UndergraduateTextsinMathematics
Serieseditors
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet UniversityofCalifornia,Berkeley,CA,USA
AdvisoryBoard
ColinAdams, WilliamsCollege,Williamstown,MA,USA
AlejandroAdem, UniversityofBritishColumbia,Vancouver,BC,Canada
RuthCharney, BrandeisUniversity,Waltham,MA,USA
IreneM.Gamba, TheUniversityofTexasatAustin,Austin,TX,USA
RogerE.Howe, YaleUniversity,NewHaven,CT,USA
DavidJerison, MassachusettsInstituteofTechnology,Cambridge,MA,USA
JeffreyC.Lagarias, UniversityofMichigan,AnnArbor,MI,USA
JillPipher, BrownUniversity,Providence,RI,USA
FadilSantosa, UniversityofMinnesota,Minneapolis,MN,USA
AmieWilkinson, UniversityofChicago,Chicago,IL,USA
UndergraduateTextsinMathematics aregenerallyaimedatthird-and fourth-yearundergraduatemathematicsstudentsatNorthAmericanuniversities.Thesetextsstrivetoprovidestudentsandteacherswithnewperspectives andnovelapproaches.Thebooksincludemotivationthatguidesthereaderto anappreciationofinterrelationsamongdifferentaspectsofthesubject.They featureexamplesthatillustratekeyconceptsaswellasexercisesthatstrengthen understanding.
Forfurthervolumes: http://www.springer.com/series/666
J.DavidLogan
AppliedPartialDifferential Equations
J.DavidLogan DepartmentofMathematics
UniversityofNebraska-Lincoln
Lincoln,NE,USA
ISSN0172-6056
ISSN2197-5604(electronic) UndergraduateTextsinMathematics
ISBN978-3-319-12492-6ISBN978-3-319-12493-3(eBook) DOI10.1007/978-3-319-12493-3
SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014955188
MathematicsSubjectClassification:34-01,00-01,00A69,97M50,97M60
c SpringerInternationalPublishingSwitzerland2015
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ToAaron,Rachel,andDavid
1.ThePhysicalOriginsofPartialDifferentialEquations
2.PartialDifferentialEquationsonUnboundedDomains
2.1CauchyProblemfortheHeatEquation
2.3Well-PosedProblems
2.4Semi-InfiniteDomains
3.OrthogonalExpansions
3.3ClassicalFourierSeries
4.PartialDifferentialEquationsonBoundedDomains
4.1OverviewofSeparationofVariables
4.2Sturm–LiouvilleProblems
4.3GeneralizationandSingularProblems
4.4Laplace’sEquation
4.5CoolingofaSphere
4.6DiffusioninaDisk
4.7SourcesonBoundedDomains
4.8Poisson’sEquation*
5.ApplicationsintheLifeSciences
5.1Age-StructuredModels
5.2TravelingWavesFronts
5.3EquilibriaandStability
6.NumericalComputationofSolutions
6.1FiniteDifferenceApproximations
6.2ExplicitSchemefortheHeatEquation
6.3Laplace’sEquation
ThePhysicalOriginsofPartial DifferentialEquations
Manyimportantideasinmathematicsaredevelopedwithintheframework ofphysicalscience,andmathematicalequations,especiallypartialdifferential equations,providesthelanguagetoformulatetheseideas.Inreverse,advances inmathematicsprovidesthestimulusfo rnewadvancementsinscience.Overthe yearsmathematiciansandscientistsextendedthesemethodologiestoinclude nearlyallareasofscienceandtechnology,andaparadigmemergedcalledmathematicalmodeling.A mathematicalmodel isanequation,orsetofequations, whosesolutiondescribesthephysicalb ehavioroftherelatedphysicalsystem. Inthiscontextwesay,forexample,thatMaxwell’sequationsformamodel forelectromagneticphenomena.Likemostmathematicalmodels,Maxwell’s equationsarebasedonphysicalobservations.Butthemodelissoaccurate, weregardthemodelitselfasdescribinganactualphysicallaw.Othermodels,forexampleamodelofhowadiseasespreadsinapopulation,aremore conceptual.Suchmodelsoftenexplainobservations,butonlyinahighlylimitedsense.Ingeneral,amathematicalmodelisasimplifieddescription,or caricature,ofrealityexpressedinmathematicalterms.Mathematicalmodeling involvesobservation,selectionofrelevantphysicalvariables,formulationofthe equations,analysisoftheequationsandsimulation,and,finally,validationof themodeltoascertainwhetherindeedit ispredictive.Thesubjectofpartial differentialequationsencompassesalltypesofmodels,fromphysicallawslike Maxwell’sequationsinelectrodynamics,toconceptuallawsthatdescribethe spreadofanplantinvasivespeciesonasavanna.
