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AnalyticSolutionsofPartialDi erentialEquations

MATH3414

SchoolofMathematics,UniversityofLeeds

15credits Taught

Semester1, Yearrunning2003/04 Pre-requisitesMATH2360orMATH2420orequivalent.

Co-requisitesNone. Objectives:Toprovideanunderstandingof,andmethodsof

solutionfor,themostimportanttypesofpartialdi erentialequations

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thatariseinMathematicalPhysics.Oncompletionofthismodule,studentsshouldbeableto:a)usethemethodofcharacteristicstosolve

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rst-orderhyperbolicequations;b)classifyasecondorderPDEaselliptic,parabolicorhyperbolic;c)useGreen'sfunctionstosolveellipticequations;d)have

abasicunderstandingofdi usion;e)obtainaprioriboundsforreaction-di usionequations.

Syllabus:Themajorityofphysicalphenomenacanbedescribedbypartialdi erential

equations(e.g.theNavier-Stokesequationofuiddynamics,Maxwell's

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equationsofelectromagnetism).Thismoduleconsidersthepropertiesof,andanalyticalmethodsofsolutionforsomeofthemostcommon

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rstandsecondorderPDEsofMathematicalPhysics.Inparticular,weshalllookindetailatellipticequations(Laplace?sequation),describingsteady-statephenomena

andthedi usion/heatconductionequationdescribingtheslowspreadofconcentratio

norheat.Thetopicscoveredare:FirstorderPDEs.Semilinearandquasilinear

PDEs;methodofcharacteristics.Characteristicscrossing.SecondorderPDEs.Classi

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cationandstandardforms.Ellipticequations:weakandstrongminimumandmaximumprinciples;Green'sfunctions.Parabolicequations:exempli

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edbysolutionsofthedi usionequation.Boundsonsolutionsofreaction-di usionequations. FormofteachingLectures:26hours.7examplesclasses.

FormofassessmentOne3hourexaminationatendofsemester(100%).

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Details:

EvyKersal e

O ce:9.22ePhone:01133435149E-mail:[email protected]:http://www.maths.leeds.ac.uk/~kersale/

Schedule:threelecturesevery

week,forelevenweeks(from27/09to10/12).

Tuesday13:00{14:00RSLT

03

Wednesday10:00{11:00RSLT04

Friday11:00{12:00RSLT06

Pre-requisite:elementarydi erentialcalculusandseveralvariablescalculus

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(e.g.partialdi erentiationwithchangeofvariables,parametriccurves,integration),elementaryalgebra(e.g.partialfractions,lineareigenvalueproblems),ordinarydi erentialequations(e.g.changeof

variable,integratingfactor),andvectorcalculus(e.g.vectoridentities,Green'stheorem).

Outlineofcourse:

Introduction:

de

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nitionsexamples

FirstorderPDEs:

linear&semilinearcharacteristicsquasilinearnonlinearsystemofequations

SecondorderlinearPDEs:

classi

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cationellipticparabolic

Booklist:

P.Prasad&R.Ravindran,\PartialDi erentialEquations",WileyEastern,1985.W.E.Williams,

\PartialDi erentialEquations",OxfordUniversityPress,1980.P.R.Garabedian,\PartialDi erentialEquations",

Wiley,1964.ThankstoProf.D.W.Hughes,Prof.J.H.Merkinand

Dr.R.Sturmanfortheirlecturenotes.

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CourseSummary

De

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nitionsofdi erenttypeofPDE(linear,quasilinear,semilinear,nonlinear) Existenceanduniquenessofsolutions SolvingPDEsanalyticallyisgenerallybasedon

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ndingachangeofvariabletotransformtheequationintosomethingsolubleoron

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ndinganintegralformofthesolution.FirstorderPDEs

@u@u

a+b=c.

@x

@y

Linearequations:changecoordinateusing (x,y),de

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nedbythecharacteristicequationdyb

=;

dxaand (x,y)independent(usually.=x)totransformthePDE

intoanODE.

Quasilinearequations:changecoordinateusingthesolutionsof

dxdydu

=a,=band=

c

dsdsds

toget

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animplicitformofthesolution (x,y,u)=F( (x,y,u)).Nonlinearwaves:regionofsolution.Systemoflinearequations:linearalgebra

todecoupleequations.

SecondorderPDEs

@2u@2u@2u@u@u

a+2b+c+d+e+fu=

g.

@x2@x@y@y2@x@y

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iii

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Classi

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cationTypeCanonicalformCharacteristicsb2-ac>0Hyperbolic@2u@ @s+...=0dydx=b

±.b2-acab2-ac=0Parabolic@2

u@ 2+...=0dydx=ba,

s=x(say)b2-ac<0Elliptic

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@2u@ 2+@2u@

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2+...=0dydx=b±.b2-aca,..=.+s

a=i(.- )

Ellipticequations:(Laplaceequation.)MaximumPrinciple.

SolutionsusingGreen'sfunctions(usesnewvariablesandtheDirac -functiontopick

outthesolution).Methodofimages.

Parabolicequations:

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(heatconduction,di usionequation.)Deriveafundamentalsolutioninintegralformormakeuseofthesimilaritypropertiesoftheequationto

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ndthesolutionintermsofthedi usionvariable

x

s==

.

2t

FirstandSecondMaximumPrinciplesandComparisonTheoremgiveboundson

thesolution,andcanthenconstructinvariantsets.

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Contents

1Introduction1

1.1Motivation.....................................1

1.2Reminder......................................1

1.3De

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nitions......................................2

1.4Examples......................................3

1.4.1WaveEquations..............................3

1.4.2Di usionorHeatConductionEquations......

..

.

.

.

.

.

.

.

.

.4

1.4.3Laplace'sEquation.............................4

1.4.4OtherCommonSecondOrderLinearPDEs....

.

.

.

.

.

.

.

.

.

.

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.4

1.4.5NonlinearPDEs..............................5

1.4.6SystemofPDEs..............................5

1.5ExistenceandUniqueness.............................6

2First

OrderEquations9

2.1LinearandSemilinearEquations.........................9

2.1.1MethodofCharacteristic

.........................9

2.1.2EquivalentsetofODEs..........................12

2.1.3CharacteristicCurves...........................

14

2.2QuasilinearEquations...............................19

2.2.1

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InterpretationofQuasilinearEquation..................19

2.2.2Generalsolution:..............................20

2.3WaveEquation...................................26

2.3.1LinearWaves................................26

2.3.2NonlinearWaves..............................27

2.3.3

WeakSolution...............................29

2.4SystemsofEquations................................31

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2.4.1LinearandSemilinearEquations.....................31

2.4.2QuasilinearEquations...........................34

3SecondOrderLinearandSemilinearEquationsinTwoVariables37

3.1Classi

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cationandStandardFormReduction...................37

3.2ExtensionsoftheTheory

.............................44

3.2.1Linearsecondorderequationsinnvariables...............44

3.2.2TheCauchyProblem............................45

i

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CONTENTS

4EllipticEquations49

4.1De

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nitions......................................49

4.2PropertiesofLaplace'sandPoisson'sEquations.............

..

.

.50

4.2.1MeanValueProperty...........................51

4.2.2Maximum-MinimumPrinciple.......................52

4.3SolvingPoissonEquationUsingGreen'sFunctions.

.

.

.

.

.

.

.

.

.

.

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.

.

.

.54

4.3.1De

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nitionofGreen'sFunctions......................55

4.3.2Green'sfunctionforLaplaceOperator.................

.55

4.3.3FreeSpaceGreen'sFunction.......................60

4.3.4MethodofImages.............................

61

4.4ExtensionsofTheory:...............................68

5ParabolicEquations69

5.1De

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nitionsandProperties.............................69

5.1.1Well-PosedCauchyProblem(InitialValueProblem)..........69

5.1.2Well-PosedInitial-BoundaryValueProblem.......

..

.

.

.

.

.

.70

5.1.3TimeIrreversibilityoftheHeatEquation

.

.

.

.

.

.

.

.

.

.

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.

.

.

.

.

.70

5.1.4UniquenessofSolutionforCauchyProblem:........

..

.

.

.

.

.71

5.1.5Uniquenessof

SolutionforInitial-BoundaryValueProblem:......71

5.2FundamentalSolutionoftheHeatEquation.......

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.

.

.

.

.

.

.

.

.

.

.

.72

5.2.1IntegralFormoftheGeneralSolution.......

..

.

.

.

.

.

.

.

.

.73

5.2.2PropertiesoftheFundamentalSolution..........

.

.

.

.

.

.

