;indeed,itcanbeshownthat,inthecontextofwaterwaves,forexample,therelevantNLSequationforthatproblemmatchestothisNLSequation—seeJohnson,1997.)Q4.27Raytheory.Awave(movingintwodimensions)slowlyevolves,onthescalesothatwhereshowthat(a)whereand(b)(theeikonalequation);(c)(sothevectorkis‘irrotational’).(GiventhattheenergyinthewavemotionisE(X,Y,T),itcanbeshownthatwhereAllthisisthebasisforraytheory,orthetheoryofgeometricaloptics,whichisusedtodescribethepropertiesofwavesthatmovethroughaslowlychangingenvironment.)Q4.28Boundary-layerproblemI.Usethemethodofmultiplescalestofind,completely,thefirsttermofauniformlyvalidasymptoticexpansionofthesolutionofwhere 195Q4.29Boundary-layerproblemII.SeeQ4.28;repeatthisfortheproblemwithQ4.30Boundary-layerproblemIII.SeeQ4.28;repeatthisfortheproblemwithQ4.31Boundary-layerproblemIV.SeeQ4.28;repeatthisfortheproblemwithQ4.32Boundary-layerproblemV.SeeQ4.28;repeatthisfortheproblemwithQ4.33Heattransferproblem.SeeQ4.28;repeatthisfortheheattransferproblem(asgiveninQ2.30)with[Takecare!]Q4.34Amoregeneralboundary-layerproblem.SeeQ4.28;repeatthisfortheproblemwhereisagiveninteger,withtheboundaryconditionsQ4.35Twoboundarylayers.SeeQ4.28;repeatthisfortheproblemwithbutnotethattwofastscalesarerequiredhere,toaccommodatethetwoboundarylayers—onenearx=0andtheothernearx=1. Thispageintentionallyleftblank 5.SOMEWORKEDEXAMPLESARISINGFROMPHYSICALPROBLEMSInthisfinalchapter,theaimistopresentanumberofworkedexampleswheremostofthedetailsaregivenexplicitly;whatlittleisleftundonemaybecompletedbytheinterestedreader(althoughnoformalexercisesareoffered).Also,wewillnotdwelluponthepurelytechnicalaspectsoffindingthesolutionofaparticulardifferentialequation.Theseexamplesaretakenfrom,orbasedon,textsandpapersthatintroduce,describe,develop,explainandsolvepracticalproblemsinvariousfields;referencestoappropriatesourcematerialwillbeincluded.Mosthavearisen—notsurprisingly—fromthephysicalsciences,butwehaveattemptedtoprovideafairlybroadspreadoftopics.Eachproblemisdescribedwithsufficientdetail(wehope)toenableittobeputintocontext,althoughitwouldbequiteimpossibletoincludeallthebackgroundideasforthosealtogetherunfamiliarwiththeparticularfield.Tothisend,theprob-lemsarecollectedundervariousheadings(suchas‘mechanical&electricalsystems,‘semiconductors’or‘chemical&biologicalreactions’)andsothereaderwithparticularinterestsmightturntospecificonesfirst.Nevertheless,thehopeisthateveryproblemisaccessible,asanexampleinsingularperturbationtheory,tothosewhohavefollowedthis(oranyothersuitable)text.Thetechniqueadoptedtoconstructtheasymptoticsolutionwillbementioned,andareferencewillbegiventoarelevantsectionorexamplefromtheearlierchaptersofthistext.Anumberoftheexamplesandexercisesthathavealreadybeendiscussedhavebeentakenfromvariousimportantapplications;insomecases,thosepresentedinthischap-terbuildonandexpandtheseearlierproblems.Thereadershouldbeaware,therefore, 1985.Someworkedexamplesarisingfromphysicalproblemsthattherelevantcalculationsinthepreviouschaptersmayneedtoberehearsedbeforeembarkingonsomeofthenewmaterialpresentedhere.Ineachgroupofproblems,everyexamplewillbelabelledbyasuitablename,andthefulllistofthesewillappearinthepreambletothatgroup.Thetitlesofthegroupsare:5.1Mechanical&elec-tricalsystems;5.2Celestialmechanics;5.3Physicsofparticles&light;5.4Semi-andsuperconductors;5.5Fluidmechanics;5.6Extremethermalprocesses;5.7Chemical&biochemicalreactions.Thesechosenheadingsareintendedsimplytoprovideageneralguidetothereader;thereisnodoubtthatsomeexamplescouldbeplacedinadifferentgroup—orappearinmorethanonegroup.Further,manyotherexamplescouldhavebeenincluded(andtheauthorapologisesifyourfavouritehasbeenomitted);thein-tentioninatextsuchasthisistogiveonlyaflavourofwhatispossible.Nevertheless,itishopedthatsufficientinformationisavailabletoencouragetheinterestedresearchertoappreciatethepowerofthetechniquesthatwehavedescribed.Althoughthephysicalbasisforeachproblemwillbeoutlined,therelevantnon-dimensional,scaledequationswillusuallybethestartingpointfortheanalysis.Thereislittletobegainedbypresentingtheoriginalphysicalproblem,inallitsdetail,togetherwiththenon-dimensionalisation,etcetera,ifonlybecauseoftherequirement,forexample,todefineallthephysicalvariablesineveryproblem.Further,thereasonablelimitationonspacealsoprecludesthis.Theinterestedreadershouldbeabletofillinthedetails,particularlywiththeaidoftheoriginalreference(s).5.1MECHANICAL&ELECTRICALSYSTEMSTheexamplescollectedunderthisheadingarebasedonfairlysimplemechanicalorphysicalprinciples;moreadvancedandspecifictopics(suchascelestialmechanics)whichmighthaveappearedinthisgroupareconsideredseparately.Theexamplestobediscussedare:E5.1Projectilemotionwithsmalldrag;E5.2Child’sswing;E5.3Meniscusonacirculartube;E5.4Drillingbylaser;E5.5ThevanderPol/Rayleighoscillator;E5.6Adiodeoscillatorwithacurrentpump;E5.7AKlein-Gordonequa-tion.E5.1ProjectilemotionwithsmalldragWeconsideraprojectilewhichismovinginthetwo-dimensional(x,z)-planeundertheactionofgravity(whichisconstantinthenegativez-direction)andofadragforceproportionaltothesquareofthespeed(andactingbackalongthelocaldirectionofmotion).Thenon-dimensionalequationsaremostconvenientlywrittenaswhereandtheinitialconditionsaregivenas 199istheangleofprojectionThesmallparameteris(>0),andformotionsuchasthatforashot-put(seeMestre,1991)itsvalueistypicallyabout0.01.Intheseprojectileproblems,themaininterestisinestimatingtherange(andmaximumrange),x,foragivenverticaldisplacement(z),whichmaybezero;here,wefindxwherez=h.Weseekasolution,followingthestraightforwardprocedure(whichmayproduceauniformlyvalidsolution;cf.§2.3):andthenfrom(5.1a,b)weobtainAtthenextorder,wehavetheequationswhichgive,afteranintegrationandonusingtheinitialconditionsont=0,Thuswehave,forexample,whichremainsvalidonlyiftissmallerthanandatthisstagewedonotknowthedomainforwhichz=hatFinally,weintegrateonceagaintofindhowthepositionoftheobject,(x,z),de-pendsontime(t),onandon 2005.SomeworkedexamplesarisingfromphysicalproblemswhereeachforNowwesupposethath=O(1)asthenthetimeatwhichthisisattainedisalsoO(1),andhenceourasymptoticexpansionsarevalidforInparticular,forz=handselectingthelargeroftherootsfori.e.furtheraway,wefind(from(5.6)),thatwhereprovidedthat(Forthecaseoftheshot-putapplicationofthismodel,thelandingpointisbelowtheprojectionpoint,soh<0andthisconditioniscertainlysatisfied.)Finally,usingthisasymptoticexpansionforin(5.5),wefindthattherangeisfromwhich,forexample,wecanestimatetheanglewhichmaximisestherange;thisisleftasanexercise.Thisexamplehasprovedtobeparticularlystraightforward;indeed,becauseofthespecificapplicationthatwehadinmind—theshot-put—alltheasymptoticexpansionsareuniformlyvalid.Ontheotherhand,ifwehadprojectedtheobjectfromthetopofahighcliff,thenwewouldencountertheproblemofi.e.andthenthevalidityoftheoriginalexpansionswouldbeindoubt.Theexpansionsarenotvalidwhenbutwestillhaveu=O(1)althoughthisinvestigationisalsoleftasanexercise.E5.2Child’sswingWeareallfamiliarwiththechild’sswing,andthetechniqueforincreasingthearc(i.e.theamplitude)oftheswing.Theprocessofswingingthelegs(coupledwitha 201Figure8.Pendulumofvariablelength,swingingthroughtheangle(inaverticalplane).smallmovementofthetorso)causesthecentreofgravityofthebodytoberaisedandloweredperiodically.Thiscanbemodelledbytreatingtheswingasapendulumwhichchangesitslength,byasmallamount;themodelequationforthis(intheabsenceofdamping)isfortheangleoftheswing,givenseefigure8.Wechoosetorepresentthechild’smovementontheswingbywhereisa(positive)constantandisaconstantfrequencytobeselected.Wewillfurthersimplifytheproblembyanalysingonlytheinitialstagesofthemotionwhenissmall,sowewritesin(Forlargeramplitudes,wemustretainsinthiscomplicatestheissuesomewhat.AnumberofmoregeneralobservationsaboutthisproblemcanbefoundinHolmes,1995.)Thusweapproximateequation(5.7)aswhichwewillsolveusingthemethodofmultiplescales(cf.E4.1).WetakethefastscaleasT=t(or,moregenerally,butthereisnoadvantageinthis,forweareledtothechoiceand,byvirtueoftheterminaslowscalethuswehavetheidentity 2025.SomeworkedexamplesarisingfromphysicalproblemsEquation(5.8),withbecomesandweseekasolutionintheformwhichisperiodicinT.Inthisproblem,wehaveyettochoosethefrequencyiswhatthechildcancontrolinordertoincreasetheamplitudeoftheswing.Withoutanydampinginthemodel,wemustanticipatethatwecanfindanwhichallowstheamplitudetogrowwithoutbound.First,with(5.10)in(5.9),weobtainandsoon.Equation(5.11a)hasthegeneralsolutionforarbitraryfunctionsandwhichleadstoequation(5.11b)intheformInordertomakecleartheforcingtermsinequation(5.12),whichmayleadtosecularitiesinT,weexpandthelasttermtogiveWeseeimmediatelythatperiodicityinTrequiresandsowewill,forsimplicity,choose(Wearenotparticularlyconcernedabouthowthemotionisinitiated,whichwouldservetoselectaparticularvalueofNowif(allsigncombinationsallowed),thenwerequireforperiod-icityinT,andtheamplituderemainsconstant:thearcoftheswingisnotincreased.However,weareseekingthatconditionwhichwillallowtheamplitudetoincreaseonthetimescalethusweexpectthatwillincreaseasincreases.Thisispossi-bleonlyifandthiscanarisehereif(allsigncombinationstobeconsidered),andsowemaychoose(andthesignofisimmaterial).ThuswithperiodicityinTrequiresthat 203andtheamplitudegrows.Weconclude,therefore,thattheadjustmentsprovidedbythechildontheswingmustbeattwicethefrequencyoftheoscillationoftheswing—whichiswhatwelearntaschildren.E5.3MeniscusonacirculartubeThephenomenonofaliquidrisinginasmall-diametertubethatpenetrates(vertically)thesurfaceoftheliquidisveryfamiliar,asarethemeniscithatforminsideandout-sidethetube.Inthisproblem,wedetermineafirstapproximationtotheshapeofthesurface(insideandoutside)inthecasewhenthesurfacetensiondominates(or,equivalently,thetubeisnarrow).Thebasicmodelassumesthatthemeancurvatureatthesurfaceisproportionaltothepressuredifferenceacrossthesurface(whichismaintainedbyvirtueofthesurfacetension).Withthetwoprincipalcurvaturesofradiiwrittenasandthenthisassumptioncanbeexpressedaswherezistheverticalcoordinateandthepressuredifferenceisproportionaltothe(local)heightoftheliquidabovetheundisturbedlevelfarawayfromthetube;thisrelationisusuallyreferredtoasLaplace’sformula.Indetail,writteninnon-dimensionalform,thisequation(forcylindricalsymmetry)becomeswherethesurfaceisristheradialcoordinatewithr=0atthecentreofthetube,andthetubewall(ofinfinitesimalthickness)isatr=1;theliquidsurfacesatisfiesas(seefigure9).Thenon-dimensionalparameterisinverselyproportionaltothesurfacetensionintheliquidandproportionaltothesquareofthetuberadius(andisusuallycalledtheBondnumber).Wewillexaminetheproblemofsolvingequation(5.13)forwiththeboundaryconditionswhereisthegivencontactanglebetweenthemeniscusandthetube(measuredrelativetotheupwardverticalsideofthetube).Forwetting,thenwehavewhichwewillassumeisthecaseforourliquid.Wesolvetheinteriorandtheexterior(r>1)problemsindependently.ThediscussionthatwepresentfortheexteriorproblemisbasedonLo(1983);anotherdescriptionofboththeinteriorandexteriorproblemsisgiveninLagerstrom(1988).Aswewillsee,thisproblemresultsintheconstructionofauniformlyvalidexpansion(interior),andascalingandmatchingproblemintheexterior(cf.