c SpringerInternationalPublishingSwitzerland2015 1 J.D.Logan, AppliedPartialDifferentialEquations, 3rdedition. UndergraduateTextsinMathematics,DOI10.1007/978-3-319-12493-3 1
1.1PDEModels
Inthisbookweexaminemodelsthatcanbedescribedbypartialdifferential equations.Thefocusisontheoriginofsuchmodelsandtoolsusedfortheir analysis.Ofparticularinterestaremodelsindiffusionandheatflow,wavepropagation,andtransportofenergy,chemicals,andothermatter.Itisimpossible tooverestimatetheroleandimportanceofPDEsinscienceandengineering. Readersshouldbefamiliarwithsystemsgovernedbyordinarydifferential equations(ODEs).Forexample,atypicalODEmodelinpopulationecologyis thelogisticmodel
du dt = ru 1 u K ,t> 0, whichisasimpleequationforpopulationgrowthwherethepercapitarate ofchangeofpopulation, u (t)/u(t),isadecreasingfunctionofthepopulation. Here t istime,and u = u(t)isthepopulationofagivensystemofindividuals. Wereferto u asthestateandsaythattheevolutionofthestatevariableis governedbythemodelequation.Thepositivenumbers r and K aregivenphysicalparametersthatrepresenttherelativegrowthrateandcarryingcapacity, respectively;presumably, r and K canbemeasuredforthepopulationunder investigation.Thesolutiontothelogisticequationiseasilyfoundbyseparation ofvariablestobe
u(t)= u0 K u0 +(K u0 ) e rt ,t> 0, where u(0)= u0 istheinitialpopulation.Thelogisticmodelaccurately describessomepopulationshavingasigmoidgrowthshape.Ingeneral,anODE modelhastheform
du dt = F (t,u; r1 ,...,rn ),t> 0,
where F isagivenfunctionalrelationbetween t, u,and m parameters r1 ,...,rm .Oftenthemodelincludesaninitialconditionoftheform u(0)= u0 , where u0 isagivenstatevalueat t =0.Moregenerally,anODEmodelmay consistofasystemof n ODEsfor n statevariables u1 (t),...,un (t).
APDEmodeldiffersfromanODEmodelinthatthestatevariable u dependsonmorethanoneindependentvariable.ODEsgoverntheevolutionof asystemintime,andobservationsaremadeintime.PDEsmodeltheevolution ofasystemin both timeandspace;thesystemcanbeobservedbothinatime intervalandinaspatialregion(whichmaybeone-,two-,orthree-dimensional). PDEmodelsmayalsobeindependentoftime,butdependonseveralspatial variables.TwoexamplesofPDEsare
utt (x,t) c 2 uxx (x,t)=0, (waveequation)
uxx (x,y )+ uyy (x,y )=0 (Laplace’sequation)
Thewaveequationdescribesthepropagationofwavesinaonedimensional medium.Theunknownfunction u = u(x,t)isafunctionofposition x andtime t.InLaplace’sequation,theunknownstateisafunction u = u(x,y ),where x and y arespatialvariables.Itmodels,forexample,equilibriumtemperatures inatwo-dimensionalregionoftheplanewithprescribedtemperaturesonits boundary.