.74

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5.2.3Behaviouratlarget............................75

5.3SimilaritySolution.................................75

5.3.1In

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niteRegion...............................76

5.3.2Semi-In

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niteRegion............................77

5.4MaximumPrinciplesandComparisonTheorems.............

..

.78

5.4.1FirstMaximumPrinciple.........................79

A

Integralofe..x2inR81

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Chapter1

Introduction

Contents

1.1Motivation................................11.2Reminder.................................11.3De

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nitions.................................21.4Examples.................................31.5ExistenceandUniqueness.......................61.1MotivationWhydowestudypartialdi erentialequations(PDEs)andinparticularanalytic

solutions?

WeareinterestedinPDEsbecausemostofmathematicalphysics

isdescribedbysuchequations.Forexample,uidsdynamics(andmoregenerallycontinuous

mediadynamics),electromagnetictheory,quantummechanics,tra cow.Typically,a

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givenPDEwillonlybeaccessibletonumericalsolution(withoneobviousexception examquestions!)andanalyticsolutionsinapracticalorresearch

scenarioareoftenimpossible.However,itisvitaltounderstandthegeneraltheory

inordertoconductasensibleinvestigation.Forexample,wemayneedto

understandwhattypeofPDEwehavetoensurethe

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numericalsolutionisvalid.Indeed,certaintypesofequationsneedappropriateboundaryconditions;withoutaknowledgeofthegeneraltheoryitispossiblethat

theproblemmaybeill-posedofthatthemethodissolutioniserroneous.

1.2ReminderPartialderivatives:Thedi erential(ordi erentialform)ofa

functionfofnindependentvariables,(x1;x2;:::;xn),isalinear

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combinationofthebasisform(dx1,dx2;:::,dxn)

n

.@f@f@f@fdf=dxi=dx1+dx2+::.

+dxn;

@xi@x1@x2@xn

i=1

wherethepartial

derivativesarede

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nedby

@ff(x1;x2;:::;xi+h,...,xn)-f(x1;x2;:::;xi;:::;xn)

=lim.

@xih!0h

1

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1.3De

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nitionsTheusualdi erentiationidentitiesapplytothepartialdi erentiations(sum,product,quotient,chainrules,etc.)

Notations:Ishalluseinterchangeablythe

notations

@f@2f

[email protected],

[email protected];

@xi@xi@xj

forthe

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rstorderandsecondorderpartialderivativesrespectively.Weshallalsouseinterchangeablythenotations

~u.u.u,

forvectors.

Vectordi erentialoperators:inthreedimensionalCartesiancoordinatesystem

(i,j,k)weconsiderf(x,y,z):R3.Rand[ux(x,

y,z);uy(x,y,z);uz(x,y,z)]:R3

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.R3.Gradient:rf=@xfi+@[email protected]:divu r·u=@xux+

@yuy+@zuz.

Curl:r×u=(@zuy-@y

uz)i+(@zux-@xuz)j+(@xuy-@yux)k.

Laplacian: f r2f=@2f+@2f+

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@2f.

xyz

Laplacianofavector: u r2u=r2uxi+r2uyj+r2uzk.

Note

thattheseoperatorsaredi erentinothersystemsofcoordinate(cylindricalorspherical,

say)

1.3De

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nitionsApartialdi erentialequation(PDE)isanequationforsomequantityu(dependentvariable)whichdependsontheindependentvariablesx1;x2;x3;:::;xn;n~2,

andinvolvesderivativesofuwithrespecttoatleastsomeofthe

independentvariables.

F(x1;:::;xn;@x1u,...,@xnu,

@2u,@2u,...,@nu)=0:

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x1x1x2x1:::xn

Note:

1.Inapplicationsxiareoftenspacevariables(e.g.x,y,z)and

asolutionmayberequiredinsomeregionGofspace.Inthis

casetherewillbesomeconditionstobesatis

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edontheboundary@;thesearecalledboundaryconditions(BCs).2.Alsoinapplications,oneoftheindependentvariablescanbetime

(tsay),thentherewillbesomeinitialconditions(ICs)tobesatis

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ed(i.e.,uisgivenatt=0everywherein

)3.Againinapplications,systemsofPDEscanariseinvolvingthe

dependentvariablesu1;u2;u3;:::;um;m~1withsome(atleast)oftheequationsinvolving

morethanoneui.

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Chapter1 Introduction

TheorderofthePDEistheorderofthehighest(partial)di erentialcoe cientintheequation.

Aswithordinarydi erentialequations(ODEs)itisimportanttobeableto

distinguishbetweenlinearandnonlinearequations.Alinearequationisoneinwhich

theequationandanyboundaryorinitialconditionsdonot

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includeanyproductofthedependentvariablesortheirderivatives;anequationthatisnotlinearisanonlinearequation.

@u@u

+c=0,

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rstorderlinearPDE(simplestwaveequation),@t@x

@2@2

uu

+=(x,y),secondorderlinearPDE(Poisson).

@x2@y2

Anonlinearequationissemilinearifthecoe cientsof

thehighestderivativearefunctionsoftheindependentvariablesonly.

@u

2@u

(x+3)+xy=u

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3;

@x@y

@2u@2u@u2@u

4

x+(xy+y2)+u+u

=u.

@x2@y2@x@y

AnonlinearPDE

ofordermisquasilinearifitislinearinthederivativesof

ordermwithcoe cientsdependingonlyonx,y,.

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.

.andderivativesoforder<m.

" 2." 2#

@u@2u@u@u@2u @u@2u

1+-2

+1+=0.

@y@x2@x@y@x@y@x@y2

Principleofsuperposition:Alinearequationhastheusefulpropertythatifu1

andu2bothsatisfytheequationthensodoes u1

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+

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u2forany ,a=R.Thisisoftenusedinconstructingsolutionstolinearequations(forexample,soastosatisfyboundary

orinitialconditions;c.f.Fourierseriesmethods).Thisisnottruefornonlinear

equations,whichhelpstomakethissortofequationsmoreinteresting,butmuch

moredi culttodealwith.

1.4Examples1.4.1

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WaveEquationsWavesonastring,soundwaves,wavesonstretchmembranes,electromagneticwaves,etc.

@2u1@2u

=;

@x2c2@t2

ormoregenerally1@2u

=r2u

c2@t2

wherecisa

constant(wavespeed).

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1.4Examples1.4.2Di usionorHeatConductionEquations@u@2u

= ;

@t@x2

ormoregenerally

@u= r2u;

@t

oreven

@u

=r·( ru)

@twheretisaconstant

(di usioncoe cientorthermometricconductivity).Boththoseequations(waveand

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di usion)arelinearequationsandinvolvetime(t).Theyrequiresomeinitialconditions(andpossiblysomeboundaryconditions)fortheirsolution.

1.4.3

Laplace'sEquationAnotherexampleofasecondorderlinearequationisthefollowing.

@2@2

uu

+=0;@x2@y2

ormoregenerally

r2u=0.Thisequation

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usuallydescribessteadyprocessesandissolvedsubjecttosomeboundaryconditions.

Oneaspectthatweshallconsideris:whydothe

similarlookingequationsdescribesessentiallydi erentphysicalprocesses?Whatisthereaboutthe

equationsthatmakethisthecases?

1.4.4OtherCommonSecondOrder

LinearPDEsPoisson'sequationisjusttheLapace'sequation(homogeneous)

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withaknownsourceterm

(e.g.electricpotentialinthepresenceofadensityofcharge):r2u=.

The

Helmholtzequationmayberegardedasastationarywaveequation:

r2

u+k2u=0.TheSchrodingerequationisthefundamentalequationof

physicsfordescribingquantummechanicalbehavior;Schrodingerwaveequationis

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aPDEthatdescribeshowthewavefunctionofaphysicalsystemevolvesovertime:

@u

..r2u+Vu=

i.

@t

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1.4.5NonlinearPDEsAnexampleofanonlinearequationistheequationforthepropagationofreaction-di usion

waves:@u@t=@2u@x2+u(1-u)(2nd

order),orfornonlinearwavepropagation:@u@t+(u+c)@u

@x=0;(1storder).Theequation

@u

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@u

x2u+(y+u)=u3

@x@y

isanexampleofquasilinearequation,and

@u@u

y+(x3+y)=u3

@x@y

isanexampleofsemilinearequation.

1.4.6SystemofPDEsMaxwellequationsconstituteasystem

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oflinearPDEs:d1@E

r·E=,r×B= j+2;

pc@t

@B

r·B=0,r×E=-.

