数学专业英语9.docx
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数学专业英语9.docx
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数学专业英语9
MathematicalEnglish
Dr.XiaominZhang
Email:
zhangxiaomin@
§2.9IntroductiontoDifferentialEquations
TEXTAIntroduction
Alargevarietyofscientificproblemsariseinwhichonetriestodeterminesomethingfromitsrateofchange.Forexample,wecouldtrytocomputethepositionofamovingparticlefromaknowledgeofitsvelocityoracceleration.Oraradioactivesubstancemaybedisintegratingataknownrateandwemayberequiredtodeterminetheamountofmaterialpresentafteragiventime.Inexamplelikethese,wearetryingtodetermineanunknownfunctionfromprescribedinformationexpressedintheformofanequationinvolvingatleastoneofthederivativesoftheunknownfunction.Theseequationsarecalleddifferentialequations,andtheirstudyformsoneofthemostchallengingbranchesofmathematics.
Differentialequationsareclassifiedundertwomainheadings:
ordinaryandpartial,dependingonwhethertheunknownisafunctionofjustonevariableortwoormorevariables.Asimpleexampleofanordinarydifferentialequationistherelation
(9.1)f'(x)=f(x)
whichissatisfied,inparticularbytheexponentialfunction,f(x)=ex.Weshallseepresentlythateverysolutionof(9.1)mustbeoftheformf(x)=Cex,whereCmaybeanyconstant.
Onetheotherhand,anequationlike
isanexampleofapartialdifferentialequation.Thisparticularone,calledLaplace’sequation,appearsinthetheoryofelectricityandmagnetism,fluidmechanics,andelsewhere.Ithasmanydifferentkindsofsolutions,amongwhicharef(x,y)=x+2y,f(x,y)=excosy,andf(x,y)=log(x2+y2).
Thestudyofdifferentialequationsisonepartofmathematicsthat,perhapsmorethananyother,hasbeendirectlyinspiredbymechanics,astronomy,andmathematicalphysics.Itshistorybeganinthe17thcenturywhenNewton,Leibniz,andtheBernoullissolvedsomesimpledifferentialequationsarisingfromproblemsingeometryandmechanics.Theseearlydiscoveries,beginningabout1690,graduallyledtothedevelopmentofalotof“specialtricks”forsolvingcertainspecialkindsofdifferentialequations.Althoughthesespecialtricksareapplicableinrelativelyfewcases,theydoenableustosolvemanydifferentialequationsthatariseinmechanicsandgeometry,sotheirstudyisofpracticalimportance.Someofthesespecialmethodsandsomeoftheproblemswhichtheyhelpussolvearediscussedneartheendofthischapter.
Experiencehasshownthatitisdifficulttoobtainmathematicaltheoriesofmuchgeneralityaboutsolutionsofdifferentialequations,exceptforafewtypes.Amongthesearetheso-calledlineardifferentialequationswhichoccurinagreatvarietyofscientificproblem.Thesimplesttypesoflineardifferentialequationsandsomeoftheirapplicationsarealsodiscussedinthisintroductorychapter.AmorethoroughstudyoflinearequationsiscarriedoutinVolume2.
Notations
lineardifferentialequationsAnordinarydifferentialequation(ODE),i.e.,anequalityinvolvingafunctionanditsderivativesofordern
issaidtobelinearifitisoftheform
AlinearODEwhereQ(x)=0issaidtobehomogeneous.
Simpletheoriesexistforfirst-order(integratingfactor)andsecond-order(Sturm-Liouvilletheory)ordinarydifferentialequations,andarbitraryODEswithlinearconstantcoefficientscanbesolvedwhentheyareofcertainfactorableforms.IntegraltransformssuchastheLaplacetransformcanalsobeusedtosolveclassesoflinearODEs.
ThesolutionstoanODEsatisfyexistenceanduniquenessproperties.
integratingfactorAnintegratingfactorisafunctionbywhichanordinarydifferentialequationcanbemultipliedinordertomakeitintegrable.Forexample,alinearfirst-orderordinarydifferentialequationoftype
wherepandqaregivencontinuousfunctions,canbemadeintegrablebylettingv(x)beafunctionsuchthat
and
Thenev(x)wouldbetheintegratingfactorsuchthatmultiplyingbyy(x)givestheexpression
usingtheproductrule.Integratingbothsideswithrespecttoxthengivesthesolution
TEXTBTerminologyandnotation
Whenweworkwithadifferentialequationsuchas(9.1),itiscustomarytowriteyinplaceoff(x)andy'inplaceoff'(x),thehigherderivativesbeingdenotedbyy",y'",etc.Ofcourse,otherletterssuchasu,v,z,etc.arealsousedinsteadofy.Bytheorderofanequationismeanttheorderofthehighestderivativewhichappears.Forexample,(9.1)isafirst-orderequationwhichmaybewrittenasy'=y.Thedifferentialequationy'=x3y+sin(xy")isoneofsecondorder.