Example1.1
(Heatflow )Considertheproblemofdeterminingthetemperatureinathin, laterallyinsulated,cylindrical,metalbaroflength l andunitcross-sectional area,whosetwoendsaremaintainedataconstantzerodegrees,andwhose temperatureinitially(attimezero)variesalongthebarandisgivenbyafixed function φ(x).SeeFigure 1.1.
Figure1.1 Alaterallyinsulatedmetalbarwithzerotemperatureatboth ends.Heatflowsintheaxial,or x-direction,and u(x,t)isthetemperatureof thecross-sectionat x attime t.Attime t =0thetemperatureatlocations x isgivenby φ(x)
Howdoesthebarcooldown?Inthiscase,thestatevariable u isthetemperature,anditdependsuponbothwhenthemeasurementistakenandwherein thebaritistaken.Thus, u = u(x,t),where t istimeand0 <x<l .Theequationgoverningtheevolutionofthetemperature u iscalledthe heatequation (wederiveitinSection1.3),andithastheform
ut = kuxx . (1.1)
Observethatthesubscriptnotationisusedtoindicatepartialdifferentiation, andwerarelywritetheindependentvariables,preferring u to u(x,t).Theequationstatesthatthepartialderivativeofthetemperaturewithrespectto t must
equalthesecondpartialderivativeofthetemperaturewithrespectto x,multipliedbyaconstant k .Theconstant k ,calledthe diffusivity,isaknownparameterandapropertyofthebar;itcanbedeterminedintermsofthedensity, specificheat,andthermalconductivityofthemetal.Valuesforthesephysical constantsfordifferentmaterialscanbefoundinhandbooksoronline.Laterwe observethat(1.1)comesfromabasicphysicallaw(energyconservation)and anempiricalobservation(Fourier’sheatconductionlaw).Theconditionsthat theendfacesofthebararemaintainedatzerodegreescanbeexpressedbythe equations
whicharecalled boundaryconditions becausetheyimposeconditionsonthe temperatureattheboundaryofthespatialdomain.Thestipulationthatthe barinitiallyhasafixedtemperature φ(x)degreesacrossitslengthisexpressed mathematicallyby
Thisconditioniscalledan initialcondition becauseitspecifiesthestate variableattime t =0.Theentiresetofequations(1.1)–(1.3)—thePDEand theauxiliaryconditions—formthemathematicalmodelforheatflowinthe bar.SuchamodelinthesubjectofPDEsiscalledan initialboundaryvalue problem.Theinventionandanalysisofsuchmodelsarethesubjectsofthis book.
Inthisheatflowmodel,thestatevariable u,thetemperature,dependsupon twoindependentvariables,atimevariable t andaspatialvariable x.Sucha modelisan evolutionmodel.Somephysicalsystemsdonotdependupon time,butratheronlyuponspatialvariables.Suchmodelsarecalled steady state or equilibrium models.Forexample,if Ω isabounded,two-dimensional spatialdomainrepresentingaplanar,laminarplate,andontheboundaryof Ω ,denotedby ∂Ω ,thereisimposedagiven,time-independenttemperature, thenthesteady-statetemperaturedistribution u = u(x,y )inside Ω satisfies theLaplaceequation,apartialdifferentialequationhavingtheform
Ifwedenotethefixedboundarytemperatureby f (x,y ),then(1.4)alongwith theboundarycondition u(x,y )= f (x,y ), (x,y ) ∈ ∂Ω, (1.5)
isanequilibriummodelfortemperaturesintheplate.InPDEsthesespatial modelsarecalled boundaryvalueproblems.SolvingLaplace’sequation
(1.4)inaregion Ω subjecttoagivencondition(1.5)ontheboundaryisa famousproblemcalledthe Dirichletproblem.