@t

Inemptyspace(freeofchargesandcurrents)thissystem

canberearrangedtogivetheequationsofpropagationof

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theelectromagnetic

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eld,

@2E@2B

=c2r2E,=c2r2B.

@t2@t2

Incompressiblemagnetohydrodynamic(MHD)equationscombine

Navier-Stokesequation(includingtheLorentzforce),theinductionequationaswellasthe

solenoidalconstraints,

@U+U rU=..r +B rB+

r2U+F;

@t

@B=r×

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(U×B)+ r2B;

@t

r·U=0,r·B=0.

Bothsystemsinvolvespaceandtime;theyrequire

someinitialandboundaryconditionsfortheirsolution.

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1.5ExistenceandUniqueness1.5ExistenceandUniquenessBeforeattemptingtosolveaprobleminvolvingaPDEwewouldliketoknowif

asolutionexists,and,ifitexists,ifthesolutionisunique.Also,

inprobleminvolvingtime,whetherasolutionexists8t>0(globalexistence)or

onlyuptoagivenvalueoft i.e.

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onlyfor0<t<t0(

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nitetimeblow-up,shockformation).Aswellastheequationtherecouldbecertainboundaryandinitialconditions.Wewouldalsoliketoknow

whetherthesolutionoftheproblemdependscontinuouslyoftheprescribeddata

i.e.smallchangesinboundaryorinitialconditionsproduceonlysmallchangesin

thesolution.

IllustrationfromODEs:

1.

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du=u,u(0)=1.

dt

Solution:u=etexistsfor0.t<.

2.

du=u2;u(0)=1.

dt

Solution:

u=1=(1-t)existsfor0.t<1

3.

du==u,u(0)=0;

dt

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hastwosolutions:u.0andu=t2=4(nonuniqueness).

WesaythatthePDEwithboundaryorinitial

conditioniswell-formed(orwell-posed)ifitssolutionexists(globally),isuniqueand

dependscontinuouslyontheassigneddata.Ifanyofthesethreeproperties(existence,

uniquenessandstability)isnotsatis

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ed,theproblem(PDE,BCsandICs)issaidtobeill-posed.Usuallyproblemsinvolvinglinearsystemsarewell-formedbutthismaynotbe

alwaysthecasefornonlinearsystems(bifurcationofsolutions,etc.)

Example:

Asimpleexampleofshowinguniquenessisprovidedby:

r2u

=FinG(Poisson'sequation).withu=0on

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@,theboundaryof

,andFissomegivenfunctionofx.

Supposeu1andu2twosolutionssatisfying

theequationandtheboundaryconditions.Thenconsiderw=u1-u2;

[email protected]

theoremgives

wrw·ndS=r·

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(wrw)dV,

@.

=wr2w+(rw)2.dV

wherenisaunitnormaloutwardsfrom

.

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(rw)2dV=w@wdS=0.

@n

.@.

Now

theintegrand(rw)2isnon-negativeinGandhencefortheequalityto

holdwemusthaverw.0;i.e.w=constantin

.Sincew=0on@Gandthesolutionis

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smooth,wemusthavew.0in

;i.e.u1=u2.Thesameproofworksif@u=@nisgivenon@G

orformixedconditions.

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1.5ExistenceandUniqueness

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Chapter2

FirstOrderEquations

Contents

2.1LinearandSemilinearEquations...................92.2QuasilinearEquations..........................192.3WaveEquation..............................26

2.4SystemsofEquations..........................312.1LinearandSemilinearEquations2.1.1MethodofCharacteristic

Weconsiderlinear

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rstorderpartialdi erentialequationintwoindependentvariables:

@u@u

a(x,y)+b(x,y)+c(x,y)u=f(x,y),

(2.1)

@x@y

wherea,b,candfare

continuousinsomeregionoftheplaneandweassumethata(x,y)

andb(x,y)arenotzeroforthesame(x,

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rstorderequation(wherethenonlinearityispresentonlyintheright-handside)suchas

@u@u

a(x,y)+b(x,

y)= (x,y,u),(2.2)

@x@y

insteadofa

linearequationasthetheoryoftheformerdoesnotrequireanyspecial

treatmentascomparedtothatofthelatter.

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Thekeytothesolutionoftheequation(2.1)isto

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ndachangeofvariables(orachangeofcoordinates)

.

. (x,y);s. (x,y)

whichtransforms(2.1)

intothesimplerequation

@w

+h( , )w=

F( , )(2.3)@.wherew( , )=u(x( , );y( , )).

9

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2.1LinearandSemilinearEquationsWeshallde

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nethistransformationsothatitisone-to-one,atleastforall(x,y)insomesetDofpointsinthe(x-y)plane.

Then,onDwecan(intheory)solveforxandyas

functionsof , .Toensurethatwecandothis,werequire

thattheJacobianofthetransformationdoesnotvanishin

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D:

J=

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@ @x@ @y@ @x@ @y

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=@ @x@ @y..@ @y@ @x6=f0;1gfor(x,y)inD.Webeginlookingforasuitabletransformationbycomputingderivativesvia

thechainrule

@u@w@.@w@s@u@w@.

@w@s

=+and=+.

@x@.

@x@s@x@y@.@y@s@y

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Wesubstitutetheseintoequation(2.1)toobtain

@w

@.@w@s

@w

@.@w@s

a++b++cw=f.

@.@x

@s@x@.@y@s@y

Wecanrearrangethisas

@.@.

.@w

8/10/2019 m3414_1


@s@s

.@w

a+b+a+b+cw=f.(2.4)

@x@y@.

@x@y@s

Thisisclosetotheformofequation

(2.1)ifwecanchooses. (x,y)sothat

@s@sa@x+b@y=0for

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(x,y)inD.

Providedthat@ =@y6

=0wecanexpressthisrequiredpropertyofsas

@x b

=-.

@y a

Suppose

wecande

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neanewvariable(orcoordinate)swhichsatis

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esthisconstraint.Whatistheequationdescribingthecurvesofconstant ?Puttings. (x,y)=k(kanarbitraryconstant),then

@s@s

ds=dx+dy=0

@x

@y

impliesthatdy=dx=..@x =@ys=b=a.So,the

equation (x,y)=kde

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nessolutionsoftheODE

dyb(x,y)

=.(2.5)

dxa(x,y)

Equation(2.5)iscalledthe

characteristicequationofthelinearequation(2.1).Itssolutioncanbewrittenin

theformF(x,y, )=0(wheresistheconstant

ofintegration)andde

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nesafamilyofcurvesintheplanecalledcharacteristicsorcharacteristiccurvesof(2.1).(Moreoncharacteristicslater.)Characteristicsrepresentcurvesalongwhich

theindependentvariablesofthenewcoordinatesystem( , )isconstant.

So,wehavemadethecoe cientof@w=@svanishinthetransformedequation

(2.4),bychoosings. (x,y),with (x,y)=

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kanequationde

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Chapter2 FirstOrderEquations

Then

J=

10

@f@f

@x

@y

@s

=,

@y

andwehavealreadyassumedthison-zero.Nowweseefromequation(2.4)

thatthischangeofvariables,

.=x,

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rstwrite (x,y);c(x,y)andf(x,y)intermsof.andstoobtain@w

A( , )+C( , )w=

( , ).

@.Finally,restrictingthevariablestoasetin

whichA( , )6

=0wehave

@wC

d

+w=;@.AA

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whichisintheformof(2.3)withC( , ) ( , )

h( , )=andF( , )=.

A( , )A( , )

Thecharacteristicmethodappliesto

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=0.

Notation:Itisveryconvenienttousethefunctionuinplaceswhererigorouslythefunctionwshouldbe

used.E.g.,theequationhereabovecanidenticallybewrittenas@u=@.=

K=A.

Example:Considerthelinear

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rstorderequation

2@u@u

x+y+xyu=1.

@x@y

Thisis

equation(2.1)witha(x,y)=x2,b(x,y)=y,c(x,y)=xy

andf(x,y)=1.Thecharacteristicequationis

dyby

==:

2

dxax

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2.1LinearandSemilinearEquationsSolvethisbyseparationofvariables

111

dy=2dxvln

y+=k,fory>0,andx=06.

yxx

Thisisanintegralofthecharacteristicequationdescribingcurvesof

constantsandsowechoose

1

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s. (x,y)=lny+.

xGraphsoflny+1=xarethecharacteristicsofthisPDE.

Choosing.=xwehavetheJacobian

@s1

J==6asrequired:

=0

@yySince.

=x,

1

..1=

s

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=lny+vy=e.