§§2.4,2.5). 2045.SomeworkedexamplesarisingfromphysicalproblemsFigure9.Circulartube(centrer=0)penetratingthesurfaceofaliquidwhoseundisturbedlevelisz=0.InteriorproblemItisafamiliarobservation,atleastforthatthenarrowerthetubethenthehighertheliquidrisesinthetube;thissuggeststhattheheightoftheliquidwillincreaseasWhenthisiscoupledwiththeproperty(theconfirmationofwhichisleftasanexercise)thatnorelevantsolutionofequation(5.13)existsifweignorethetermweareledtowritethesolutionintheformwhereh(0)=O(1)(andh(0)>0).Nowweseekanasymptoticsolutionandsotheleading-orderproblembecomeswith(Theprimedenotesthederivativewithrespecttor.)Oneintegrationof(5.14)givesdirectlythat 205andthen(5.15b)requiresthattheconstantofintegrationbezero.Thecondition(5.15c)nowshowsthatwhichdefinestheheightofthecolumnoftheliquid(measuredatthecentre-lineofthetube).Thesolutionforcanbewrittendownimmediately;itisconvenientlyexpressedintermsofaparameteraswhichsatisfiescondition(5.15a).Wehavefoundthefirstapproximationtotheheightoftheliquidatr=0(namely,andtheshapeofthesurfaceoftheliquidinsidethetube:asectionofasphericalshell.Furthertermsintheasymptoticexpansionscanbefoundquiteroutinely;theseexpansionsareuniformlyvalidforExteriorproblemFindingthesolutionfortheshapeofthesurfaceoutsidethetubeistechnicallyamoredemandingexercise.First,thedeviationofthesurfacefromitslevel(z=0)atinfinityisobservedtobenotparticularlylarge—ascomparedwithwhathappensinside.Thissuggeststhatweattempttosolveequation(5.13)directly,subjecttoLetuswritesothatwearenotcommittingourselves,atthisstage,tothesizeofthenexttermintheasymptoticexpansion;infact,asweshallsee,logarithmictermsarise,althoughwewillnotpursuethedetailshere.TheequationforissimplywherethearbitraryconstantAisdeterminedfrom(5.16a)asOnefurtherintegrationof(5.17)thenyields 2065.SomeworkedexamplesarisingfromphysicalproblemswhereBisasecondarbitraryconstant.Thecomplicationsalludedtoearlierarenowevident:Inraswhichcanneveraccommodatethecondi-tionsatinfinity,(5.16b),foranychoiceofB.Nowanysolutionofequation(5.13)whichadmits(5.16b)mustbalancecontribu-tionsfromeachsideoftheequation;becausethedifficultiesin(5.18)ariseasandthisiswheretheseotherboundaryconditionsaretobeapplied,wewritewhereasWhetherzalsoneedstobescaledisunclearatthisstage;letusthereforewriteandthen(5.13)becomesandso,providedthataswemustselectWechooseandhenceobtaintheequationvalidfarawayfromthetube.Thesolutionofthisequationistosatisfytheboundaryconditionsatinfinityandalsotomatchtothesolutionvalidforr=O(1)i.e.to(5.18).Weseekasolutionofequation(5.19)intheformwherewhichhassolutionsandthemodifiedBesselfunctions;butgrowsexponentiallyas(anddecays),soweselectthesolutionwhereCisanarbitraryconstant.NowthismodifiedBesselfunctionhastheproperties:sotheconditionsatinfinity,(5.16b),aresatisfied.Finally,wematchthefirsttermvalidfor(5.20),tothefirsttermvalidforr=O(1),(5.18).Thelattergives 207andsowherewehaveretainedthelogarithmictermin(aswehavelearntpreviouslyisnecessary;see§1.9).From(5.20)weobtainor,revertingtotheR-variable,simplyFor(5.21)and(5.22)tomatch,wechooseWenotethattheleft-handsideofequation(5.13)allowsztobeshiftedbyaconstant,whichatthisorderisB,andthiscontainsatermInTheasymptoticprocedureforthefullequationmayproceedprovided(theright-handside)0forr=O(1),andthisisthecaseeveninthepresenceofthislogarithmicterm,sinceInasOfcourse,theappearanceofthistermindicatestheneedtoincludelogarithmictermsthroughouttheasymptoticexpansions.Thiscompletesthedescriptionoftheexteriorproblemsofarasweareconcernedhere;muchmoredetailcanbefoundinLo(1983).E5.4DrillingbylaserThisproblemisaone-dimensionalmodelfortheprocessofdrillingthroughathickblockofmaterialusingalaser.Thelaserheatsthematerialuntilitvaporises,andweassumethatthevapouriscontinuouslyremoved;theessentialcharacterofthisproblemisthereforeoneofheattransferataboundary—thebottomofthedrillhole—whichismoving.Insuitablenon-dimensionalvariables,thetemperaturerelativetoambientconditionssatisfiestheclassicalheatconductionequationwith(whichdescribestheinitialstateandtheconditionatinfinity,respectively),and 2085.SomeworkedexamplesarisingfromphysicalproblemsFigure10.Sketchofalaserdrillingthroughablockofmaterial;thebottomofthedrillholemovesaccordingtothevaporisationconditionatthebottomofthedrillholeatseefigure10.ThespeedofthedrillingprocessiscontrolledbywhichisbasedonFourier’slawappliedatthebottomofthehole.Thisset,(5.23)–(5.26),isanexampleofaStefanproblem;seeCrank(1984)formoredetails(andimportantadditionalreferences)aboutthisproblemandothermoving-boundary,heat-transferexamples.(AdiscussionofthisparticularproblemcanalsobefoundinAndrews&McLone,1976,and,inoutline,inFulford&Broadbridge,2002.)OurintentionistoseekasolutionofthissetofequationsforFormanycommonmetals,isfairlysmall(about0.2);smallsignifiesthatrathermore(latent)heatthanheatcontentisrequiredtovaporisethematerial,onceithasreachedthevaporisationtemperature.Weshallapproachthesolutionbyseekingastraightforwardexpansioninpowersofbutwewillneedtotakecareovertheevaluationonthemovingboundary.Wewillfindthattheresultingsolutionisnotuniformlyvalidasandthewayforwardrequiresacarefulexaminationofwhatishappeningintheearlystagesoftheheatingprocess.Wewritesothatequations(5.25)and(5.26)yieldand 209respectively,wherealltheevaluationsonx=XhavebeenmappedtobyallowingTaylorexpansionsabout(andconstructedforThesubscriptsinxdenotepartialderivativesandtheover-dotisthetimederivative.Theleading-orderproblemisthendescribedbytheequationswithaswhereThusandthenthecompletesolutionfor(obtainedbyusingtheLaplacetransform,forexample)iswhereerfcisthecomplementaryerrorfunction:Allthisappearstobequitesatisfactory,atthisstage.Thesolutionfor(5.30),cannowbeusedtoinitiatetheprocedureforfindingthenextterm.Inparticular,(5.29b)becomeswhichmeansthatandthisasymptoticexpansionisnotuniformlyvalidasindeed,weseethatitbreaksdownwhenButwestillhavesofurther,forthegeneralinequation(5.26)tobeO(1)then,retainingT=O(1)whichisnecessaryinordertoaccommodate(5.25),weseethatinthisregion.Thuswedefinethenewvariableswhichproducestheprobleminthisregionas 2105.SomeworkedexamplesarisingfromphysicalproblemswithwhereOneessentialdifficultythatwehaveoverlookedthusfaristhattheprocessofheatingthematerial,startingfromambientconditionsatt=0,requiresthetemperaturetoberaisedbeforeanyvaporisationoccurs,andthereforebeforetheholecanbegintoform.Becausewehaveafailureofouroriginalasymptoticexpansiononlywhenthisshouldcontainthetimeperiodoverwhichtheinitialheatingphaseoccurs.Lettheholebeunformedfor(andweassumethatifitissmaller,wewillrescale),theninthistimeintervalwehaveandsotheboundarycondition(5.35)nowreadsTheresultingproblemfor(whichexcludes(5.34c)anduses(5.36)inplaceof(5.35))nolongercontainsanditcanbesolvedexactly:(ManyofthesestandardsolutionsthatweuseintheoriesofheatconductioncanbefoundinCarslaw&Jaeger,1959.)Theadditionalboundarycondition(5.35c)nowbecomestheconditionwhich,whenattained,heraldstheformationofthehole.From(5.37),weseethatonatthetimeandafterthistimetheholedevelops.Finally,wecompletetheformulationoftheproblemforbutfortimeslargerthanweintroduceandwritewithtogivewithand 211Theleading-orderproblem,asistherefore(withequations(5.39),(5.40)and(5.41),and(5.42a,b)replacedbyThesolutionforcanbefound(byusingtheLaplacetransformagain)andthenusedtodeterminesomedetailsofthiscalculationaregiveninCrank(1984).Althoughtheformofiscumbersome,theresultingexpressionforonisverystraightforward,yieldingwhichmatchespreciselywith(5.31)whenthisisexpandedforThedifficultiesinashavebeenovercome.Thisexamplehasrequiredustoundertakesomequiteintricateanalysisintermsofsingularperturbationtheory,coupledwithacarefulappreciationofthedetailsofthephysicalprocessesinvolved.Thisproblem,perhapsmorethanthepreviousthree,showshowpowerfulthesetechniquescanbeinilluminatingthedetails.Wenowturntoafarmoreroutinetypeofcalculation,althoughtheequationandphysicalbackgroundareimportant,andtheresultingsolutionhasfar-reachingconsequences.E5.5ThevanderPol/RayleighoscillatorThisclassicalexamplerequiresafairlyroutineapplicationofthemethodofmultiplescalestoanearlylinearoscillator(cf.E4.2),althoughthesolutionthatweobtaintakesaquitedramaticform.TheequationfirstcametoprominencefollowingtheworkofvanderPol(1922)ontheself-sustainingoscillationsofatriodecircuit(forwhichtheanodecurrent-voltagelawtakestheformofacubicrelation).However,essentiallythesameequationhadalreadybeendiscussedbyRayleigh(1883),asamodelfor‘maintained’vibrationsin,forexample,organpipes.