Ingeneral,asecond-orderevolutionPDEinonespatialvariableandtime isanequationoftheform
where I isagivenspatialinterval,whichmaybeaboundedorunbounded. Theequationinvolvesanunknownfunction u = u(x,t),thestatevariable,and someofitspartialderivatives.The order ofaPDEequationistheorderof thehighestderivativethatoccurs.ThePDEisalmostalwayssupplemented withinitialand/orboundaryconditionsthatspecifythestate u attime t =0 andontheboundary.Oneormoreparameters,whicharenotexplicitlyshown, mayalsooccurin(1.6).
PDEsareclassifiedaccordingtotheirorderandotherproperties.Forexample,asisthecaseforODEs,theyareclassifiedaslinearornonlinear.Equation (1.6)is linear if G isalinearfunctionin u andinallofitsderivatives;how theindependentvariables x and t appearisnotrelevant.Thismeansthatthe unknown u anditsderivativesappearaloneandtothethefirstpower.Otherwise,thePDEis nonlinear.Alinearequationis homogeneous ifeveryterm contains u orsomederivativeof u.Itis nonhomogeneous ifthereisaterm dependingonlyontheindependentvariables, t and x
Example1.2
Bothsecond-orderequations
ut + uuxx =0and utt ux +sin u =0
arenonlinear,thefirstbecauseoftheproduct uuxx andthesecondbecausethe unknown u istiedupinthenonlinearsinefunction.Thesecond-orderequation
ut sin(x 2 t)uxt =0 islinearandhomogeneous,andtheequation
ut +3xuxx = tx2 islinearandnonhomogeneous.
Inmanydiscussionsitisconvenienttointroduce operatornotation.For example,wecanwritetheheatequation ut kuxx =0
as
=0where L =
Here L isadifferentialoperator,andwewriteitsactiononafunction u as aseither Lu or L(u).Itactsontwicecontinuouslydifferentiablefunctions u = u(x,t)toproduceanewfunction.Wesayadifferentialoperator L is linear if,andonlyif,itsatisfiesthetwoconditions
L(u + v )= Lu + Lv,L(cu)= cLu
forallfunctions u and v ,andallconstants c.If L isalinear,thentheequation Lu =0issaidtobe homogeneous,andtheequation Lu = f is nonhomogeneous
OnecannotoverstatethesignificanceofthepartitionofPDEsintothetwo categoriesoflinearandnonlinear.Linearequationshavealgebraicstructureto theirsolutionsets: thesumoftwosolutionstoahomogeneouslinearequationis againasolution,asareconstantmultiplesofsolutions.Anotherwayofsaying thisisthatsolutions superimpose.Thus,if u1 , u2 ,..., un aresolutionsto Lu =0,and c1 , c2 ,...,cn areconstants,thenthe linearcombination
c1 u1 + c2 u2 + + cn un
isalsoasolutionto Lu =0 Asweseelater,this superpositionprinciple extendsinmanycasestoinfinitesumsandeventoacontinuumofsolutions.
Forexample,if u(x,t,ξ )isaone-parameterfamilyofsolutionsto Lu =0,for all ξ inaninterval J ,thenwecanoftenprove
c(ξ )u(x,t,ξ ) dξ
isasolutionto Lu =0forspecialconditionsonthedistributed‘constants’ (i.e.,thefunction) c(ξ ).Thesesuperpositionprinciplesareessentialinthis text.Everyconceptweuseinvolvessuperpositioninonewayoranother.
Anotherresultbasedonlinearityisthattherealandimaginarypartsof acomplex-valuedsolution w toahomogeneousdifferentialequation Lw =0 arebothrealsolutions.Specifically,if w iscomplex-valuedfunction,then w = u + iv ,where u =Re w and v =Im w arereal-valuedfunctions.Then,by linearity,
Lw = L(u + iv )= Lu + iLv =0
Thisimplies Lu =0and Lv =0,becauseifacomplexfunctionisindentically zerothenbothitsrealandimaginarypartsarezero.
Nonlinearequationsdonotsharetheseproperties.Nonlinearequationsare hardertosolve,andtheirsolutionsaremoredifficulttoanalyze.Evenwhen
J
naturepresentsuswithanonlinearmodel,weoftenapproximateitwithamore manageablelinearone.