Nowweapplythetransformation.=x,s=lny+1=xwith

w( , )=u(x,y)andwehave

@u@w@.@w

@s@w@w1@w1@w

=+=+ -=

-;

@x@.@x@s@x@.@s

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x2@. 2@s@u@w@.@w@s@w11@w

=+=0+·=.

..1=

@y@.@y@s@y@sye@s

ThenthePDE

becomes @[email protected]@w

..1=

2-

+e+ e ..1=.w=1;

2 ..1=

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@.@se@s

whichsimpli

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esto

2@w@w11 ..1=

+ e ..1=.w=1thento+ew=.@.@. 2

Wehavetransformedtheequationintotheformofequation(2.3),forany

regionof( , )spacewith.6

=0.

2.1.2

EquivalentsetofODEsThepointofthistransformationis

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thatwecansolveequation(2.3).Thinkof

@w

+h( , )w=F( , )@.

as

alinear

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rstorderordinarydi erentialequationin ,withscarriedalongasaparameter.Thusweuseanintegratingfactormethod

R

RR

h( ; )d.@wh( ; )d h( ; )d.

e+h( , )ew

=F( , )e;

@.

RR

.

h( ; )d h( ; )d.

ew=F( , )e.

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@.

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Nowweintegratewithrespectto .Sincesisbeingcarriedasaparameter,

theconstantofintegrationmaydependons

R.R

h( ; )d h( ; )d.d.+g( )

ew=F( , )e

inwhichgisanarbitrarydi erentiablefunction

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ofonevariable.Nowthegeneralsolutionofthetransformedequationis

R.RR

-h( ; )d h( ; )d.d.+g( )

e-h( ; )d.

w( , )=eF( , )e.

Weobtainthegeneralformoftheoriginalequationbysubstitutingback (x,

y)and (x,y)toget

u(x,y)=

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e (x;y)[

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(x,y)+g( (x,y))].(2.6)

Acertainclassof

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rstorderPDEs(linearandsemilinearPDEs)canthenbereducedtoasetofODEs.Thismakesuseofthegeneralphilosophythat

ODEsareeasiertosolvethanPDEs.

Example:Considertheconstant

coe cientequation

@u@u

a+b+cu

=0

@x@ywherea,b,c=

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R.Assumea=0,thecharacteristicequationis6dy=dx=b=awithgeneralsolutionde

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nedbytheequationbx-ay=k,kconstant.SothecharacteristicsofthePDEarethestraightlinegraphsofbx

-ay=kandwemakethetransformationwith

.

=x,s=bx-ay.Usingthesubstitutionwe

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ndtheequationtransformsto@wc

[email protected]

Theintegratingfactormethodgives

.

c =a

ew=0

@.andintegratingwithrespect

to.gives

c =a

ew=g( ),wheregis

anydi erentiablefunctionofonevariable.Then

..c =a

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w=g( )eandintermsofxandywebacktransform

..cx=a

u(x,y)=g(bx-

ay)e.

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2.1LinearandSemilinearEquationsExercise:VerifythesolutionbysubstitutingbackintothePDE.

Note:Considerthedi erencebetweengeneral

solutionforlinearODEsandgeneralsolutionforlinearPDEs.ForODEs,the

generalsolutionof

dy

+q(x)y=p(x)dx

containsanarbitraryconstantofintegration.Fordi erent

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constantsyougetdi erentcurvesofsolutionin(x-y)-plane.Topickoutauniquesolutionyouusesomeinitialcondition(sayy(x0)=y0)to

specifytheconstant.ForPDEs,ifuisthegeneralsolutiontoequation

(2.1),thenz=u(x,y)de

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nesafamilyofintegralsurfacesin3D-space,eachsurfacecorrespondingtoachoiceofarbitraryfunctiongin(2.6).Weneedsomekind

ofinformationtopickoutauniquesolution;i.e.,tochosethearbitrary

functiong.

2.1.3CharacteristicCurvesWeinvestigatethesigni

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canceofcharacteristicswhich,de

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nedbytheODE

dyb(x,y)

=;

dxa(x,y)

representaoneparameterfamily

ofcurveswhosetangentateachpointisinthedirectionofthe

vectore=(a,b).(Notethattheleft-handsideofequation(2.2)is

thederivationofuinthedirectionofthevector

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e,e ru.)Theirparametricrepresentationis(x=x(s);y=y(s))wherex(s)andy(s)satisfythepairofODEs

dx

dy

=a(x,y),=b(x,y).(2.7)

dsds

Thevariationofuwithrespectx=.alongthesecharacteristiccurves

isgivenby

du@udy@u@ub@u

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=+=+;

dx@xdx@y@xa@y (x,y,u)

=fromequation(2.2),

a(x,

y)

suchthat,intermofthecurvilinearcoordinates,the

variationofualongthecurvesbecomes

dududx

== (x,y,u).

dsdxds

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Theoneparameterfamilyofcharacteristiccurvesisparameterisedbys(eachvalueofsrepresentsoneuniquecharacteristic).Thesolutionof

equation(2.2)reducestothesolution

ofthefamilyofODEs

duds= (x,y,u).orsimilarlydudx=du

d.= (x,y,u)a(x,y).(2.8)

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alongeachcharacteristics(i.e.foreachvalueof ).Characteristicequations(2.7)havetobesolvedtogetherwithequation(2.8),calledthecompatibility

equation,to

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ndasolutiontosemilinearequation(2.2).

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CauchyProblem:Consideracurve.in(x,y)-planewhoseparametricformis(x=

x0( );y=y0( )).TheCauchyproblemistodetermineasolutionofthe

equation

F(x,y,u,@xu,@yu)=0

in

aneighbourhoodof.suchthatutakesprescribedvalues

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u0( )calledCauchydataon...

=csthu o(1)u o(2)(x , y )oo=cstxxyGNotes:

1.ucan

onlybefoundintheregionbetweenthecharacteristicsdrawnthroughtheendpoint

of...2.Characteristicsarecurvesonwhichthevaluesofucombined

withtheequationarenotsu cienttodeterminethenormal

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derivativeofu.3.Adiscontinuityintheinitialdatapropagatesontothesolutionalongthecharacteristics.Thesearecurvesacrosswhichthederivatives

ofucanjumpwhileuitselfremainscontinuous.Existence&Uniqueness:Why

dosomechoicesof.in(x,y)-spacegiveasolutionandother

givenosolutionoranin

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nitenumberofsolutions?ItisduetothefactthattheCauchydata(initialconditions)maybeprescribedonacurve.which

isacharacteristicofthePDE.Tounderstandthede

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nitionofcharacteristicsinthecontextofexistenceanduniquenessofsolution,returntothegeneralsolution(2.6)ofthelinearPDE:

u(x,y)=e (x;y)[

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(x,y)+g( (x,y))].

ConsidertheCauchydata,u0,prescribedalongthecurve.whoseparametricformis(x=x0( );y

=y0( ))andsupposeu0(x0( );y0( ))=q( ).If.isnotacharacteristic,

theproblemiswell-posedandthereisauniquefunctiongwhichsatis

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esthecondition

q( )=e (x0( );y0( ))[

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(x0( );y0( ))+g(x0( );y0( ))].

Ifontheotherhand(x=x0( );y=y0( ))istheparametrisationofacharacteristic( (x,y)=

k,say),therelationbetweentheinitialconditionsqandgbecomes

q( )=e (x0( );y0( ))[

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(x0( );y0( ))+G],(2.9)

whereG=g(k)isaconstant;theproblemisill-posed.Thefunctions (x,y)and

8/10/2019 m3414_1


(x,y)aredeterminedbythePDE,soequation(2.9)placesaconstraintonthegivendatafunctionq(x).Ifq( )isnotof

thisformforanyconstantG,thenthereisnosolutiontakingon

theseprescribedvalueson...Ontheotherhand,ifq( )isof

thisformforsomeG,thentherearein

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nitelymanysuchsolutions,becausewecanchooseforganydi erentiablefunctionsothatg(k)=G.

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2yand.=x.

xy=cstx=csth=cstx(Wecanseethatansand.crossonlyoncethey

areindependent,i.e.J6

=0;sand.havebeen

properlychosen.)

Thisgivesthesolutionu(x,y)=e..4xg(3x

-2y)

wheregisadi erentiablefunction

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de

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nedovertherealline.Simplyspecifyingthesolutionatagivenpoint(asinODEs)doesnotuniquelydetermineg;weneedto

takeacurveofinitialconditions.Supposewespecifyvaluesofu(x,y)

alongacurve.intheplane.Forexample,let'schoose.as

thex-axisandgivesvaluesofu(x,y)atpoints

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on..,say

u(x,0)=sin(x).