(AsimpletransformationtakesRayleigh’sequa-tionintothevanderPolequation.)WewillwritetheequationintheRayleighformforinthecontextofthevanderPolproblem,isproportionaltothegridvoltage,V.[Acircuitdiagramisgiveninfigure11,andthegoverningequationsforthistriodecircuitarewhereandarepositiveconstants.] 2125.SomeworkedexamplesarisingfromphysicalproblemsFigure11.Circuitdiagramforthetriodeoscillator.Themethodofmultiplescalesleadsustointroduceandthenequation(5.44),forbecomesWeseekasolutionintheformwhichisperiodicinTanduniformlyvalidasTheequationsforthefirstthreetermsareThegeneralsolutionofequation(5.45a)is(forarbitraryfunctionsandandthenequation(5.45b)canbewritten 213whereFortobeperiodicinT,i.e.inwerequirewhichmaybeintegratedtogivewhereHence,foranyinitialamplitudeasthesolutionexhibitsalimitcycle(ultimatelyanoscillationinTwithamplitude2).Thisistheraisond’êtreofthetriodecircuit.Ifweproceedwiththeanalysis(thedetailsofwhichareleftasanexercise)wefind,first,thatwhereandareadditionalarbitraryfunctions.(TheconstantisfixedbytheinitialdataAtthenextorder,wededucethattheamplituderemainsboundedasonlyifAnotherproblemwithanelectricalbackground,butwitharatherdifferentasymp-toticstructure,willnowbedescribed.E5.6AdiodeoscillatorwithacurrentpumpInthisproblem,whichcontainstwosmallparameters(oneofwhichisusedtosimplifysomeoftheintermediateresults,asexpedient),weseektheinitialconditionwhichleadstoaperiodicsolution.Thecircuit(figure12)isrepresentedbytheequationsandthenKirchhoff’slawgiveswhichleadstothenon-dimensionalequationTypicalvaluesoftheparametersare:(Thisequationwasbroughttotheauthor’sattentionbyacolleague,DrArmstrong.) 2145.SomeworkedexamplesarisingfromphysicalproblemsFigure12.Circuitdiagramforthediodeoscillatorwithacurrentpump.Weseekthefirsttermofanasymptoticexpansionforforfixed(andwemaytakeadvantageofsmalltosimplifysomeofthedetails,butthisisnotessential).Thustheproblemforfrom(5.47),becomeswhichcanbesolvedexactly(bywritingitisconvenienttoexpresstheinitialvalueasandthenweobtainThenatureofthissolutionis,perhaps,notimmediatelyapparent,butitbecomesmoretransparentifweinvokeWeprovideapproximationstoforvarioust,below(thedetailsofwhichareleftasanexercise): 215Figure13.Sketchofthesolution(5.49)forsmall(Mostoftheseresultsrequiretheuseofasymptoticestimatesofforvarioust,asseee.g.Olver,1974,orCopson,1967;cf.Q1.16.)This(approximate)solutionissketchedinfigure13.Clearlyandsothisfirsttermisnotperiodic;however,forthesolutionisexponentiallysmall(asandifthisbecomesassmallasthenthetermomittedin(5.48)cannotbeignored.(Thisobservationalsofollowswhenwefindthenexttermintheasymptoticexpansionandseekabreakdown;thisisarathertiresomeprocesshere,sowetreattheproblemasoneofrescalingthedifferentialequation,asdiscussedinForalthoughtheprecisedomainisnotyetknown,wewriteandthen(5.47)becomesWeseekasolution 2165.Someworkedexamplesarisingfromphysicalproblemsandso,from(5.51),weobtaintheequationwhichcanbeintegrateddirectly:whereisanarbitraryconstant.Thissolutionistomatchtotheasymptoticexpan-sionforand,inparticular,tothefirstterm,(5.49).Further,weareseek-ingthecondition(s)thatensuretheexistenceofaperiodicsolution,sowealsoimposeInprinciple,weareabletomatch(5.49)and(5.52)forarbitrarybuttheresultisnotparticularlyuseful;further,theperiodicityconditionimpliesthat,forsmallthenmustbesmall.Thusweagaininvokeandfrom(5.52)weobtain(forwhichmatcheswith(5.50f)ifwechooseIndeed,thismatchisvalidforTheperiodicityrequirement,(5.53),nowbecomesandthusfrom(5.52)(withand(5.50a)weobtainThus(5.54)and(5.55)implythatwemusthavewhereandsoi.e.forthegivenwehave 217Thusforaperiodicsolutiontoexist,werequiretheinitialamplitude,a,toberestrictedinvalue,aconclusionthatcanalsobereachedonthebasisofanexamina-tionofthedirectionfieldforthisequation.Foragiventheamplitudeischosenbyselectingandwhichsatisfy(5.54)and(5.55),whereisgivenby(5.56).Theproblemoftheexistenceofsolutionstothisset—solutionsdoexist!—isleftasanadditionalinvestigation.Thisexamplehasdemonstratedhowwecanextractfairlysimpleestimatesfromasolutionwithacomplicatedstructure,eventhoughthegoverningdifferentialequationmayhavepersuadedusthatnoseriousdifficultieswouldbeencountered.Finally,weapplythemethodofmultiplescalestoapartialdifferentialequationofsomeimportance.E5.7Klein-GordonequationThegeneralformoftheKlein-Gordonequation,writteninonespatialdimension,iswhereV(u)(whichcanbetakenasapotential,inquantum-mechanicalterms)is,typically,afunctionwithnonlinearitymoreseverethanquadratic.WewillconsidertheproblemforwhichandsointroduceaparameterwhichwewillallowtosatisfyTheequa-tionwiththischoicearises,forexample,inthestudyofwavepropagationinacoldplasma.(ThechoiceV(u)=–cosugivesrisetotheso-calledsine-Gordonequation—apunontheoriginalname—andthisequationhasexact‘soliton’solutions;seee.g.Drazin&Johnson,1992.)Wefollowthetechniquedescribedin§4.4,andsoweintro-duceandthenwithweobtaintheequationWeseekasolutionwhichistobeperiodicinintheform 2185.Someworkedexamplesarisingfromphysicalproblemsandsoweobtainthesetofequationswherewehaveelectedtotakek(andhenceasconstants,forthepurposesofthisdiscussion.Asuitablesolutionofequation(5.58)iswithandarearbitraryfunctions.Equation(5.59),fornowbecomeswhereWeuseandthenobservethatisperiodicini.e.inonlyifwhichhavethegeneralsolutionsforarbitraryfunctionsFandThustheleadingtermintheasymptoticexpansioniswhereandweobservethatwemaywritetheoscillatorytermasBoththewavenumber(k)andthefrequencyhavesmall-amplitudecorrections—weassumethatisaboundedfunction—whichproducesanapproximatespeedofthecarrierwaveofwhichshowsthattheinclusionofthenonlinearityintheequationistochangethe 219speedofthewave.Indeed,largerwavestravelfaster—atypicalobservationinmanywavepropagationphenomena.Notealsothattheamplitudefunction,representspropagationatthespeedwhichispreciselythegroupspeed,see§4.4.Theinitialdataforthissolutionmusttaketheformwherekisagivenconstant(ratherthanappearinthemoregeneralformsinandforasuitableamplitudefunction.Thisconcludesthesetofexamplesthathavebeentakenfromaratherbroadspectrumofsimplemechanicalandelectricalsystems.Wewillnowconsiderthemorespecialisedbranchofclassicalmechanics.5.2CELESTIALMECHANICSWepresentthreetypicalproblemsthatarisefromplanetary,orrelated,motions:E5.8TheEinsteinequation(forMercury);E5.9Planetaryrings;E5.10Slowdecayofasatelliteorbit.E5.8TheEinsteinequation(forMercury)ClassicalNewtonian(Keplerian)mechanicsleadstoanequationforasingleplanetaroundasunoftheformwhereisthepolarangleoftheorbit,uisinverselyproportionaltotheradialcoordinateoftheorbitandhmeasurestheangularmomentumoftheplanet.However,whenacorrectionbasedonEinstein'stheoryofgravitationisadded,theequationbecomeswhereisasmallparameter(aboutforMercury,theplanetforwhichtheequationwasfirstintroduced).Theplanistofindanasymptoticsolutionofequation(5.60),usingthemethodofmultiplescales(cf.E4.2),subjectto(Ithappensthatequation(5.60)canbeintegratedasitstands,intermsofJacobianellipticfunctions,butthistendstoobscurethecharacterofthesolutionandistherefore;hardlyworththeeffortsinceissosmall.) 2205.SomeworkedexamplesarisingfromphysicalproblemsWeintroduceandthentheequationforbecomesWeseekasolution,periodicinTandboundedininthefamiliarformandsoobtainandsoon.TheinitialconditionsgiveThegeneralsolutionforfrom(5.61a),isforarbitraryfunctionsandtheinitialconditions,(5.62a,c),requirethatandsoweselectEquation(5.61b)thenbecomeswhereisperiodicinT,i.e.inonlyif 221Then,withconditions(5.63),weseethatwhichleavesthesolutionforasItisleftasanexercisetoshowthatthesolutionofequation(5.61c),forisperiodicinTandboundedasifwithandThisexamplehasprovedtobeaparticularlystraightforwardapplicationofthemethodofmultiplescales;thenextisaratherlessroutineproblemthatcontainsaturningpoint(seeE4.6).E5.9PlanetaryringsInastudyofamodelfordifferentiallyrotatingdiscs(Papaloizou&Pringle,1987),theradialstructureoftheazimuthalvelocitycomponentforlargeazimuthalmodenumber(essentiallyhere)satisfiesanequationoftheformThisequationclearlypossessesaturningpointat(see§2.8);letusexaminethesolutionnearthispointfirst.Fortheneighbourhoodofwesetwhereasandignorethescalingofv(becausetheequationislinear),soequation(5.64)becomes(withThusweselectandwithweobtaintheleading-orderequationwhichisanAiryequation(seeequations(4.52),(4.54))withaboundedsolution 2225.SomeworkedexamplesarisingfromphysicalproblemswhereCisanarbitraryconstant.ThesolutioninandisnowfoundbyusingtheWKBmethod(seeE4.5).(Inthisexample,wewillfindanapproximationtothesolution,ineachofthethreeregions,byusingonlytheappropriatelocalvariable;aswehaveseenin§4.3,allthiscouldbeexpressedusingformalmultiplescales.)ForR>0,isoscillatoryandsoinweseekasolutionofequation(5.64)intheformwhere(asisascalingtobedetermined,isaconstantandistobewrittenasasuitableasymptoticexpansionwhenweknowThusweobtainwhereandtheprimedenotesthederivativewithrespecttor.Thuswerequireandsowewriteandhenceweobtainandsoon.Thesetwoequationsarereadilysolved,togivewhereAisanarbitraryconstant.Thecorrespondingsolutionfor(thedetailsofwhichareleftasanexercise)is 223whereBisasecondarbitraryconstant.Thematchingof(5.68)and(5.69)with(5.65)(usingtheresultsquotedin(4.54))followsdirectly.From(5.68)and(5.65)wefindthatmatchingispossibleiffrom(5.69)and(5.65)weobtainwhichimpliestheconnectionformulaA=2B.