EquallyimportantinclassifyingPDEsisthespecificnatureofthephysicalphenomenathattheydescribe.Forexample,aPDEscanbeclassifiedas wave-like, diffusion-like,or equilibrium,dependingonwhetheritmodelswave propagation,adiffusionprocess,oranequilibriumstate.Forexample,Laplace’s equation(1.4)isasecond-order,linearequilibriumequation;theheatequation (1.1)isasecond-order,lineardiffusionequationbecauseheatflowisadiffusion process.Inthelastsectionofthischapterwegiveamoreprecise,mathematical characterizationoftheseproperties.
Bya solution tothePDE(1.6)wemeanafunction u = u(x,t)defined onthespace–timedomain t> 0,x ∈ I ,thatsatisfies,uponsubstitution, theequation(1.6)identicallyonthatdomain.Implicitinthisdefinitionisthe stipulationthat u possessasmanycontinuouspartialderivativesasrequiredby thePDE.Forexample,asolutiontoasecond-orderequationshouldhavetwo continuouspartialderivativessothatitmakessensetocalculatethederivatives andsubstitutethemintotheequation.WhereasthegeneralsolutiontoanODE involvesarbitraryconstants,thegeneralsolutiontoaPDEinvolvesarbitrary functions.SometimesthegeneralsolutiontoaPDEcanbefound,butitis usuallynotnecessarytohaveittosolvemostproblemsofinterest.
Example1.3
Oneshouldcheck,bydirectsubstitution,thatbothfunctions u1 (x,t)= x 2 +2t and u2 (x,t)= e t sin x aresolutionstotheheatequation ut uxx =0
Therearemanyothersolutionstothisequation.Auxiliaryconditions,like initialandboundaryconditions,generallysingleouttheappropriatesolution toaproblem.
Example1.4
Considerthefirst-order,linear,nonhomogeneousPDE
ux = t sin x.
Thisequationcanbesolvedbydirectintegration.Weintegratewithrespectto x,holding t fixed,toget u(x,t)= t cos x + ψ (t),
where ψ isanarbitraryfunctionof t.InPDEs,integrationwithrespecttoone variableproducesanarbitraryfunctionoftheothervariable,notanarbitrary constantasinone-dimensionalcalculus.Thislastequationdefinesthegeneral solution.Onecancheckthatitisasolutionforanydifferentiablefunction ψ (t).Usually,PDEshavearbitraryfunctionsintheexpressionfortheirgeneral solutions;thenumberofsuchfunctionsoftenagreeswiththeorderofthe equation.
Example1.5
Thesecond-orderPDEfor u = u(x,t), utt 4u =0
isjustanlikeanODEwith x asaparameter.Sothe‘constants’dependon x. Thesolutionis u(x,t)=
, where φ and ψ arearbitraryfunctionsof x.
Figure1.2 Asolutionsurface u = u(x,t).Across-section u(x,t0 )ofthe surfaceattime t0 isinterpretedasawaveprofileat t = t0
Geometrically,asolution u = u(x,t)canbethoughtofasa surface in xtuspace.RefertoFigure 1.2.Thesurfaceliesoverthespace–timedomain: x ∈ I , t> 0.Alternately,onecouldregardthesolutionasacontinuoussequenceof
timesnapshots.Thatis,foreachfixedtime t0 , u(x,t0 )isafunctionof x alone andthusrepresentsatimesnapshotofthesolution.Indifferentwords, u(x,t0 ) isthetraceofthesolutionsurface u = u(x,t)takeninthe t = t0 -plane.In somecontexts, u(x,t0 )isinterpretedasa waveprofile,orsignal,attime t0 .In thiswayasolution u(x,t)of(1.6)canberegardedacontinuoussequenceof evolvingwaveformsevolvingintime.