Thenweneed

..4x4x

u(x,0)=eg(3x)

=sin(x)i.e.g(3x)=sin(x)e,

andputtingt=3x,g(t)

=sin(t=3)e4t=3.

Thisdeterminesgandthesolutionsatisfying

theconditionu(x,0)=sin(x)on.is

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edalongthex-axis.Wedonothavetochooseanaxis say,alongx=y,u(x,y)=u(x,x)=x4.

Fromthegeneralsolutionthisrequires,

..4x444x

u(x,x)=eg(x)=x,sog(x)=xe

togivetheunique

solution

8(x..y)

u(x,y)=(3x-

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2y)4e

satisfyingu(x,x)=x4.

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However,noteverycurveintheplanecanbeusedtodetermineg.Supposewe

choose.tobetheline3x-2y=1andprescribe

valuesofualongthisline,say

u(x,y)=u(x,(3x

-1)=2)=x2.

Nowwe

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mustchoosegsothat

..4x2

eg(3x-(3x-1))=x.

Thisrequiresg(1)=

x2e4x(forallx).Thisisimpossibleandhencethereisno

solutiontakingthevaluex2atpoints(x,y)onthisline.Last,

weconsideragain.tobetheline3x-

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2y=1butchoosevaluesofualongthislinetobe

u(x,y)=u(x,(3x-1)=2)=e..4x

.

Nowwemustchoosegsothat

..4x

..4x

eg(3x-(3x-1))=e.

This

requiresg(1)=1,conditionsatis

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edbyanin

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nitenumberoffunctionsandhencethereisanin

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nitenumberofsolutionstakingthevaluese..4xontheline3x-2y=1.Dependingontheinitialconditions,thePDEhas

oneuniquesolution,nosolutionatalloranin

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nitenumberorsolutions.Thedi erenceisthatthex-axisandtheliney=xarenotthecharacteristicsofthePDEwhile

theline3x-2y=1isacharacteristic.

Example

2:

x@u@x-y@u@y=uwith

u=x2ony=x,1.

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y.2Characteristics:dydx=-yxvd(xy)=0vxy=c,constant.

So,takes

=xyand.=x.Thentheequationbecomes@w@w@w

@w@w

xy+x-xy=wv.

-w=0v=0.

@s@.@s

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@[email protected],w= g( )orequivalentlyu(x,y)=xg(xy).Wheny=x

22

with1.y.2,u=x;sox=xg(x2)v

g(x)==xandthesolutionis=

u(x,y)=x

xy.

=consth=constxGThis

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2.1LinearandSemilinearEquationsAlternativeapproachtosolvingexample2:

@u@u

x-y=uwith

u=x2ony=x,1.y.2

@x@y

Thismethodisnotsuitablefor

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ndinggeneralsolutionsbutitworksforCauchyproblems.Theideaistointegratedirectlythecharacteristicandcompatibilityequationsincurvilinearcoordinates.(See

also\alternativemethodforsolvingthecharacteristicequations forquasilinearequationshereafter.)The

solutionofthecharacteristicequations

dxdy

=x

and=..y

dsdsgivestheparametricform

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ofthecharacteristiccurves,whiletheintegrationofthecompatibilityequation

du

=u

dsgivesthesolution

u(s)alongthesecharacteristicscurves.Thesolutionofthecharacteristicequationsis

x=c1esandy=c2e..s,

wheretheparametricformofthedatacurve

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.permitsusto

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ndthetwoconstantsofintegrationc1&c2intermsofthecurvilinearcoordinatealong...Thecurve.isdescribedby

x0( )=ßandy0( )=ßwithß=[2,1]

andweconsiderthepointson.tobetheoriginofthe

coordinatesalongthecharacteristics

(i.e.s=

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0on..).So,on.(s=0)

v

s

=ß=c1

x(s, )=

ße

x0

,8ß=[0,1].

..s

=ß=c2

y(s, )=ße

y0

Forlinearorsemilinearproblems

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wecansolvethecompatibilityequationindependentlyofthecharacteristicequations.(Thispropertyisnottrueforquasilinearequations.)Alongthecharacteristicsuis

determinedby

du

s

=uv

u=c3e.

dsNowwecanmakeuse

oftheCauchydatatodeterminetheconstantofintegration

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c3,on..,ats=0,u0(x0( );y0( )).u0( )= 2=c3.Then,wehavetheparametricformsofthecharacteristiccurvesand

thesolution

s..ss

x(s, )=ße;y(s,

)=ßeandu(s, )= 2e,

interms

oftwoparameters,sthecurvilinearcoordinatealongthecharacteristic

8/10/2019 m3414_1


curvesandßthecurvilinearcoordinatealongthedatacurve...Fromthetwo

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rstoneswegetsandßintermsofxandy.

=

x

x

2s

xy= 2vß=

vs

=ln

and

xy

(ß~0).

=e

yyThen,wesubstitute

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sandßinu(s, )to

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nd

=

x

x

u(x,y)=xyexpln

=xy

=xxy.

yy

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2.2QuasilinearEquationsConsiderthe

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z=u(x,y),in(x,y,z)-space.De

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netheMongedirectionbythevector(a,b,c)andrecallthatthenormaltotheintegralsurfaceis(@xu,@yu,..1).

Thusquasilinearequation(2.10)saysthatthenormaltotheintegralsurfaceis

perpendiculartotheMongedirection;i.e.integralsurfacesaresurfacesthatateach

pointaretangenttotheMongedirection,

01.

8/10/2019 m3414_1


.

a@xu

@u@u

.b.·.@yu.=a(x,y,u)+b(x,y,

u)-c(x,y,u)=0.

@x@y

c..1

Withthe

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eldofMongedirection,withdirectionnumbers(a,b,c),wecanassociatethefamilyofMongecurveswhichateachpointaretangent

tothatdirection

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eld.Thesearede

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nedby

.101.1

dxacdy-bdz

dxdydz

.dy.

×.b.=.adz-cdx.=0

8==(=ds),

a(x,y,u)b(x,y,u)c(x,

y,u)

dzcbdx-ady

8/10/2019 m3414_1


wheredl=(dx,dy,dz)isanarbitraryin

8/10/2019 m3414_1


nitesimalvectorparalleltotheMongedirection.Inthelinearcase,characteristicswerecurvesinthe(x,y)-plane(see§2.1.3).Forthequasilinear

equation,weconsiderMongecurvesin(x,y,u)-spacede

8/10/2019 m3414_1


nedby

dx

=a(x,y,u);

dsdy

=b(x,y,u);

dsdu

=c(x,y,u).

ds

Characteristicequations(dfx,

yg=ds)andcompatibilityequation(du=ds)aresimultaneous

8/10/2019 m3414_1


rstorderODEsintermsofadummyvariables(curvilinearcoordinatealongthecharacteristics);wecannotsolvethecharacteristicequationsandcompatibilityequation

independentlyasitisforasemilinearequation.Notethat,incaseswhere

c.0,thesolutionremainsconstantonthecharacteristics.Theroughidea

insolvingthePDEisthustobuildupthe

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integralsurfacefromtheMongecurves,obtainedbysolutionoftheODEs.

Notethatwemakethedi erencebetweenMongecurveor

directionin(x,y,z)-spaceandcharacteristiccurveordirection,theirprojectionsin

(x,y)-space.

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2.2QuasilinearEquations2.2.2Generalsolution:Supposethatthecharacteristicandcompatibilityequationsthatwehavede

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nedhavetwoindependent

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rstintegrals(function,f(x,y,u),constantalongtheMongecurves)

(x,y,u)=c1and(x,y,u)=c2.

Thenthesolutionofequation(2.10)satis

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esF( ,)=0forsomearbitraryfunctionF(equivalently,.=G( )forsomearbitraryG),wheretheformofF

(orG)dependsontheinitialconditions.

Proof:Since.and

.are

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rstintegralsoftheequation, (x,y,u)= (x(s);y(s);u(s)),= (s)=c1.Wehavethechainrule

[email protected]@.dy

@.du

=++=0;

ds@xds@yds

@uds

andthenfromthecharacteristicequations

@.

@.@.

a+b+c=0.

8/10/2019 m3414_1


@x@y@uAndsimilarlyfor.

@.@.@.

a+b+c=0.

@x@y@u

Solvingforcgives

@.

@.@.@.

@.

@.@.@.

a-+b-=0

@x

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@u@x@u@y@u@y@u

oraJ[u,x]=bJ[y,u]whereJ[x1;x2]=

@p@p@x1@x2@.@.

@x1@x2

.