Inthischapterweshallbeginourstudywithfirst-orderequationswhichcanbesolvedfory'andwrittenasfollows:
(9.2)y'=f(x,y),
wheretheexpressionf(x,y)ontherighthasvariousspecialforms.Adifferentiablefunctiony=Y(x)willbecalledasolutionof(9.2)onanintervalIifthefunctionYanditsderivativeY'satisfytherelation
Y'(x)=f(x,Y(x))
foreveryxinI.Thesimplestcaseoccurswhenf(x,y)isindependentofy.Inthiscase,(9.2)becomes
(9.3)y'=Q(x)
say,whereQisassumedtobeagivenfunctiondefinedonsomeintervalI.Tosolvethedifferentialequation(9.3)meanstofindaprimitiveofQ.TheSecondfundamentaltheoremofcalculustellsushowtodoitwhenQiscontinuousonanopenintervalI.WesimplyintegrateQandaddanyconstant.Thus,everysolutionof(9.3)isincludedintheformula.
(9.4)
whereCisanyconstant(usuallycalledanarbitraryconstantofintegration).Thedifferentialequation(9.3)hasinfinitelymanysolutions,oneforeachvalueofC.
Ifitisnotpossibletoevaluatetheintegralin(9.4)intermsoffamiliarfunctions,suchaspolynomials,rationalfunctions,trigonometricandinversetrigonometricfunctions,logarithms,andexponentials,stillweconsiderthedifferentialequationashavingbeensolvedifthesolutioncanbeexpressedintermsofintegralsofknownfunctions.Inactualpractice,therearevariousmethodsforobtainingapproximateevaluationsofintegralswhichleadtousefulinformationaboutthesolution.Automatichigh-speedcomputingmachinesareoftendesignedwiththiskindofprobleminmind.
EXAMPLE.Linearmotiondeterminedfromthevelocity.Supposeaparticlemovesalongastraightlineinsuchawaythatitsvelocityattimetis2sint.Determineitspositionattimet.
Solution.IfY(t)denotesthepositionattimetmeasuredfromsomestartingpoint,thenthederivativeY'(t)representsthevelocityattimet.Wearegiventhat
Y'(t)=2sint.
Integrating,wefindthat
ThisisallwecandeduceaboutY(t)fromaknowledgeofthevelocityalone;someotherpieceofinformationisneededtofixthepositionfunction.WecandetermineCifweknowthevalueofYatsomeparticularinstant.Forexample,ifY(0)=0,thenC=2andthepositionfunctionisY(t)=2-2cost.ButifY(0)=2,thenC=4andthepositionfunctionisY(t)=4-2cost.
Insomerespectstheexamplejustsolvedistypicalofwhathappensingeneral.Some-whereintheprocessofsolvingafirst-orderdifferentialequation,anintegrationisrequiredtoremovethederivativey'andinthisstepanarbitraryconstantCappears.ThewayinwhichthearbitraryconstantCentersintothesolutionwilldependonthenatureofthegivendifferentialequation.Itmayappearasanadditiveconstant,asinEquation(9.4),butitismorelikelytoappearinsomeotherway.Forexample,whenwesolvetheequationy'=yinSection9.3,weshallfindthateverysolutionhastheformy=Cex.
Inmanyproblemsitisnecessarytoselectfromthecollectionofallsolutionsonehavingaprescribedvalueatsomepoint.Theprescribedvalueiscalledaninitialcondition,andtheproblemofdeterminingsuchasolutioniscalledaninitial-valueproblem.Thisterminologyoriginatedinmechanicswhere,asintheaboveexample,theprescribedvaluerepresentsthedisplacementatsomeinitialtime.
SUPPLEMENTEquationsofmathematicalphysics
Apartialdifferentialequation(PDE)isanequationinvolvingfunctionsandtheirpartialderivatives;forexample,thewaveequation
Ingeneral,partialdifferentialequationsaremuchmoredifficulttosolveanalyticallythanareordinarydifferentialequations.TheymaysometimesbesolvedusingaBäcklundtransformation,characteristics,Green'sfunction,integraltransform,separationofvariables,or--whenallelsefails(whichitfrequentlydoes)--numericalmethodssuchasfinitedifferences.
Thefollowingthreegeneralexamplesofimportantpartialdifferentialequationsthatcommonlyariseinproblemsofmathematicalphysics.
waveequationThewaveequationistheimportantpartialdifferentialequation
thatdescribespropagationofwaveswithspeedV.Theformabovegivesthewaveequationinthree-dimensionalspacewhere2istheLaplacian,whichcanalsobewritten
Theone-dimensionalwaveequation(D'Alembert,vibratingstringequation)isgivenby
Aswithallpartialdifferentialequations,suitableinitialand/orboundaryconditionsmustbegiventoobtainsolutionstotheequationforparticulargeometriesandstartingconditions.
WaveEquation--Disk,WaveEquation--RectangleandWaveEquation--Triangleareexamplesoftwo-dimensionalwaveequation.
HeatconductionequationApartialdifferentialequationoftheform
Physically,theequationcommonlyarisesinsituationswhereisthethermaldiffusivityandTthetemperature.
Theone-dimensionalheatconductionequationis
Notethatinthesteadystate,thatiswhen
=0,weareleftwiththeLaplacianofT:
2T=0.
Laplace'sequationisthepartialdifferentialequation
where2istheLaplacian,notethattheoperator2iscommonlywrittenasbymathematicians.
Laplace'sequationi
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