(Wealsoseethat,ifC=O(1),thenAandBareasOurfinalexampleunderthisheadingisrelatedtoproblemE5.1,butnowplacedinacelestialcontext.OurpresentationisbasedonthatgivenbyKevorkian&Cole(1981,1996).E5.10SlowdecayofasatelliteorbitTheequationsforasatelliteinorbitaroundaprimary(intheabsenceofallothermasses),withadragproportionaltothearefirstwrittendownintermsofpolarcoordinates,Thesearethentransformedtoand(wheretistime),andfinally—thisisLaplace’simportantobservation—toandweobtainthenon-dimensionalequationswhereisameasureofthedragcoefficientonthesatellite.Weseekasolutionofthispairofequations,forsubjecttotheinitialconditionsi.e.conditionsprescribedatwhatwewillcallHere,vati.e.t=0)istheinitialcomponentofthevelocityvectorintheweassumethatthisisgivensuchthatv>1andthatitisindependentofTheformofequations(5.70),andourexperiencewithproblemsofthistype,suggeststhatweshouldintroducenewvariables(multiplescales)andmoregeneralchoicesforT(e.g.orareunnecessaryinthisproblem.Beforeweproceed,observethatequations(5.70)containonlythrough 2245.Someworkedexamplesarisingfromphysicalproblemssoitisconvenienttosolveforuandfirst(andthentfollowsafteranintegrationor,atleast,byquadrature).Thus,forthepurposesofconstructingasolution,letussetwith(andtheconditionontatisredundantatthisstage).Thusourequations(withbecomeandweseekasolutionwhichisperiodicinT.Fromequations(5.71)weobtainandsoon.Theexactsolutionofequations(5.72),whichdescribeaKeplerianellipse,isusuallywrittenintheformwhereistheeccentricity,denotesthepositionofthepericentre(i.e.atandistheangularmomentum.(Wemaywritewhereisthesemi-majoraxis,ifthisisuseful.)Equation(5.73b)nowbecomes(withappropriateuseofequations(5.74))whichmaybeintegrated,atleastformally,togive 225whereAisanarbitraryfunction.Thisexpressioncanbeuseddirectlyin(5.73a)togiveandweimposetheconditionthatbeperiodicinT.Thiscanbedonequitegener-allybywritingsolvingforandthenimposingperiodicity.However,byvirtueoftheintegraltermin(5.75),thisproducesasomewhatinvolvedandfar-from-transparentresult.Inordertomakesomeheadway,andtoproduceusefulsolutions,letussupposethatthesatelliteorbitisinitiallyalmostcircular—afairlycom-monsituation—sothatthevalueofissmall.Inparticular,theinitialconditionsgiveandsonowweareassumingthatviscloseto1;itwillsoonbecomeclearthatthisapproximationholdsasbecausewewillshowthatdecreasestozerofromitsinitialvalue.Forsmalle,wefindthatandsoweexpandequation(5.75)aswhereÂisanewarbitraryfunction(replacingA).Nowthesolution,ofequation(5.76)isperiodicinTifwhichhavesolutions(correcttothisorderine):where(whichissmall). 2265.SomeworkedexamplesarisingfromphysicalproblemsWesee,therefore,thatthesmalldraginthismodelleavesthepericentreunaffectedtheeccentricitydecreasestowardszero(fromitsalreadyassumedsmallvalue)andthesemi-majoraxisalsoapproacheszeroasThustheorbitgraduallyspiralsinand,asitdoesso,itbecomesmorecircular.Wehaveseenhowwemighttackletheproblemoforbitsthatgrazetheatmosphereofaplanet,althoughhereitwasexpedienttoassumethattheorbitwasinitiallynearlycircular.Ifthisisnotthecase,thenwillnotbesmallandwefaceamoreexactingcalculation,althoughtheessentialprinciplesareunaltered.5.3PHYSICSOFPARTICLESANDOFLIGHTInthissection,wewillexaminesomeproblemsthatarisefromfairlyelementaryphysics;thesewilltouchonquantummechanics,lightpropagationandthemove-mentofparticles.Inparticular,wediscuss:E5.11PerturbationoftheboundstatesofSchrödinger’sequation;E5.12Lightpropagatingthroughaslowlyvaryingmedium;E5.13Ramanscattering:adampedMorseoscillator;E5.14Quantumjumps:theiontrap;E5.15Low-pressuregasflowthroughalongtube.E5.11PerturbationoftheboundstatesofSchrödinger’sequationThisisaclassicalprobleminelementaryquantummechanics;itinvolvesthetime-independent,one-dimensionalSchrödingerequationwhereisthegivenpotentialandEistheenergy(i.e.theeigenvaluesofthedifferentialequation).Weseeksolutionsforwhichasandisfinite(andconventionally,wechoosetoprovideanormalisationoftheeigenfunction,Inthisexample,wechooseandthisistobeauniformlyvalidapproximationasforTheproblemthenbecomeswithasandWeseekasolutionbyassumingastraightforwardexpansion,andwewillcommentontheconditionsthatensureauniformexpansionvalidforallx;see§2.3.Thuswewrite 227andsoequation(5.77)giveswithandWeassumethatasolutionexistsforwillgiveanexampleshortly—then(5.78b)canbewrittenwhichisintegratedoverallx.Afterusingintegrationbypartsonthefirstterm,andinvokingthedecayconditionsatinfinity,weobtainwhichreducestowhenwemakeuseof(5.78a)and(5.80b).Thusthecorrectiontotheenergyisknown(andweassumethatandaresuchastoensurethatthiscorrectionisfinite).Thesameprocedureappliedtoequation(5.78c)yieldsAsimplepotentialisthatassociatedwiththeharmonicoscillator,namelythenequation(5.78a)becomeswhichcanberewrittenintermsoftogive 2285.SomeworkedexamplesarisingfromphysicalproblemsThisisHermite’sequationwithsolutionsthatguaranteeatinfinityonlyifInparticular,andeachofthesesolutionshasbeenchosentosatisfythenormalisationconditionon(5.80a).Atypicalchoiceforisbutinordertoensurethatisuniformlyvalidforallx,then(toavoidabreakdownasand(toavoidabreakdownasforp>2;thecasep=2andisequivalenttoLetuscalculateform=0(sowehaveandandforaperturbationwithp=2andthusandsotheenergybecomesObservethat,atthisorder,wedonotneedtodeterminetofindCalculationsofthistype,forvariousandcanbefoundinanygoodtextonquantummechanics.E5.12LightpropagatingthroughaslowlyvaryingmediumFermat’sprinciplestatesthatlighttravelsbetweenanytwopointsonapathwhichminimisesthetimeofpropagation.Ifthepath,intwodimensions,iswrittenasy=y(x),andthespeedoflightatanypointisc(x,y),theny(x)mustsatisfyThisequationcanbeobtainedeitherfromtheeikonalequation(seeQ4.27)forraysorastherelevantEuler-Lagrangeequationinthecalculusofvariations.[Inthespecialcasewherethemediumvariesonlyinx,sothatc=c(x),weobtain 229if,foralightray,wesetthenwhichisSnell’slaw.]Letussupposethatthepropertiesofthemediumslowlychangeonthescaleintheformandsotheinherentdifficultyofthisproblemisnowevident:weseekyetyappearsinthefunctionc.Inthecasethatc=constant,thelightraysarestraightlinese.g.letusseekasolutionof(5.81)whichsatisfiespreciselytheseconditionsi.e.Forthegivenc,(5.82),andwithweseethatbothandareandso,from(5.81),wehavethatacursoryanalysisoftheproblemsuggeststhatwewritethesolutionintheimplicitformwhereThusandwewillassumethatissuchthatA,andallitsrelevantderivatives,leadtouniformasymptoticexpansionsinthedomainwherethematerialexists.Thisensures,forexample,thatwemaywriteforallX,Yinthedomain.Differentiationof(5.84)yieldswhichcanbesolvedfor(givenfrom(5.85)).Weseekasolutionintheform 2305.Someworkedexamplesarisingfromphysicalproblemsandthen(5.86)and(5.85)inequation(5.81)produces,atleadingorder,theequationforThisequationcanbesolvedingeneral(byintroducingandexpressingthesolutionintermsofonceisexpressedinthesesamevariables);thisisleftasanexercise.Wepresentasimpleexample:forwhichthesolutioncanbewritten(onintroducingwhichsatisfiesonasrequiredbytheconditionattheorigin.Thisproblemoffindingthepathofalightrayhasusedtheideaofmultiplescalesinalessroutineway;wenowexamineanequationforwhichamorefamiliarapproach(§4.2)isapplicable.E5.13Ramanscattering:adampedMorseoscillatorUndercertaincircumstances,asmallfractionoftheincidencelightpropagatingthroughamediummaybescatteredsothatthewavelengthofthislightdiffersfromthatoftheincidentlight—usuallyitisofgreaterwavelength.ThisiscalledRamanscattering.Anexampleofthis(Lie&Yuan,1986),whichincorporatestheMorse(exponential)modelforthepotentialenergyofatomsasafunctionoftheirseparation,isThisequationalsoincludesa(weak)lineardampingterm,whichwewillcharacterisebywewishtofindasolution,subjecttotheinitialconditionsBecauseequation(5.87)isnonlinear(althoughtheunderlyingsolution—validforbeexpressedintermsofelementaryfunctions),wemustexpectadevelopmentalongthelinesofthatdescribedinE4.3.Weintroducefastandslowscalesaccordingtoandthenseekasolutionwhichhasaconstantperiod(inT).Equation(5.87)therefore 231becomes,withwherethesolutionistobeTheequationsforthearethereforeandsoon.Themostgeneralsolutionofequation(5.89),withaconstantperiodisforarbitraryandnotethat,foraconstantperiod,wewillrequirethatTheinitialconditions,(5.88),nowonaresatisfiedbythechoicesandtheexistenceofareal,oscillatorysolution(oftheform(5.91))requiresthat0>a>–ln2.Theperiodicitycondition,whichwilldefinecanbeobtainedfrom(5.90)byfirstwritingandthenusingtheT-derivativeof(5.89);thisyieldsThiscanbeintegratedoncedirectly,whenmultipliedbysothatwenowhavei.e.Fortobeperiodic,thensomustbebothFandwiththeperiodprescribedas 2325.SomeworkedexamplesarisingfromphysicalproblemswethereforeobtaintheperiodicityconditionFinally,theevaluationoftheintegral(whichisleftasanexercise)yieldsandsowehave,aftersolvingforwhereWeobservethatasandthatwithThisdescribestheevolution(shift)ofthefrequency,asthedampingprogressivelyaffectsthesolution.E5.14Quantumjumps:theiontrapInthestudyofthediscontinuousemissionorabsorptionofenergy(quantumjumps),asingleionistrapped(inanelectromagneticdevicecalledaPaultrap;seeCook,1990)anditsmotionisgovernedbyanequationoftheformHere,isaparameter,v(x)isagivenfunction(sufficientfortheexistenceofandweseekthecomplex-valuedfunctionforInthiscase,weseethattheoscillatorytermontherightoscillatesrapidlyandsoweusethemethodofmultiplescalesintheformwhichgivestheequationWeseekasolution 233whichisuniformlyvalidasfrom(5.92)weobtainandsoon.Thesolutionof(5.93a)isimmediatelyseentobewhereisanarbitraryfunction;then(5.93b)becomesThiscanbewrittenasandhencefortoremainboundedasalsoensuresperiodicityinrequireAsolutionofthisequationcanbeexpressedasbutverylittleheadwaycanbemade,atthisstage,withoutsomeknowledgeofv(x).Wedonote,however,thatfork<0thesolutionsforareessentiallyexponential(growinganddecaying),butforboundedandlargeenough,thesolutionsareoscillatory.Ourfinalexampleisaproblemofagasflowbut,becausethedensityissolow,themodelisbasedonanapproachthatinvokestheideasofstatisticalmechanics.E5.15Low-pressuregasflowthroughalongtubeTheflowofagasthroughalongcirculartube,i.e.radius/lengthissmall,wheremolecularcollisionsareassumedtooccuronlywiththewallofthetube,canberepresentedbytheClausingintegralequation 2345.SomeworkedexamplesarisingfromphysicalproblemsHere,istherateofmolecularcollisions(withthewall)betweenxandandtheratecontributedbythosemoleculesthathavetheirfirstcollisionbetweenthesesamestations.Thekernel,K(x–y),measurestheprobabilitythatamoleculewhichhascollidedwiththewallatx=ywillcollideagainbetweenthestations.