Andsimilarly,solvingfora,bJ[x,

y]=cJ[u,x].

Thus,wehaveJ[u,x]=J[x,y]b=c

andJ[y,u]=J[u,x]a=b=J[x,y]a=c.

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NowconsiderF( ,)=0 rememberF( (x,y,u(x,y)); (x,y,u(x,y))) anddi erentiate

@F@F

dF=dx+dy=0

@x@yThen,thederivative

withrespecttoxiszero,

@F@F

@.

@.@u

@F

@.

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@.@u

=+++=0;

@x@.@x@u@x@.@x@u@xaswellasthederivative

withrespecttoy

@F@F

@.

@.@u

@F

@.

@.@u

=+++=0.

@y@.@y@u

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@y@.@y@u@y

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Foranon-trivialsolutionofthiswemusthave

@.. @.

@.@.@u@.@u @.@.@u@.@u

++-++=0;

@x@u@x@y@u@y@x@u

@x@y@u@y @.@.@.@..@u

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@.@.@.@..@u@.@.@.@.

)-+-=-;

@y@u@y@u@x

@u@x@u@x@y@x@y@x@y@u@u

vJ[y,u]+J[u,x]=J[x,y].

@x@yThenfrom

thepreviousexpressionsfora,b,andc@u@u

8/10/2019 m3414_1


a+b=c;

@[email protected].,F( ,)=0de

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nesasolutionoftheoriginalequation.

Example1:

@u@u

(y+u)+y=x-

yiny>0,...<x<1;

@x@ywithu=1+

xony=1.

We

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rstlookforthegeneralsolutionofthePDEbeforeapplyingtheinitialconditions.Combiningthecharacteristicandcompatibilityequations,

dx

=y+u,(2.11)

ds

dy

=y,(2.12)

ds

du

=x-

y(2.13)

dsweseektwoindependent

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rstintegrals.Equations(2.11)and(2.13)gived

(x+u)=x+u;

ds

andequation(2.12)1dy

=1.

ydsNow,considerd.x+

u.1dx+udy

=(x+u)-

2;

dsyydsydsx+ux+

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u

=-=0.

yy

So,(x+u)=y=c1isconstant.Thisde

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nesafamilyofsolutionsofthePDE;so,wecanchoose

x+u

(x,y,u)=,

y

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2.2QuasilinearEquationssuchthat.=c1determinesoneparticularfamilyofsolutions.Also,equations(2.11)and(2.12)gived

(x-y)=u;

ds

andequation(2.13)ddu

(x-y)(x-y)=u.

dsdsNow,consider

ddd

2 .2

(x-

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y)2-u=(x-y)2.-u;

dsdsdsddu

=2(x-y)(x-y)-2u=0.

dsds

2

Then,(x-y)2-

u=c2isconstantandde

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nesanotherfamilyofsolutionsofthePDE.So,wecantake

2

(x,y,u)=(x-y)2-u.

ThegeneralsolutionisF.x+u,(x-

y)2-u2.=0or(x-y)2-u

2=G.x+u.,y

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y

forsomearbitraryfunctionsForG.Nowtogetaparticularsolution,applyinitialconditions(u=1+xwheny

=1)

(x-1)2-(x+1)2=G(2x

+1)vG(2x+1)=..4x.Substituteß=2x+1,i.e.

x=(ß-1)=2,soG( )=2(1- ).

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Hence,

x+u2

(x-y)2-u2=21-=(y-x-u).

yyWecanregardthisasaquadraticequationforu:2

.x-y.

u2-u-2

+(x-y)2=0,

yy

2

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211

u2-u-x-y++=0:

2

yyy

Then,

.

2

111111

u±=±

2+x-y+-2=±x-y

+.

yyyyyyConsideragaintheinitialcondition

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u=1+xony=1u (x,y=1)=1±(x-1+1)=1±xvtakethepositive

root.Hence,

2

u(x,y)=x-y+

.

y

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Example2:usingthesameproceduresolve

@u@u

x(y-

u)+y(x+u)=(x+y)uwithu=x2+1

ony=x.

@x@y

Characteristicequationsdx

=x(y-u),(2.14)

dsdy

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=y(x+u),(2.15)

dsdu

=(x+y)u.(2.16)

ds

Again,weseekto

independent

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rstintegrals.Ontheonehand,equations(2.14)and(2.15)give

dxdy2

y+x=xy2

-xyu+yx+xyu=xy(x+y);

ds

ds1du

=xyfromequation(2.16).

uds

Now,consider1dx1dy1dud xy.

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+=vln=0.xdsydsudsdsuHence,xy=u=c1isconstantand

xy

(x,y,

u)=

u

isa

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rstintegralofthePDE.Ontheotherhand,dxdydu

-=xy-xu-xy-yu=

..u(x+y)=-;

dsdsdsd

v

(x+u-y)=0.

dsHence,x+u-

y=c2isalsoaconstantontheMonge

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curvesandanother

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rstintegralisgivenby

(x,y,u)=x+u-y,

sothegeneralsolutionis

xy

=G(x+u-y).

uNow,

wemakeuseoftheinitialconditions,u=x2+1ony

=x,todetermineG:

2

x

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2=G(x2+1);

1+x

2

setß=x2+1,i.e.x=ß-

1,thenß-1

G( )=;

ß

and

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nallythesolutionisxyx+u-y-1

=.Rearrangeto

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nish!

ux+u-y

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2.2QuasilinearEquationsAlternativeapproach:Solvingthecharacteristicequations.Illustrationbyanexample,

2@u@u

x+u

=1,withu=0onx+y=1.

@x@y

Thecharacteristicequationsaredx2dydu

=

x,=uand=1;

dsdsdswhich

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wecansolvetoget1

x=,(2.17)

c1-s2

s

y=+c2

s+c3,(2.18)

2u=c2+s,forconstants

c1;c2;c3.(2.19)Wenowparameterisetheinitiallineintermsof :ß

=x,y=1- ,andapplytheinitial

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dataats=0.Hence,11

(2.17)givesß=vc1=c1 ,

(2.18)gives1

-ß=c3vc3=1- ,(2.19)gives0=

c2vc2=0.Hence,wefoundtheparametricformofthesurface

integral,2

s

x=;y=+1

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-ßandu=s.

1-sß2

Eliminatesand , x

x=v

ß=;

1-sß1+sxthen

2

ux

y=+1-.

21+

uxInvariants,or

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rstintegrals,are(fromsolution(2.17),(2.18)and(2.19),keepingarbitraryc2=0).=u2=2-yand.=x=(1+

ux).

Alternativeapproachtoexample1:

@u@u

(y+u)+y=x-yiny>0,...

<x<1;

@x@ywithu=1+xon

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y=1.Characteristicequationsdx

=y+u,(2.20)

dsdy

=y,(2.21)

dsdu

=x-y.(2.22)

ds

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s

=c1e+c2es-x;

ds11

..ss

..ss

so,x

=c3e+

(c1+c2)eandu

=..c3e+(c2-c1)e.22Now,ats=

0,y=1andx= ,u=1+

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ß(parameterisinginitialline..),

c1=1;c2=1+2ßandc3=..1.

Hence,theparametricformofthesurfaceintegral

is,

ss..s

x=-e..s+

(1+ )e,y=eandu=e+ es

.

Theneliminateßands:

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111

x=-+(1+ )yvß=x-y+,

yyy

so

111

u=+x-

y+y.

yyy

Finally,

2

u=x-y+,as

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before.

y

To

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ndinvariants,returntosolvedcharacteristicsequationsandsolveforconstantsinterms

s

ofx,yandu.Weonly

needtwo,soputforinstancec1=1andsoy=

e.Then,

c31c31

x=+

(1+c2)yandu=-

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+

(c2-1)y.

y2y2

Solveforc2

x+ux+

u

c2=,so.=,

yy

andsolveforc3

1

c3=

(x-u-y)y,so.=(x-

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u-y)y.2Observe.isdi erentfromlasttime,butthisdoesnotasweonlyrequiretwoindependentchoicesfor.and

.Infactwecanshowthatourprevious.isalsoconstant,

(x-y)2-u2=(x-y+u)(x

-y-u),.

=( y-y)

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,

y=(.-1).whichisalsoconstant.

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2.3WaveEquationSummary:Solvingthecharacteristicequations twoapproaches.