BibliographicNotes. TherearedozensofexcellentelementaryPDEbooks writtenataboutthesamelevelasthis one.WeespeciallymentionFarlow (1993)andStrauss(1992).AmoreadvancedtreatmentisgivenbyMcOwen (2003).NonlinearPDEsatthebeginninglevelaretreatedindetailinDebnath (1997)orLogan(2008).PDEmodelsoccurineveryareaofthepureandapplied sciences.GeneraltextsinvolvingmodelinginengineeringandscienceareLin &Segel(1989),Holmes(2011),andLogan(2013).
EXERCISES
1.Verifythatasolutiontotheheatequation(1.1)onthedomain −∞ <x< ∞,t> 0isgivenby u(x,t)= 1 √
Forafixedtime,thereadershouldrecognizethissolutionasabell-shaped curve.(a)Pick k =0 5.Usesoftwaretosketchseveraltimesnapshotson thesamesetofcoordinateaxestoshowhowthetemperatureprofileevolves intime.(b)Whatdothetemperatureprofileslooklikeas t → 0?(c)Sketch thesolutionsurface u = u(x,t)inadomain 2 ≤ x ≤ 2,0 1 <t< 4.(d) Howdoeschangingtheparameter k affectthesolution?
2.Verifythat u(x,y )=ln x2 + y 2 satisfiestheLaplaceequation
uxx + uyy =0 forall(x,y ) =(0, 0).
3.Findthegeneralsolutionoftheequation uxy (x,y )=0intermsoftwo arbitraryfunctions.
4.Derivethesolution u = u(x,y )= axy + bx + cy + d (a,b,c,d constants), ofthePDE
u 2 xx + u 2 yy =0. Observethatthesolutiondoesnotexplicitlycontainarbitraryfunctions.
5.Findafunction u = u(x,t)thatsatisfiesthePDE
uxx =0, 0 <x< 1,t> 0, subjecttotheboundaryconditions u(0,t)= t2 ,u(1,t)=1,t>
6.Verifythat
isasolutiontothewaveequation utt = c2 uxx ,where c isaconstantand g is agivencontinuouslydifferentiablefunction.Hint:Hereyouwillneedtouse Leibniz’srule fordifferentiatinganintegralwithrespecttoaparameter thatoccursinthelimitsofintegration:
7.Forwhatvaluesof a and b isthefunction u(x,t)= eat sin bx asolutionto theheatequation ut = kuxx .
8.Findthegeneralsolutiontotheequation uxt +3ux =1.Hint:Let v = ux andsolvetheresultingequationfor v ;thenfind u
9.Showthatthenonlinearequation ut = u2 x + uxx canbereducedtotheheat equation(1.1)bychangingthedependentvariableto w = eu
10.Showthatthefunction u(x,y )=arctan(y/x)satisfiesthetwo-dimensional Laplace’sequation uxx + uyy =0.
11.Showthat e ξy sin(ξx), x ∈ R,y> 0,isasolutionto uxx + uyy =0for anyvalueoftheparameter ξ .Deducethat u(x,y )= ∞ 0 c(ξ )e ξy sin(ξx)dξ isasolutiontothesameequationforanyfunction c(ξ )thatisbounded andcontinuouson[0, ∞).Hint:Thehypotheseson c allowyoutobring aderivativeundertheintegralsign.[Thisexerciseshowsthattakingintegralsofsolutionssometimesgivesanothersolution;integrationisawayof superimposing,oradding,acontinuumofsolutions.]