(Thistypeofprocessiscalledafree-molecularorKnudsenflow.)Whenweintroducetheappropriatemodelsforandnon-dimensionaliseandusethesymmetryofn(x)(i.e.n(x)+n(–x)=0sothatn(0)=0),weobtaintheequationwiththenormalisedboundarycondition(SeePao&Tchao,1970,andDeMarcus,1956&1957,andformoregeneralbackgroundinformation,Patterson,1971.)Atfirstsight,equation(5.94)looksquitedauntingandverydifferentfromanythingwehaveexaminedsofarinthistext.However,thefirsttermsontherightdoindicatethepresenceofboundarylayersnearx=±1/2,soperhapsourfamiliartechniquescanbeemployed.Forxawayfromtheendsofthedomain,theexpansionofthefirsttermsin(5.94)leadstotheasymptoticformoftheequation:Nowwemustestimatetheintegral,forwhichweuseideasdiscussedin§2.2andexerciseQ2.8.ThisisaccomplishedbyexpressingthedomainofintegrationasandwherebutsuchthatItisleftasanexercise(whichinvolvesconsiderableeffort)toshowthat(5.95)eventuallycanbewrittenasandsoasmustsatisfy 235Butsowehavesimplythisisthefirsttermintheasymptoticexpansion,whereisanarbitraryconstant,validawayfromtheboundarylayers(see§2.6).(Ifweweretoapplytheboundaryconditiononx=1/2,thenwewoulddeducethatwhichturnsouttobecorrect,asweshallseebelow.)Fortheboundarylayernearx=1/2,wewriteandwhichgives(from(5.94))ButthedominantcontributiontotheintegralwillcomefromthebehaviourofNoutsidetheboundarylayersi.e.weuseWhenwedothis,theintegraltermyieldstheresultandthisisusedin(5.96),togetherwiththeboundaryconditiontogiveThecorrespondingsolutionintheboundarylayerattheotherendisobtainedfromthisresultbyformingwhereNotethat,becausetheboundaryconditionhasbeenusedhere,isnowdetermined(inawayanalogoustomatching)andsoawayfromtheboundarylayers.Thisconcludesallthatwewillwriteaboutthisverydifferenttypeofboundary-layerproblem;seePao&Tchao(1970)formoredetails.5.4SEMI-ANDSUPERCONDUCTORSThestudyofsemiconductorsandofsuperconductors,asithasunfoldedoverthelast50yearsorso,hasthrownupanynumberofinterestingandimportantequationsthatdescribetheirpropertiesanddesigncharacteristics.Wewilllookatthreefairlytypicalexamples:E5.16Josephsonjunction;E5.17Ap-njunction;E5.18Impuritiesinasemiconductor. 2365.SomeworkedexamplesarisingfromphysicalproblemsE5.16JosephsonjunctionTheJosephsonjunctionbetweentwosuperconductors,whichareseparatedbyathininsulator,canproduceanACcurrentwhenaDCvoltageisappliedacrossthejunc-tion(thisbyvirtueofthetunnellingeffect).Anequationthatmodelsanaspectofthisphenomenon(Sanders,1983)iswhereaandbaregivenconstants,andWewillconstructtheasymptoticsolution,usingthemethodofmultiplescales,forNotethat,intheabsenceofthetermthenisasolutionofthecompleteproblem.Weanticipatethatthepresenceofwillforceanon-zerosolutionwhich,ifitremainsbounded,shouldbeforalltime(t);thuswewriteFurther,weintroduceandsosatisfiestheequationWeassumeabounded,periodicsolutioncanbewrittenasandsowemayexpandThusweobtainthesetofequationsandsoon.Theseequationsfollowthepatternforanearlylinearoscillator;see§4.1.Thegeneralsolutionof(5.98a)iswithinitialconditions 237thenequation(5.98b)becomeswhereImmediatelyweseethatisperiodicinT,i.e.inonlyifandsowehavewhichshowsthataboundedsolution(asrequiresThusthesolution,tothisorder,istheconstructionofhigher-ordertermsisleftasafairlyroutineexercise.E5.17Ap-njunctionAp-njunctioniswheretwosemiconductingmaterialsmeet;suchjunctionsmayperformdifferentfunctions.Theonethatwedescribeisadiode.Weanalysethedeviceforwherethejunctionsitsatx=0(and,bysymmetry,itextendsintoandanohmiccontactisplacedatx=1.Insuitablenon-dimensional,scaledvariableswehavewhereeistheelectrostaticfield,ptheholedensityandntheelectrondensity.Theterm‘+1’in(5.99a)isaconstant‘doping’densityandwewillassumethatthecurrentdensity,I(x)(appearingin(5.99b,c)),isgiven;indeed,inthissimplemodel,wetakeI(x)=constant.Theboundaryconditionsareandisoursmallparameter(typicallyabout0.001).(SeeShockley,1949;Roosbroeck,1950;Vasil’eva&Stelmakh,1977;Schmeisser&Weiss,1986.)Itisevidentthattheset(5.99)exhibitsthecharacteristicsofaboundary-layerproblem(§§2.6,2.7)becausethesmallparametermultipliesthederivativesineachequation.However,aneatmanoeuvreallowsoneequationtobeindependentof 2385.SomeworkedexamplesarisingfromphysicalproblemsLetusintroduceu=np,thenequations(5.99)canberewrittenaswithNowawayfromtheboundarylayer(whosepositionisyettobedetermined)wewriteeachofandasasymptoticexpansionsandthentheleadingorder(from(5.100))yieldsThuswemustselectthesolutionitisnowevidentthatthecondition(see(5.101a))cannotbeattainedbythissolution,sotheboundarylayermustbeatx=0—thepositionofthejunction—andthuswearepermitted(inthissoluiton)tousetheboundaryconditionsatx=1,togiveNowfromequations(5.100a,b),itisclearthattheboundary-layerthicknessisandsoweintroduceandwriteforeachofe,pandu,toobtainthesetwithatX=0andmatchingconditionsforThus,fromequations(5.103),theleading-orderterms(zerosubscripts)inthestraightforwardasymptoticexpansionssatisfytheequations 239andthelastequationsimplyrequiresthat(whichisequivalenttotheobservation,from(5.100c),thatthereisnoboundary-layerstructureinthesolutionforThefirsttwoequations,(5.104a,b),giveanequationforwhichcanbeintegratedonce(bysettingtogivewhereAisanarbitraryconstant.Sadly,wecannotintegrateonceagain(soanumericalapproachmightbeconsidered),butwecanmakeafewobservations.TheboundaryconditiononX=0becomesand,inaddition,thematch-ingconditionissatisfiedif(see(5.102c)whichrequiresthechoice(ItisleftasanexercisetoshowthatthereisasolutionforwhichHowever,moresuccessinthedevelopmentofusefulanalyticaldetailispossibleifweuse(5.104a,b)toproduceanequationforOneintegrationthenproducestheresultwherethearbitraryconstantmust,inordertosatisfythematchingconditionatinfinity, 2405.SomeworkedexamplesarisingfromphysicalproblemstakethesamevalueasAabove:Thus,althoughweareunabletowritedownanexpressionforintermsofelementaryfunctions,wedohaveasimplerelationbetweenandinparticular,weseethattheelectrostaticfieldatthejunction,x=0.Moredetailscanbefoundinthereferencescitedabove;alsoadiscussionofsimilarproblemsisgiveninSmith(1985)andO’Malley(1991).E5.18ImpuritiesinasemiconductorAsignificantissueinthedesignandoperationofsemiconductorsisthepresence,andmovement,ofimpurities.Inparticular,thelevelofimpuritiesthatdiffusefromtheoutersurfaceofthematerialandmovetooccupyvacantlocationswithinthestructurecanbemodelled(King,Meere&Rogers,1992)bytheequationsHere,istheconcentrationoftheimpurities,theconcentrationofvacancies(holes),andandarepositiveconstants;theboundaryandinitialconditionsareAlltheseboundaryvaluesareconstants—andispositive—and,further-more,theappearanceofthesamevaluesatt=0andassuggeststhatwecouldusearelevantsimilaritysolution.Thesmallparameter,isassociatedonlywiththeandsowemayanticipatetheexistenceofaboundary-layerstructureinv,butnotinc;cf.E3.3.Indeed,itshouldbeclearthatthisnecessarilymustbenearx=0andusedtoaccommodatetheboundaryvaluegivenby(5.106d).Awayfromx=0,weseekasolutionwith 241which,from(5.105),mustthereforesatisfytheequationsandthen(5.107b)giveswhereisanarbitraryfunction.Wemayimposetheinitialconditions,(5.106a,b),andsothen(5.107a)becomessimplyandthe(similarity)solutionwhichsatisfies(5.106a,c,e)is(providedthatt=0isinterpretedasThuswhichdoesnotsatisfytheboundaryvalueonx=0andsowerequiretheboundarylayernearhere.Letusintroduceandwritethenequations(5.105)becometheleading-orderproblem(zerosubscript)thereforesatisfiesThesolutionofthispairistosatisfythematchingconditions 2425.Someworkedexamplesarisingfromphysicalproblemsandthus(5.108a)givesdirectlyandthen(5.108b)becomesTheappropriatesolutionofthisequation,whichsatisfiesboththematchingconditionand(5.106d),isHigher-orderterms,inboththeouterandboundary-layersolutions,canbefoundaltogetherroutinely(althoughthecalculationsarerathertedious).Thiscompletesourfewexamplesinthisgroup;wenowturntooneoftheareaswheresingularperturbationtheoryhasplayedaverysignificantrôle.5.5FLUIDMECHANICSThestudyoffluidmechanicsisbroadanddeepanditoftenhasfar-reachingconse-quences.Manyoftheclassicaltechniquesofsingularperturbationtheorywerefirstdevelopedinordertotackleparticulardifficultiesthatwereencounteredinthisfield.Examplesthatareavailablearenumerous,andanynumbercouldhavebeenselectedfordiscussionhere(andsomehavealreadyappearedasexamplesinearlierchapters).Wewillcontentourselveswithjustfourmoreverydifferentproblemsthatgiveaflavourofwhatispossible,buttheseareallfairlyclassicalexamplesoftheirtype.Manyotherscanbefoundinmostofthetextsalreadycitedearlier.Wewilldiscuss:E5.19Viscousboundarylayeronaflatplate;E5.20Veryviscousflowpastasphere;E5.21Apistonproblem;E5.22Avariable-depthKorteweg-deVriesequationforwaterwaves.E5.19ViscousboundarylayeronaflatplateThesolutionofthisproblem(about1905),withPoincaré’sworkoncelestialmechanics,togetherlaidthefoundationsforsingularperturbationtheory.Inthisexample,weconsideranincompressible,viscousfluid(iny>0)flowingoveraflatplate,theflowdirectionatinfinitybeingparalleltotheplate.ThegoverningequationsaretheNavier-Stokesequation(intheabsenceofgravity)andtheequationofmassconservation: 243Figure14.Sketchoftheviscousboundarylayeronaflatplate.whereistheReynoldsnumber(andwehaveusedsubscriptsthroughouttodenotepartialderivatives).Wewillconsidertheproblemofsteadyflowwiththeboundaryconditionsforuniformflowatinfinityarewhichimplythatthepressurepconstantawayfromtheplate(andwewillnotanalysethenatureoftheflownearx=0);theplatewillextendtoinfinityThepresenceofthesmallparametermultiplyingthehighestderivatives,isthehallmarkofaboundary-layerproblem.Inparticular,the(inviscid)problemcansatisfyv=0ony=0,butnotas(inx>0),soweexpectaboundary-layerscalinginy;seefigure14.Outsidetheboundarylayer,thesolutioniswrittenandsoequations(5.109)givesubjecttotheboundaryconditions(5.110a,b,d),forzero-subscriptedvariables;thishasthesolution(Itisclearthatthissolutionhasadditionalproblemsforony=0,wherethestagnationpointexistsattheleadingedgeoftheplate.)Notethatthissolutioncanbeexpressedintermsofthestreamfunction:(where,ingeneral,Theregionoftheboundarylayerisdescribedbythescaledvariablewhereasandxisunscaled.