1.Manipulatetheequationstogetthemina'directly

integrable form,e.g.1d(x+u)=1

x+uds

and

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ndsomecombinationofthevariableswhichdi erentiatestozero(

8/10/2019 m3414_1


rstintegral),

e.g.dx+u

=0.

dsy

2.Solvetheequationswithrespectto

thedummyvariables,andapplytheinitialdata(parameterisedby )at

s=0.Eliminateßands;

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ndinvariantsbysolvingforconstants.2.3WaveEquationWeconsidertheequation

@u@u

+(u+c)=0

withu(0;x)=f(x);@t@xwherecissomepositiveconstant.

2.3.1LinearWavesIfuissmall(i.e.u2.u),thenthe

equationapproximatetothelinearwaveequation@u@u

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+c=0withu(x,0)=f(x).@t@x

Thesolutionoftheequationofcharacteristics,dx=dt=c,

givesthe

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rstintegralofthePDE, (x,t)=x-ct,andthengeneralsolutionu(x,t)=g(x-ct),wherethefunctiongis

determinedbytheinitialconditions.Applyingu(x,0)=f(x)we

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ndthatthelinearwaveequationhasthesolutionu(x,t)=f(x-ct),whichrepresentsawave(unchangingshape)propagatingwithconstantwave

speedc.

h=x-ct=csth=0xGtxu(x,t)t=0 t=t1f(x)x =ct11

cNotethatuisconstantwherex-ct=constant,i.e.on

thecharacteristics.

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2.3.2NonlinearWavesForthenonlinearequation,

@u@u

+(u+

c)=0;@t@x

thecharacteristicsarede

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nedbydtdxdu

=1,=c+uand=0;

dsdsds

whichwecansolve

togivetwoindependent

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rstintegrals.=uand.=x-(u+c)t.So,

u=f[x-(u+c)t],

accordingtoinitialconditionsu(x,0)=f(x).Thisissimilarto

thepreviousresult,butnowthe\wavespeed involvesu.However,thisform

ofthesolutionisnotveryhelpful;itismore

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instructivetoconsiderthecharacteristiccurves.(ThePDEishomogeneous,sothesolutionuisconstantalongtheMongecurves thisisnot

thecaseingeneral whichcanthenbereducedtotheirprojections

inthe(x,t)-plane.)Byde

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nition,.=x..(c+u)tisconstantonthecharacteristics(aswellasu);di erentiate.to

8/10/2019 m3414_1


ndthatthecharacteristicsaredescribedby

dx

=u+c.

dt

Thesearestraight

lines,x=(f( )+c)t+ ,

expressedintermsofa

parameter .(Ifwemakeuseoftheparametricformofthedata

curve

..:fx= ,t=0;ß=

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R}andsolvedirectlytheCauchyproblemintermsofthecoordinates=t,wesimilarly

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nd,u=f( )andx=(u+c)t+ .)Theslopeofthecharacteristics,1=(c+u),variesfromonelineto

another,andso,twocurvescanintersect.t minxtG q f( )<0qq q q{x= ,t=0}u=f( ) & x-(f( )+c)t=

Considertwocrossingcharacteristicsexpressedintermsof 1and 2,

i.e.x=(f( 1)+c)t+ 1,x=(f( 2)+c)t+

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2.(Thesecorrespondtoinitialvaluesgivenatx= 1andx= 2.)Thesecharacteristicsintersectatthetime

1

- 2

t=-;

f( 1)-f( 2)

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2.3WaveEquationandifthisispositiveitwillbeintheregionofsolution.Atthispointuwillnotbe

single-valuedandthesolutionbreaksdown.Byletting 2. 1we

canseethatthecharacteristicsintersectat

1

t

=-;

f0( )

andtheminimumtime

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forwhichthesolutionbecomesmulti-valuedis

1

tmin=;

max[..f0( )]

i.e.thesolutionissingle

valued(i.e.isphysical)onlyfor0.t<tmin.Hence,whenf0( )

<0wecanexpectthesolutiontoexistonlyfora

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nitetime.Inphysicalterms,theequationconsideredispurelyadvective;inrealwaves,suchasshockwavesingases,whenverylargegradients

areformedthendi usiveterms(e.g.@xxu)becomevitallyimportant.xu(x,t)t

multi-valuedf(x)breakingToillustrate

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nitetimesolutionsofthenonlinearwaveequation,considerf( )= (1- ),(0.ß.1),f0( )=1-2 .So,f0( )<

0for1=2< <1andwecanexpectthesolutionnotto

remainsingle-valuedforallvaluesoft.(max[..f0( )]=1sotmin=1.

Now,u=f(x-(u+c)t),sou

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=[x-(u+c)t]×[1-x+(u+c)t],(ct.x.1+ct),whichwecanexpressas

22

tu2+(1+t-2xt+2ct2)u+(x

2-x-2ctx+ct+ct2)=0,andsolvingforu

(wetakethepositiverootfrominitialdata)

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1.p.

u=2t(x-ct)-(1+t)+(1+t)2-4t(x-ct).

2t2

Now,att=1,

=u=x-

(c+1)+1+c-x,

sothesolutionbecomessingular

ast.1andx.1+c.

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2.3.3WeakSolutionWhenwavebreakingoccurs(multi-valuedsolutions)wemustre-thinktheassumptionsin

ourmodel.Consideragainthenonlinearwaveequation,

@u@u

+(u+c)=0;@t@xandputw(x,t)=u(x,t)+

c;hencethePDEbecomestheinviscidBurger'sequation@w

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@w

+w=0;@t@x

orequivalentlyinaconservativeform

.2

@w@w

+=0;@t@x2wherew2=2istheuxfunction.We

nowconsideritsintegralform,

.x2.2 ..

x2.x2.2

@w@wd@w

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+dx=08w(x,t)dx=-dx@t@x2dt@x2

x1x1x1

wherex2>x1arereal.

Then,d.x2w2(x1;t)w2(x2;t)

w(x,t)dx=-.

dtx122

Letusnowrelaxtheassumption

regardingthedi erentiabilityoftheoursolution;supposethatw

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hasadiscontinuityinx=s(t)withx1<s(t)<x2.

xw(x,t)w(s ,t)w(s ,t)2x1 xs(t)+-Thus,splittingtheinterval

[x1;x2]intwoparts,wehave

.s(t).x2

w2(x1;t)w2(x2;t)dd

-=w(x,t)dx+w(x,t)dx;

22dtdt

x1s(t)

.s(t)

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.x2

@w@w

=w(s..;t)_s(t)+dx-w(s+;t)_s(t)+dx;

@t@t

x1s(t)

wherew(s..(t);t)andw(s+(t);t)arethevaluesofwasx.s

frombelowandaboverespectively;

_s=ds=dt.

Now,takethelimit

x1.s..(t)andx2.s+(t).Since@w=@tis

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bounded,thetwointegralstendtozero.Wethenhave

..+

w2(s;t)w2(s;t)...+

-=s_w(s;t)-

w(s;t).

22

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2.3WaveEquationThevelocityofthediscontinuityoftheshockvelocityU=s_.If[]indicatesthejumpacrossthe

shockthenthisconditionmaybewrittenintheform

.

2.

w

..U[w]=.

2

TheshockvelocityforBurger'sequationis

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1w2(s+)-w2(s..)w(s+)+w(s..)

U=

=.

2w(s+)-w(s..)2

Theproblem

thenreducesto

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ttingshockdiscontinuitiesintothesolutioninsuchawaythatthejumpconditionissatis

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edandmulti-valuedsolutionareavoided.Asolutionthatsatis

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estheoriginalequationinregionsandwhichsatis

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estheintegralformoftheequationiscalledaweaksolutionorgeneralisedsolution.

Example:ConsidertheinviscidBurger'sequation

@w@w

+w=0;@t@x

with

initialconditions

.

1forß.0;

.

w(x= ,t=0)=

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f( )=1-ßfor0.ß.1,

.

0forß~1.

Asseen

before,thecharacteristicsarecurvesonwhichw=f( )aswellas

x-f( )t=ßareconstant,whereßistheparameter

oftheparametricformofthecurveofinitialdata,

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...Forallß=(0,1),f0( )=..1isnegative(f==0elsewhere),sowecanexpectthatallthe

characteristicscorrespondingtothesevaluesofßintersectatthesamepoint;the

solutionoftheinviscidBurger'sequationbecomesmulti-valuedatthetime

tmin=1/max[..f0( )]=1,8ß=(0,1).

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Then,thepositionwherethesingularitydevelopsatt=1is

x=..f( )t+ß=1-ß

+ß=1.

t

w(x,t) 1t=0 t=1w=1 & x-t=cstw=1 & x-t=0w=0 & x=cst

x=1 xweak solutionw(x,t) (shock wave)

1

multi-valued

solution

U

t=0

t>1x

x

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Astimeincreases,theslopeofthesolution,

.>.>.