12.Linear,homogeneousPDEswithconstantcoefficientsadmitcomplexsolutionsoftheform
u(x,t)= Aei(kx ωt) , whicharecalled planewaves.Therealandimaginarypartsofthiscomplexfunction,
Re(u)= A cos(kx ωt), Im(u)= A sin(kx ωt), giverealsolutions.Theconstant A isthe amplitude, k isthe wavenumber,and ω isthe temporalfrequency.Whentheplanewaveformis substitutedintoaPDEthereresultsa dispersionrelation oftheform ω = ω (k ), whichstateshowthefrequencydependsuponthewavenumber.Forthe followingPDEsfindthedispersionrelationanddeterminetheresulting planewave;sketchwaveprofilesatdifferenttimes.
a) ut = Duxx .
b) utt = c2 uxx .
c) ut + uxxx =0.
d) ut = iuxx . (Here, i isthecomplexnumber i2 = 1.)
e) ut + cux =0
13.Second-orderlinearhomogeneousequationswithconstantcoefficientsare oftenclassifiedbytheirdispersionrelation ω = ω (k )(seeExercise12). If ω (k )iscomplex,thePDEiscalled diffusive,andif ω (k )isrealand ω (k ) =0,thePDEiscalled dispersive.Thediffusionequationisdiffusive;thewaveequationisneitherdiffusiveordispersive.Thetermdispersivemeansthatthespeed ω (k )/k ofaplanewave u = Aei(kx ω (k)t) travelsdependsuponthewavenumber k .Sowavesofdifferentwavelength travelatdifferentspeeds,andthus disperse.ClassifythePDEsin(a)–(e) ofExercise12accordingtothisscheme.
14.FindplanewavesolutionstotheKuromoto–Sivashinskyequation
ut = u δuxx uxxxx ,δ> 0.
Findthedispersionrelationandclassifytheequationaccordingtothe schemeoftheprecedingexercise.Describethesolutionsandplot δ asa functionofthewavenumber k todeterminewhenthegrowthrateofa solutioniszero.Forwhichwavenumberswillthesolutiondecay?
1.2ConservationLaws
ManyPDEscomefromabasicbalance,orconservationlaw.A conservation law isamathematicalformulationofthefactthattherateatwhichaquantity changesinagivendomainmustequaltherateatwhichthequantityflows acrosstheboundary(inminusout)plustherateatwhichthequantityis createdwithinthedomain.Forexample,considerapopulationofacertain animalspeciesinafixedgeographicalregion.Therateofchangeoftheanimal populationmustequaltherateatwhichanimalsmigrateintotheregion,minus therateatwhichtheymigrateout,plusthebirthrate,minusthedeathrate. Suchastatementisaverbalexpressionofabalance,orconservation,law.One canmakesimilarkindsofstatementsformanyquantities—energy,themassof achemicalspecies,thenumberofautomobilesonafreeway,andsoon.
Figure1.3 Tubewithcross-sectionalarea A shownwitharbitrarycrosssectionat x (shaded ).Thelateralsidesareinsulated,andthephysicalquantities varyonlyinthe x-directionandintime.Allquantitiesareconstantoverany cross-section
Toquantifysuchstatementswerequiresomenotation.Letthestatevariable u = u(x,t)denotethedensityofagivenquantity(mass,energy,animals, automobiles,etc.);densityisusuallymeasuredinamountperunitvolume,or sometimesamountperunitlength.Forexample,energydensityismeasuredin energyunitspervolume.Weassumethatanyvariationinthestateberestricted toonespatialdimension.Thatis,weassumeaone-dimensionaldomain(say,a tube,asinFigure 1.3 whereeachcross-sectionislab eledbythespatialvariable x;werequirethattherebenovariationof u(x,t)withinthecross-sectionat x.Implicitistheassumptionthatthequantityinthetubeisabundantand continuousenoughin x sothatitmakessensetodefineitsdensityateach sectionofthetube.Theamountofthequantityinasmallsectionofwidth dx is u(x,t)Adx,where A isthecross-sectionalareao fthetube.Further,welet φ = φ(x,t)denotethe flux ofthequantityat x,attime t.Thefluxmeasures theamountofthequantitycrossingthesectionat x attime t,anditsunits aregiveninamountperunitarea,perunittime.Thus, Aφ(x,t)istheactual
amountofthequantitythatiscrossingthesectionat x attime t.Byconvention, fluxispositiveiftheflowistotheright,andnegativeiftheflowistotheleft. Finally,let f = f (x,t)denotethegivenrateatwhichthequantityiscreated,or destroyed,withinthesectionat x attime t.Thefunction f iscalleda source term ifitispositive,anda sink ifitisnegative;itismeasuredinamount perunitvolumeperunittime.Thus, f (x,t)Adx representstheamountofthe quantitythatiscreatedinasmallwidth dx perunittime.