(Wewouldneedtoscalexnearexcep-tionalpointssuchastheleadingedge,apointofseparationandthetrailingedgeofa 2445.Someworkedexamplesarisingfromphysicalproblemsfiniteplate.)Byvirtueoftheexistenceofastreamfunction,weseethatwemustalsoscalewewriteandthenwemustchoosetogiveThissetistobesolved,subjecttotheboundaryconditions(5.110c,d),writteninboundary-layervariables,andmatchingconditionsforTheleading-orderproblem(zerosubscripts)satisfieswithandthematchingconditionsTosolveequations(5.111),wenote,first,thatandthenthematchingconditionrequiresthroughoutthisregion.Nextweuse(5.111c)toallowtheintroductionofastreamfunctionandthen(5.111a)canbewrittenwithandTherelevantsolution(Blasius,1908)takesasimilarityform:directsubstitutionthenyieldstheordinarydifferentialequationforwithThisequationmustbesolvednumerically;thepropertiesofthesolutionagreewellwithexperimentaldataforlaminarflows.Itisleftasanexercisetoshowthatthereare 245solutionsoftheformandthevaluesoftheconstants,andareobtainedfromthenumericalsolutionasFromthebehaviourasweseethatthesolutionoutsidetheboundarylayermustnowmatchtowhichshowsthatwerequireatermintheasymptoticexpansionvalidintheouterregion.Thusweseekasolutionoftheset(5.109)intheformwhereqrepresentseachofu,vandp.Theproblemforthesecondtermsinthisregionthereforebecomesthesetwithand,intermsofthestreamfunctionThisisaclassicalproblemininviscidflowtheory,wheretheexteriorflowisdistortedbythepresenceofaparabolicsurface—theeffectoftheboundarylayerwhichgrowsontheplate.Theexactsolutioncanbeexpressedintermsofthecomplexvariable(anddenotestherealpart):whichgives(Itcanbeshownthat,inordertomatch,theboundary-layersolutionmustnowcontainatermnotasmighthavebeenexpected.Formoregeneralsurfacesthanaflatplate,thenexttermintheboundary-layerexpansionisindeed 2465.SomeworkedexamplesarisingfromphysicalproblemsWewillnotproceedfurtherwiththisanalysishere(butfarmoredetailisavailableinmanyothertextse.g.vanDyke,1975),althoughweshouldaddonewordofwarning.Thedetailsthatwehavepresentedsuggestthatwemaycontinue,fairlyroutinely,tofindthenexttermintheboundary-layerexpansion,andthenthenextintheouter,andsoon,andthatthesewilldevelopaccordingtotheasymptoticsequenceHowever,thisisnotthecase:atermlnappearsandthisconsiderablycomplicatestheprocedure(again,seee.g.vanDyke,1975).Thetwoessentialtypesofproblemthatareusuallyofmostinterestinfluidmechanicsareassociatedwith(a)(thepreviousexample)and(b)(thenextexample).ProblemsforsmallReynoldsnumber(sometimesreferredtoasStokesfloworslowflow)havebecomeofincreasinginterestbecausethislimitrelatestoimportantproblemsin,forexample,abiologicalcontext.Thusthemovementofplateletsintheblood,andthepropulsionofbacteriausingciliaryhairs,areexamplesofthesesmall-Reynoldsnumberflows.Wewilldescribeasimple,classicalproblemofthistype.E5.20VeryviscousflowpastasphereInthisexample,wetake(andsincewhereUisatypicalspeedoftheflow,datypicaldimensionoftheobjectintheflowandvthekinematicviscosity,thislimitcanbeinterpretedas‘highlyviscous’or‘slowflow’orflowpasta‘smallobject’).Weconsidertheaxisymmetricflow,producedbyauniformflowatinfinityparalleltothechosenaxis,pastasolidsphere;seefigure15.(Thiscouldbeusedasasimplemodelforflowpastaraindrop.)Itisconvenienttointroduceastreamfunction(usuallycalledaStokesstreamfunction,inthiscontext),eliminatepressurefromtheNavier-Stokesequationandhenceworkwiththe(non-dimensional)equationforandwherewithandthislatterconditionensuringthatthereisanaxisymmetricflowofspeedoneatinfinity.(Thesubscriptsheredenotepartialderivatives;wehavemixedthenotationbecause,wesubmit,thisistheneatestwaytoexpressthisequation.)Thevelocitycomponents 247Figure15.Coordinatesandvelocitycomponentsfortheuniformflowpastasphere.(seefigure15)aregivenbyNotethatequation(5.114)doesnotexhibittheconditionsforaboundary-layerstruc-ture,asbecausethehighestderivativesareretainedinthislimit—indeed,thistermdominates.Itisthereforeunclearwhatdifficultieswemayencounter.Letusseekasolutionthenfrom(5.114)wesimplyhavethatandalltheboundaryconditionsappeartobeavailable.Indeed,thereisanexactsolution(Stokes,1851)whichsatisfiesallthegivenconditions:andforanumberofyearsthiswasthoughttobeacceptable,andthathigher-ordertermswouldsimplyprovidesmallcorrectionsinthecaseHowever,difficultieswereencounteredwhenamorecarefulanalysiswasundertaken,andalittlethoughtsuggestswhythisshouldbeso.Atinfinitythemotion(convectiveterms)dominate,i.e.theleft-handsideoftheequation,butnearthespheretheviscoustermsdominate(theright-handside);thusanapproximationwhichusesonlytheright-handside(asabovedoes)cannotbeuniformlyvalid—itmustbreakdownasWeintroducewhereasandthenfrom(5.117)weseethatwemustalsoscaleEquation(5.114)yieldsimmediately 2485.Someworkedexamplesarisingfromphysicalproblemsthattheappropriatechoiceis(Oseen,1910),butunfortunatelythisscalingrecoversthefullequation—thesmallparameterisremovedidentically!However,thegoodnewsisthatthisscaling(obviously)isassociatedwiththeregionfarawayfromthesphere(the‘farfield’),wheretheuniformflowexistsand,presumably,thisshouldbethefirstterminanasymptoticsolutionvalidhere;weexpect,therefore,thatOfcourse,(5.118)and(5.117)matchdirectly,andweshouldnowregard(5.117)asvalidonlyforr=O(1)(the‘nearfield’)andthen(5.118)isvalidforWhenweexpress(5.117)infar-fieldvariables,weobtainandsowerequireaterminthefar-fieldexpansion;letuswriteTheequationforfrom(5.114),iswherewithandtheformerconditionbeinggivenbythematching,andthelatterensuringthattheflowatinfinityisunchanged.Therelevantsolutionofequation(5.120a)iswhereA–Earearbitraryconstants,andthenwritteninnear-fieldvariables,givesTheterminisunmatchable,andsoitmustberemoved,andotherwisethis 249expressionistomatchto(5.119);thisrequiresthati.e.Finally,tobeconsistentwiththedevelopmentofthisasymptoticexpansion,wesetandthensatisfieswhereisgivenby(5.117).Itisleftasanexercisetoshowthatthesolutionofthisequation,whichsatisfiestheboundaryandmatchingconditions,isAnd,tocompleteourpresentation,wecommentthatexpandedforproducesandthetermhereincontributes,apparently,toachangeintheuniformflowatinfinity—whichisimpossible—andhencetheneedforamatchedsolutioninthefarfield.Itwasthisobservationthatfirstalertedtheearlierresearcherstothedifficultiesinherentinthisproblem;thiscomplicationistypicalofflowsinthelimitAnothergeneralareaofstudyinfluidmechanicsisgasdynamics,wherethecompress-ibilityofthefluidcannotbeignored.Wehavealreadyseensomeoftheseproblems(E3.2,E3.5andQ3.9–3.11);wenowlookatanotherclassicalexample.E5.21ApistonproblemWeconsidertheone-dimensionalflowofagasinalong,open-endedtube.Thegasisbroughtintomotionbytheactionofapistonatoneend,whichmovesforwardataspeedwhichismuchlessthanthesoundspeedinthegas.(Thisisusuallycalledtheacousticproblem.)Thegasismodelledbytheisentropiclawforaperfectgas(pressureandisdescribedbytheequations 2505.SomeworkedexamplesarisingfromphysicalproblemsFigure16.Sketchofapistonmovingagas(accordingtoinanopentube.whereaisthe(local)soundspeedinthegasanduitsspeedalongthetube.TheinitialandboundaryconditionsareandwithandV(0)=0;seefigure16.(Theproblemofaclosedtube,describedinaLagrangianframeworkandusingthemethodofmultiplescales,isdiscussedbyWang&Kassoy,1990.)Wearealreadyfamiliarwiththeresult,insmall-disturbancetheories—whichthisisforthesimple,near-fieldwave-propagationproblemisnotuniformlyvalidast(orseeE3.2.Inparticular,fordisturbancespropagatingdownthetube(intowhicharedescribedbye.g.u(x,t)~F(x–t),therewillbeabreakdownwhere(or,equivalently,whenwithx–t=O(1).Notethat,inthisproblem,becausethetubeisopen,therecanbenodisturbancespropagatingbacktowardsthepiston.Weintroduceand(tomaketheevaluationonassimpleaspossible)andthenwriteandtogivewithandWeseekanasymptoticsolution,whichistobeuniformlyvalidasintheform 251whichgives(from(5.121)andsoon.Further,weassumethattheboundaryconditionatthepistoncanbeexpressedasaTaylorexpansionaboutt:thevalidityofwhichcertainlyrequiresthatremainsfiniteas(andsomustbefinite).From(5.123)wefindthatwhereisanarbitraryfunction,butandareotherwiseundeterminedatthisstage.From(5.124),wemultiplythefirstbyandthenaddtoit(5.124b),whicheliminatesandtoproduceThetermsinarenowreplacedbyusing(5.126)togivetheequationforThisisanonlinearequationwhich,forgivenisreadilysolved.However,thissolutionisincompletewithouttheweakacousticshockwavethatpropagatesaheadofthissolution;wemustthereforewritedowntheconditionsfortheinsertionofashock(discontinuity).First,from(5.122),thisinitialconditionrequiresthatTofurtherdeter-mineweimposetheRankine-Hugoniotconditionsthatdefinethejumpconditionsacrosstheshock.Theconditionsaheadareundisturbed;lettheconditionsbehindtheshockbedenotedbythesubscript‘s’andwritethespeedoftheshockasInthis 2525.Someworkedexamplesarisingfromphysicalproblemsproblem,theseconditions(seee.g.Courant&Friedrichs,1967)canbewrittenasandso(Here,istheperturbationofthedensity.)Thislatterresultconfirms,from(5.126),thatbehindtheshockand,sinceaheadoftheshock,wehaveforThusfrom(5.127)weobtaintheimplicitresultwhereFisanarbitraryfunctionwhich,from(5.125),canbedeterminedtogive(since,atthepiston,Thusthenear-fieldsolutionisrecovered,althoughthisneedstobewrittentoaccommodatetheexistenceofthewavefronttherei.e.whereHistheHeavisidestepfunction:Weconcludewiththeobservationthattheshockwavetravelsfasterthanthelocalsoundspeedbehindtheshock;thatis,from(5.128),ascomparedwith(andrememberthat).Muchmoredetailcanbefoundinanygoodtextongasdynamics.Asourfinalexample,weuseasimilartechniquetothatemployedinthepreviousproblem,butnowinaquitedifferentcontext:wavesonthesurfaceofwater.