Aconservationlawisaquantitativerelationbetween u, φ,and f .Wecan formulatethelawbyconsideringafixed,butarbitrary,section a ≤ x ≤ b of thetube(Figure 1.3)andrequiringthattherateofchangeofthetotalamount ofthequantityinthesectionmustequaltherateatwhichitflowsinat x = a, minustherateatwhichitflowsoutat x = b,plustherateatwhichitiscreated within a ≤ x ≤ b.Inmathematicalsymbols,
Thisequationisthefundamentalconservationlaw;itisanintegralexpression ofthebasicfactthattheremustbeabalancebetweenhowmuchgoesin,how muchgoesout,andhowmuchischanged.Because A isconstant,itmaybe canceledfromtheformula.
Equation(1.7)isanintegrallaw.However,ifthefunctions u and φ are sufficientlysmooth,thenitmaybereformulatedasaPDE,whichisalocal law.Forexample,if u hascontinuousfirstpartialderivatives,thenthetime derivativeontheleftsideof(1.7)maybebroughtundertheintegralsignto obtain
If φ hascontinuousfirstpartials,thenthefundamentaltheoremofcalculuscan beappliedtowritethechangeinfluxastheintegralofaderivative,or
Therefore,(1.7)maybewritten
Because a ≤ x ≤ b canbeanyintervalwhatsoever,andbecausetheintegrand iscontinuous,itfollowsthattheintegrandmustvanishidentically,or
Equation(1.8)isalocalversionof(1.7),obtainedundertheassumptionthat u and φ arecontinuouslydifferentiable;itisaPDEmodeldescribingtherelation betweenthedensitythequantity,itsflux,andtherateatwhichthequantity iscreated.WecallthePDE(1.8)the fundamentalconservationlaw.The f -termiscalledthesourceterm,andthe φ-termiscalledthefluxterm.In (1.8)weusuallydroptheunderstoodnotationaldependenceon x and t and justwrite ut + φx = f forsimplicity. Beforestudyingsomeexamples,wemakesomegeneralcomments.Theflux φ andsource f arefunctionsof x and t,buttheirdependenceon x and t maybe throughdependenceuponthedensity u itself.Forexample,thesourceterm f maybegivenasafunctionofdensityvia f = f (u),where,ofcourse, u = u(x,t). Similarly, φ maydependon u.Thesedependenciesleadtononlinearmodels. Next,weobservethat(1.8)isasingleequation,yettherearetwounknowns, u and φ (theformofthesource f isassumedtobeprescribed).Thisimpliesthat anotherequationisrequiredthatrelates u and φ.Suchequationsarecalled constitutiverelations (orequationsofstate),andtheyarisefromphysical assumptionsaboutthemediumitself.
TheMethodofCharacteristics
Inthissection,inthecontextoftheadvectionofmaterialsthrougha medium,weintroducethebasicmethodforsolvingfirstorderPDEs,the methodofcharacteristics.
Example1.6
(Advection)Amodelwherethefluxisproportionaltothedensityitself,that is, φ = cu,
where c isaconstant,iscalledan advection model.Noticethat c musthave velocityunits(lengthpertime).Inthiscasetheconservationlaw(1.8)becomes, intheabsenceofsources(f =0),
Equation(1.9)iscalledthe advectionequation.Thereadershouldverify, usingthechainrule,thatthefunction
isasolutionto(1.9)foranydifferentiablefunction F .Suchsolutions(1.10) arecalled right-travelingwaves becausethegraphof F (x ct)isthegraph of F (x)shiftedtotheright ct spatialunits.So,astime t increases,thewave profile F (x)movestotheright,undistorted,withitsshapeunchanged,atspeed