(SeeQ3.4foramuchsimplerbutrelatedexercise.)E5.22Avariable-depthKorteweg-deVriesequationforwaterwavesWeconsidertheone-dimensionalpropagationofwavesoverwater(incompressible),whichismodelledbyaninviscidfluidwithoutsurfacetension.Thewaterisstationaryintheabsenceofwaves,butthelocaldepthvariesonthesamescalethatisusedtomeasuretheweaknonlinearityanddispersiveeffectsinthegoverningequations.Forright-runningwaves,theappropriatefar-fieldcoordinatesareand 253Figure17.Wavepropagationinstationarywaterovervariabledepth.whereistobedetermined.Thenon-dimensionalequationsarewithandHere(u,w)arethevelocitycomponentsoftheflow,pthepressureinthefluidrelativetothehydrostaticpressure(withpressureconstantatthesurface)andisthesurfaceofthewater.Thebottomisrepresentedbythefunctionseefigure17.Weseekasolutioninthefamiliarform:whereqrepresentseachofu,w,pandTheleading-orderproblemgives,from(5.129),withThissetiseasilysolved;therelevantsolution(inwhichisnotafunctionofz)is 2545.Someworkedexamplesarisingfromphysicalproblemswherethesurfaceboundaryconditionfinallygivesandisarbitraryatthisstage.Forrightwardpropagation,weselectsothatthecharacteristicvariablebecomeswhich,withD=1(constantdepth),recoversthestandardresult:Notethat,inthiscalculation,wehavetakentheevaluationatthesurfacetobeonz=1;seebelow.Atthenextorder,equations(5.129)givewithandTheboundaryconditionsatthesurface,havebeenwrittenbyinvokingTaylorexpansions,andsobecomethecorrespondingboundaryconditionsevaluatedonz=1,validas(andforsufficientlysmoothsurfacewaves).Again,thissetisfairlyeasilysolved;thedetailsareleftasanexercise,butsomeoftheintermediateresultsareand 255Finally,thesurfaceboundaryconditionforgivestheequationforidenticallycancels—intheformofavariable-coefficientKorteweg-deVriesequation(seeE3.1andQ3.4):ThisisusuallyexpressedintermsoftogivewhichrecoverstheclassicalKorteweg-deVriesequationforwaterwaveswhenwesetD=1.Thisequationisthebasisformanyofthemodernstudiesinwater-wavetheory;morebackgroundtothis,andrelatedproblemsinwaterwaves,canbefoundinJohnson(1997).5.6EXTREMETHERMALPROCESSESThisnextgroupofproblemsconcernsphenomenathatinvolveexplosions,combus-tionandthelike.Thetwoexamplesthatwewilldescribeare:E5.23Amodelforcombustion;E5.24Thermalrunaway.E5.23AmodelforcombustionAmodelthataimstodescribeignition,followedbyarapidcombustion,requiresaslowdevelopmentoverareasonabletimescalethatprecedesamassivechangeonaveryshorttimescale,initiatedbytheattainmentofsomecriticalcondition.Asimple(non-dimensional)modelforsuchaprocess(Reiss,1980)istheequationwhereistheconcentrationofanappropriatechemicalthattakespartinthecombustivereaction.Thewholeprocessisinitiatedbythesmalldisturbanceattimet=0.(Itshouldbefairlyapparentthatthisequationcanbeintegratedcompletelytogivethesolutionforc,butinimplicitformandsothedetailedstructureasisfarfromtransparent;thisintegrationisleftasanexercise.)Byvirtueoftheinitialvaluewefirstseekasolutionintheform 2565.Someworkedexamplesarisingfromphysicalproblemsandthenfrom(5.130)weobtainwithThissetisveryeasilysolvedtogivetheasymptoticsolutionanditisimmediatelyevidentthatthisexpansionbreaksdownwhenletuswriteandtoproducethenewequationAgain,weseekastraightforwardsolutionsothat(5.132)givesandsoon,togetherwiththerequirementtomatchto(5.131).Thegeneralsolutiontoequation(5.133a)iswhereAisanarbitraryconstant;withweobtainandhencematchingtothefirsttermin(5.131)requiresthechoiceA=1.Thuswehavethenexttermcanbefoundsimilarlyandleadsto 257whichclearlyexhibitsacatastrophicbreakdownasThuswehaveagradualaccelerationoftheprocess,untilthetimeisreachedandthen—presumably—combustionoccurs.Thisbreakdownisnotsimple:itisatatimegivenbyWhenlogarithms(orexponentials)arise,wehavelearnt(§2.5)toreturntotheoriginalequationandseekarelevantscaling(althoughthepresenceofalogarithmhereindicatesthatlntermsarelikelytoappearintheasymptoticexpansion).Letussetthenfrom(5.134),whereasthuswewriteanditisimmediatelyclearfrom(5.132)thatwemustchooseTheequationforisthereforetheoriginalequation!Inordertoproceed,weneedanappropriatesolution—butthisisnolongerrequiredtosatisfytheinitialcondition.Thegeneralsolutiontoequation(5.135)canbewrittenasandforsmallthisgivesandsotomatchwerequirethearbitraryconstant,B,tobezero(butwewillreturntothisoriginshiftbelow).Inpassing,wenotethatforclosetounity,weobtainandsothestateoffullcombustionisattainedasFinally,wereconsiderthematchingof(5.136),withB=0,totheexpansion(5.134).From(5.136)weobtainwhichwithgivesandthematchingisnotpossible,asitstands,becauseofthepresenceofthelnterm.However,thissuggeststhatthevariableusedinthisregionofrapidcombustionshouldincludeanoriginshift.Ifwewrite,now,Then(5.137)producesandmatchingwith(5.134)requiresthatThusthecombustionoccursinanO(1)neighbourhoodofthetimetheappear-anceofshiftsexpressedintermsoflnarequitetypicaloftheseproblems. 2585.SomeworkedexamplesarisingfromphysicalproblemsE5.24ThermalrunawayAphenomenonthatcanbeencounteredincertainchemicalreactionsinvolvesthereleaseofheat(exothermic)whichincreasesthetemperature,andthetemperaturenor-mallycontrolsthereactionrate.Itispossible,therefore,toinitiateareaction,thenheatisreleasedwhichraisesthetemperatureandsoincreasestherateofreactionwhichreleasesevenmoreheat,andsoon;thisiscalledthermalrunaway.Inthemostextremecases,thereisnotheoreticallimittothetemperature,althoughphysicalrealityinter-venese.g.thecontainingvesselmightmeltortheproductsexplode.Astandardmodelusedtodescribethis(Szekely,Sohn&Evans,1976;seealsoFowler,1997,whichprovidesthebasisforthediscussionpresentedhere)istheequationwhereTisthetemperatureandisaparameter.Inthisexample,wewillexaminethenatureofthesteady-statetemperatureinonedimensioni.e.thesolutionofasforvariousTheboundaryconditionisthattheexternaltemperatureismaintained;wewillrepresentthisbyT=0onandsoweseekasolutionfor(andwewillassumesymmetryofthetemperaturedistributionaboutx=0).Animportantpropertyofthesolutionof(5.138)canbederivedbyfirstwritingwheresatisfiestheequationwiththeboundaryconditionsThisproblemhastheexactsolutionwhereisthemaximumtemperature(attainedatx=0)definedby 259whichfollowsimmediatelywhenevaluationonx=±1isimposed.Itisleftasanexercise(whichmayrequireagraphicalapproach)toconfirmthat(5.140)haszero,oneortwosolutionsforgivendependingonwhetherorrespectively,wherethecriticalvalueisthesolutionofitcanbeshownthattheredoesexistjustoneItturnsoutthattheconsequencesofthisarefundamental:foranyiftheinitialtemperatureishighenough,orforanytemperatureifthenthetime-dependentproblemproducesatemperaturethatincreaseswithoutbound—indeed,inafinitetime.Whatwewilldohereistoexaminethetemperatureattainedaccordingtothesteady-stateequation,(5.138),forvariousalthoughwewillapproachthisbyconsideringdifferentsizesoftemperature(asmeasuredbyItisimmediatelyapparentthattheapproximationthatledto(5.139)cannotbevalidifthetemperatureisaslargeasseeequation(5.138).Letusthereforewriteandthen(5.138)becomesandsoifweseekasolution(withas)weobtainsimplywhereAandBarearbitraryconstants.Suchasolutionisunabletoaccommodateamaximumtemperatureatx=0(ifthesolutionistobedifferentiable,andconstantforallxdoesnotsatisfy(5.141)).Thus(5.142)candescribethesolutiononlyawayfromx=0,butthenwemayimposetheboundaryconditionsonx=±1,soandthesinglearbitaryconstant,A,maybeusedinbothsolutionsbyvirtueofthesymmetry.Nearx=0,letandseekasolutionwhereisthe(scaled)maximumtemperatureattainedinthelimitnotethat,atthisstage,wedonotknowthescalingsandEquation(5.141)becomes 2605.Someworkedexamplesarisingfromphysicalproblemsandforanappropriatesolutiontoexist,toleadingorderforX=O(1),wemusthavewhichimpliesthatisexponentiallysmallasLetuswritethenfrom(5.144)with(5.145)included,weobtaintheequationforasandthisgivesameaningfulfirstapproximation,independentofonlyifwechoosee.g.ThisequationthenhasthegeneralsolutionforarbitraryconstantsandC.ThissolutionistobesymmetricaboutX=0,so(whichissatisfiedwith),andistomatchto(5.143).Thuswemusthaveandthenweobtainwhichmatchesonlyif,first,wechoosethescalingandthen(validinX>0,X<0,respectively).Thus,inparticular,wefindthatandsothemaximumtemperaturebecomeswhere,from(5.145),wehaveforgivenwemaywritethisequivalentlyaswhereandsodeterminesistheinterpretationthatweemploy.Thecalculationthusfarindicatesthat,forsuitabletheresultingsteady-statetemperature(ifitcanbeattainedthroughatime-dependentevolution)isalreadyverylarge,namelyButitisalsoclearfrom(5.147)thatevensmallerexistthatgive 261andthenthetemperatureexpansion,(5.146),isnotuniformlyvalid;inpar-ticularthisexpansionbreaksdownwhenthatis,forLetusthereforerescalewriteandthenequation(5.141)becomeswithatx=±1.Thisbranchofthesolution,interpretedasafunctionofisusuallycalledthehotbranch.Weseekasolutiontogivebutwecannotusetheboundaryconditionsherebecauseoftheevidentnon-uniformityasinequation(5.148).However,ifwesetatx=0(wherebysymmetry)weobtain(wherewehavesetandthislatterintegralcanbeexpressedintermsofanexponentialintegral,ifthatisuseful.Asolutionwiththepropertythatas(whichisnecessaryifmatchingistobepossibletothesolutionvalidnearx=±1,whereisexponentiallysmalli.e.mustsatisfywhereasor–1.(Ofcourse,matchingisthentrivial,forwesimplychooseA=B.)Thuswemayintegrate(5.149)fromx=0to,say,x=1:andnowwefindthatincreasesasincreases.Sooncewehavereachedthis‘hotbranch’,whichisaccessedbyusingthetemperaturewillincreasewithoutbound(or,rather,untilsomeotherphysicsintervenes).Evenmoredetailsofthisproblem,andrelatedthermalprocesses,canbefoundintheexcellenttextonmodellingbyFowler(1997).Thefinalgroupofproblemsbearsomerelationtothosejustconsidered,fortheyalsoinvolvechemicalprocesses,butweincludeinthissectionsomementionofbiochemicalprocessesaswell. 2625.Someworkedexamplesarisingfromphysicalproblems5.7CHEMICALANDBIOCHEMICALREACTIONSTheexamplesthatarepresentedhereareintendedtoshowthatitispossibletomodel,anddescribeusingperturbationtheory,someverycomplexprocessesthat,perhapsfiftyyearsago,werethoughttobemathematicallyunresolvable.Certainly,someextensivesimplificationisnecessaryinthedevelopmentofthemodel—andthisrequiresconsid-erableskillandknowledge—buttheresultingdifferentialequationsremain,generally,quitedaunting.Wewilldescribe:E5.26Kineticsofacatalysedreaction;E5.27Enzymekinetics;E5.28TheBelousov-Zhabotinskiireaction.E5.26KineticsofacatalysedreactionInamodel(theLangmuir-Hinshelwoodmodel;seeKapila,1983)forthekineticsofaparticulartypeofcatalysedreaction,theconcentrationofthereactantvariesaccordingtotheequationwith(whereisagivenconstant.Theparameterisarateconstantand,forweobservethattheequationreducestoapurelyalgebraicproblemwhichisreadilysolved:WehaveselectedthepositivesignsothatOnx=1,thisgivesthevalueandsowewillrequireaboundarylayernearx=1;see§§2.6,2.7.(Notethat,fromthissolution,c=1onx=0,sowecanexpectthatasinthesolutionofthefullequation.)LetusintroduceandwritetheequationforCisthenWeseekasolution 263andsosatisfieswithThegeneralsolutionofthisequationcanbefound,albeitinimplicitform,aswhereandAisanarbitraryconstantwhichisevaluatedaswhentheboundaryconditiononX=0isimposed.As(whichwillgivethebehaviouroutsidetheboundarylayer),weseethatwhichthereforeautomaticallymatcheswiththesolutionalreadyfoundintheregionawayfromtheboundarylayer,(5.151).Itisleftasastraightforwardexercisetofindhigher-orderterms,ortowritedownacompositeexpansionvalidforsee§1.10.Thisfirstexamplehaspresenteduswithaveryroutineexerciseinelementaryboundary-layertheory;thenexthasasimilarstructure,butinonlyoneofthetwocomponents.E5.27EnzymekineticsAstandardprocessinenzymekineticsconcernstheconversionofasubstrate(x)intoaproduct,bytheactionofanenzyme,viaasubstrate-enzymecomplex(y);thisistheMichaelis-Mentonreaction.AmodelforthisprocessisthepairofequationswhereandarepositiveconstantsandtheinitialconditionsareTheparametermeasurestherateoftheproductionofy;weconsidertheproblemposedabove,withItisclearthatweshouldexpectaboundary-layerstructure 2645.Someworkedexamplesarisingfromphysicalproblemsinthesolutionfory,butnotforx.Asuitableboundary-layervariableis(ratherthansimplyandwewilltaketheopportunitytousethemethodofmultiplescaleshere,usingT=tand(see§4.6).Weintroduceandandthenequations(5.152)becomewithTheasymptoticsolutionsareexpressedintheusualway:whichgivesimmediatelythatonly;thenweobtainandsoon.In(5.155a)wechoosetomaketheassimpleaspossible—therealpurposebehindtheintroductionoff(T)=f(t)—andsowewriteandthenwehavesimplyThisissolvedwithease,toproducewhereisanarbitraryfunction.Theinitialconditions,(5.154),requirethat 265From(5.155c)wenowhavewhichintegratestogiveandthenuniformityasrequiresthatwhichdefinesThusorwhereCisanarbitraryconstant.Theinitialcondition,(5.154),thenyieldstheresultC=1;theimplicitsolutionforisthereforedescribedby(Althoughthisequationappearsunsatisfactoryandratherinvolved,itssolutionhasasimpleinterpretation:startsatanddecreasestozero,eventuallyexpo-nentiallylikeasFinally,welookatequation(5.155b),whichcanbewrittenastheimportantinformationwerequirefromthisequationistheconditionthatdefines(in(5.156)).Thisrequiresthattheterminontheright-handside,isremoved(becauseotherwisewillcontainatermwhichleadstoanon-uniformityasTherelevanttermis 2665.Someworkedexamplesarisingfromphysicalproblemsbutthiscontainssonowwemustcallonequation(5.155d)tofindthisterm.Inthisequation,theavoidanceofnon-uniformitiesrequirestheremovalofalltermsthatdependononlyT(forotherwisewewillhaveItisleftasanexercisetoshowthatthisconditionproducestheequationfororthis,andthentheequationforcanbeintegrated,andsolutionsexpressedintermsofInourfinalexample,weareabletouseideasfromsingularperturbationtheoryinonlyarathersuperficialway,butthisissuchanimportantproblemthatwecouldnotignoreit.E5.28TheBelousov-ZhabotinskiireactionThisisafamousandmuch-studiedphenomenon,firstdemonstratedbyBelousovin1951.Hediscoveredthatasteadyoscillationoftheconcentrationofacatalyst,betweenitsoxidisedandreducedstates,waspossible.Inasuitablemedium,thiscanbeexhibitedasadramaticchangeincolour—acolourbeingassociatedwithastate—withaperiodofaminuteorso.Asetofmodelequationsforthischemicalreactioniswheretheconcentrationofthecatalystisrepresentedbyz(Tyson,1985).Theconstantsandvarepositiveandindependentofmeasurestherateconstantfortheproductionofx,andgivesthesizeofthecorrespondingconstantforyandofthenonlinearityin(5.158a).WewillconsidertheproblemwithTheinitialconditionsarerelativelyunimportanthere,butwewillassumethattheyaresufficienttostarttheoscillatoryprocess.Itwouldbeimpossibleinatextsuchasours,withitsemphasisonsingularperturbationtheory,togiveacomprehensivedescriptionoftherelevantsolutionoftheset(5.158).Thiswouldinvolve,forexample,adetailed(butlocal)stabilityanalysis.Wewillcontentourselveswithabriefoverviewthatemphasisesthevariousscalesthatareimportant.Thefirstpointtonoteisthatequations(5.158a,b)haveaboundary-layerstructure,sothatthereisashorttimeduringwhichtheinitialvaluesarelostandthesolutionsettlestoa(local)steadystate(sometimesreferredtoasquasi-equilibrium).Note,however,that(5.158c)allowsarelativelylongevolutiontime,sofortimesof 267sizeO(l)wehave(withThesewewilllabelstateI.FromIitisclearthat,ifthenbothxandzmustincrease,buttheneventuallythenonlinearityinequations(5.158)willbecomeimportant(andmostparticularlythetermin(5.158a)).Thisobservationprovidesthebasisforthescalinginthissituation:wewriteandandsoequations(5.158)becomeThesethreeequationseachhaveadifferenttimescale,soanotherstoryunfoldshere.OnaveryshorttimeintervalYevolvesfromitsinitialvalue(closeto1)toa(local)steadystategovernedbyThenonalonger—butstillshort!—timeintervalXnowevolvessothatandthenonanO(1)timescalewehavethesetogetherconstitutestateII:Intermsoftherelevantscales,thisissufficientfortheoscillatoryprocess.Ofcourse,howthesolutionactuallyevolves,andjumpsbetweenstatesIandII,needsmorediscussionthanweareabletopresenthere.Acarefulanalysisdemonstratesthat,pro-videdthenthesolutionslowly(O(1)scales)evolvesinstateI,becomesunstableandjumpstostateII;thisalsoslowlyevolvesuntilitbecomesunsta-bleandrevertstostateI.Thisistheessenceoftheoscillatoryprocess(sometimescalledarelaxationoscillation,becausethesolution‘relaxes’backtoaformerbranchofthesolution).WehavechosentoignoremanyimportantelementsinourpresentationoftheBelousov-Zhabotinskiireaction,mainlybecausetheyrequiremuchthatgoeswellbeyondthemethodsofsingularperturbationtheory.AnilluminatingdiscussionofthisprocesscanbefoundinFowler(1997)andthistext,inconjunctionwithMurray(1993),provideanexcellentintroductiontomanychemical,biochemicalandbiolog-icalmodelsandtheirsolutions.Inthisfinalchapter,wehavepresentedanddescribedanumberofexamplestakenfromthephysicalandchemicalsciences,andineachtheideasofsingularperturbation 2685.Someworkedexamplesarisingfromphysicalproblemstheoryplayasignificantrôle.Asweimpliedearlier,suchacollectioncouldnotbeexhaustive—indeed,wecanhopetogiveonlyanindicationofwhatispossible.Eveniftheparticularapplicationsofferedhereareofnospecificinteresttosomereaders,theydoprovideasetofadditionalworkedexamplesthatshouldhelptoreinforcetheideasthatcontributetosingularperturbationtheory.Otherexamples,somedescribedindetailandsomesetasexercises,areavailableinmanyofthetextspreviouslycited.Inaddition,interestedreadersareencouragedtoinvestigatethereferencestorelatedmaterialthathavebeenprovidedthroughoutthischapter. APPENDIX:THEJACOBIANELLIPTICFUNCTIONSGiventheintegralwheremisusuallycalledthemodulus,wethendefinethejacobianellipticfunctionsWeseeimmediatelythatwehavetheidentitiesFurther,asandasTherelevantdifferentialrelationsaresimilarlywehave 270AppendixandThe(real)periodoftheJacobianellipticfunctionsis4K(m)wherearethecompleteellipticintegralsofthefastandsecondkinds,respectively.(TheJacobianellipticfunctionsaredoubly-periodicinthecomplexplane;forexample,theotherperiodofcn(u;m)is2iK(1–m).)Theinterestedreadercanfindmoreinformationintextsthatspecialiseinthefunc-tions,suchasLawden(1989)orByrd&Friedman(1971). ANSWERSANDHINTSTheanswer,whereoneisgiven,isdesignatedbytheprefixA;forexample,theanswertoQ1.1isA1.1.Insomecasesahinttothemethodofsolutionisincluded;inafewofthemoreinvolvedcalculations,someintermediatestepsaregiven.CHAPTER1A1.1(a)for(b)for(c)for(d)for(e)for(f)(cf.(d))for(g)forx>0.[N.B.asA1.2(a)for(b)for(c)for 272AnswersandhintsA1.3(a)(b)whichisperiodicbutoflargeamplitudeas(c)(d)(e)andnotethebehaviournearx=0andnearx=1;(f)andonlynearx=0isinteresting(cf.(e));(g)andtheP.I.dominatesforx=O(1)as(h)firstandthen(i)where(j)(k)(l)(m)(n)A1.4(a)bothlimitingprocessesgive0—uniform;(b)forfirst,thelimitis0;forfirst,thelimitis1—non–uniform;(c)forfirst,thefunctiontendstoforfirst,thefunctiontendsto(d)forfirst,thelimitis0;forfirst,thelimitis1—non–uniform.A1.5(a)simplyperformtheintegrations;(b)multiplytheresultin(a)by(d)raisetheresultin(b)tothepowerA1.6(a)yes;(b)yes;(c)yes;(d)(e)(f)yes—seeQ1.5(e);(g)yes;(h)yes;(i)(j)yes;(k)andsoA1.7(a)(b)–1/lnx;(c)(d)(e)xlnx;(f)—lnx;(g)A1.8(a)supposethatthen(b)asin(a),butwithA1.9(a)(b)(c)A1.10asrequiredfor(a);(b)writeassop(x)=x–1.A1.11(a)andratiotestgivesasforallfinitex,soconvergentforthesexs;(b)andtheratiotestfailsforallfinitex—divergent.Considerwithx=2,then0.99521.Ai(X),whereisanarbitraryconstant.A4.19Forx>1(boundedsolution):forx<0:for0