1forx.t,1-x

w(x,t)=

fort.x.1,with0.t<1,

1-t0for

x~1,

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becomessteeperandsteeperuntilitbecomesverticalatt=1;thenthesolutionismultivalued.Nevertheless,wecande

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neageneralisedsolution,validforallpositivetime,byintroductingashockwave.Supposeshockats(t)=Ut+ ,withw(s..;t)=1and

w(s+;t)=0.Thejumpconditiongivestheshockvelocity,

w(s+)+

w(s..)1

U==

;22furthermore,theshockstarts

atx=1;t=1,so.=1-1=2=1=2.

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Hence,theweaksolutionoftheproblemis,fort~1,

0forx<s(t),1

w(x,t)=

wheres(t)=

(t+1).

1forx>s(t),

2

2.4SystemsofEquations2.4.1LinearandSemilinear

EquationsTheseareequationsoftheform

@u

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n

(j)(j)x+biju

=ci,

i=1,2;:::;n

aiju

=

ux;

y

@x

j=1

(1)(j)

fortheunknownsu;u;:::;u(n)andwhenthecoe cientsaijandbijare

functionsonlyofxandy.(Thoughthecicould

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alsoinvolveu(k).)Inmatrixnotation

Aux+Buy=c,

where

2

3

.

a11::.a1n

.

b11::.b1n

6.

..

.

...

.

.

.

7.

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,

B=(bij)=

6.

..

.

...

.

.

.

7.

A=(aij)=

,

an1::.annbn1::.bnn

2

3.

c1

.

(1)

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u

(2)666.

c2

.

.

.

777.

and

u=

666.

777.

u

c=

.

.

.

.

(n)cnu

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E.g.,

(1)(2)(1)(2)u-2u+3u-u

=x+y,

x

x

y

y

(1)(2)(1)(2)22u+u-

5u+2u=x+y;

xxyy

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2.4SystemsofEquationscanbeexpressedas

.(1)#. .(1)#.

1..2ux3..1uyx

+y

+=;(2)(2)2211x..52yx+

y

uu

orAux+Buy=cwhere

1..23..1x+y

A=;B

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=andc=22.

11..52x+y

1=32=3

IfwemultiplybyA..1=

..1=31=3

A..1Aux+A..1Buy=A..1c,weobtain

ux+Duy=d,

. . .

1=32=33

..17=31

whereD=A..1B==and

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d=A..1c.

..1=31=3

..52

..8=31WenowassumethatthematrixAisnon-singular(i.e.,theinverseA..1exists)

atleastthereissomeregionofthe(x,y)-planewhereitis

non-singular.Hence,weneedonlytoconsidersystemsoftheform

ux+Duy=d.Wealsolimitourattention

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tototallyhyperbolicsystems,i.e.systemswherethematrixDhasndistinctrealeigenvalues(oratleastthereissomeregionofthe

planewherethisholds).Dhasthendistincteigenvalues 1; 2;:::; nwheredet( iI

-D)=0(i=1;:::;n),with i66sothat

= j(i=

j)andthencorrespondingeigenvectorse1,e2;:::,en

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Dei= iei.ThematrixP=[e1,e2;:::,en]diagonalisesDviaP..1DP= ,

23

10::.

::.0.0 20::.0.6.

.

...:

=.0::..::.0.

6.

.0::.0 n..10

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.

0::.::.0 nWenowputu=Pv,thenPvx+Pxv+DPvy+

DPyv=d,andP..1Pvx+P..1Pxv+P

..1DPvy+P..1DPyv=P..1d,whichisoftheform

vx+ vy=q,where

q=P

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..1d-P..1Pxv-P..1DPyv.

Thesystemisnowoftheform

(i)(i)vx+ ivy

=qi(i=1;:::;n),

(1)(2)

whereqicaninvolvefv;v;:::;v(n)}

andwithncharacteristicsgivenbydy

= i.

dxThisisthecanonicalformoftheequations.

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Example1:Considerthelinearsystem

(1)(2).ux+4uy=0,

withinitialconditionsu=[2x,3x]Tony=0.

(2)

(1)ux+9uy=0,

04

Here,ux+

Duy=0withD=.

90

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Eigenvalues:det(D- I)=0v 2-36=0v.= 6.

Eigenvectors:

.. . .

..64x0x2

=v=for.

=6;

9..6y0y3

. .

64x0x2

=v

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=for.=..6.

96y0y..3Then,

2213260

P..1

P

=,=andP..1DP=.

3..3123

..20..6

2260

Soweputu=

vandvx+vy=0,whichhasgeneral

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solution

3..30..6

(1)(2)v=f(6x-y)andv=g(6x+y),

i.e.(1)=2v(2)

(1)=3v(2)u(1)+2vandu(1)-3v.

Initialconditionsgive

2x=2f(6x)+2g(6x),3x=3f(6x)-3g(6x),

so,f(x)=x=6and

g(x)=0;then

1

(1)u=

8/10/2019 m3414_1


(6x-y);31

(2)u=

(6x-y).2

Example2:Reducethe

linearsystem

.4y-x2x-2y.

ux+ux=0

2y-2x4x-y

tocanonicalformintheregionofthe

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(x,y)-spacewhereitistotallyhyperbolic.Eigenvalues: 4y-x-.2x-2y.

det=0v.2f3x,

3yg.

2y-2x4x-y-.Thesystem

istotallyhyperboliceverywhereexpectwherex=y.

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2.4SystemsofEquationsEigenvalues:

1=3xve1=[1,2]T, 2=3yve2=[2,1]T.

So,P= 1221.,P..1=1

3 ..122..1.andP..1DP= 3x00

3y.. 12.Then,withu=2

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1vweobtain

3x0

vx+vy=0.

03y

2.4.2QuasilinearEquationsWeconsider

systemsofnequations,involvingnfunctionsu(i)(x,y)(i=1;:::;n),oftheform

ux+Duy=d,

whereDaswell

asdmaynowdependonu.(Wehavealready

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shownhowtoreduceamoregeneralsystemAux+Buy=ctothatsimplerform.)Again,welimitourattentiontototallyhyperbolic

systems;then

=P..1DP(vD=P P..1

,

usingthesamede

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nitionofP,P..1andthediagonalmatrix ,asforthelinearandsemilinearcases.So,wecantransformthesystem

initsnormalformas,

P..1ux+ P..1uy

=P..1d,

suchthatitcanbewrittenincomponent

formas

n.n.P..1

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.

.(j).(j).P..1

u+ iu=dj,(i=1;:::;n)

ijij

@x@y

j=1j=1

where iistheitheigenvalueofthe

matrixDandwhereintheithequationinvolvesdi erentiationonlyinasingle

direction thedirectiondy=dx= i.Wede

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netheithcharacteristic,withcurvilinearcoordinatesi,asthecurveinthe(x,y)-planealongwhich

dxdydy

=1,

= iorequivalently= i.

dsidsidx

Hence,thedirectionalderivativeparalleltothecharacteristicis

d@@

(j)(j)(j)u=u+ iu;

dsi@x@y

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andthesysteminnormalformreducestonODEsinvolvingdi erentcomponentsofu

nn

.d

.

P..1(j)P..1

ijudj(i=1;:::;n):

=

ij

dsi

j=1j=1

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Example:Unsteady,one-dimensionalmotionofaninviscidcompressibleadiabaticgas.Considertheequationofmotion

(Eulerequation)

@u@u1@P

+u=

-;@t@x @x

andthecontinuityequation

@d@u@d

+d+u=0.

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@t@x@xIftheentropyisthesameeverywhereinthemotionthenP ...=constant,andthemotionequationbecomes

2

@u@uc@d

+u+=0;@t@x

@x

wherec2=dP=dd=P=disthesoundspeed.We

havethenasystemoftwo

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rstorderquasilinearPDEs;wecanwritetheseas

@w@w

+D=0;@t@x

with

uuc

w=andD=

2=d

.

d u

Thetwocharacteristicsof

thishyperbolicsystemaregivenbydx=dt=.where

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.aretheeigenvaluesofD;

det(D- I)=

u- c2=d

u-.

2

=0v(u- )2=c

and ±=

u±c.

Theeigenvectorsare[c,.. ]Tfor -and

[c, ]Tfor +,suchthattheusualmatricesare

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,

suchthat

=T..1DT

=

1

d..c

u-c0

cc

;T..1

T

=

=

.

.. d

d

c

0

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u+c

2 c

Put.andathecurvilinearcoordinatesalongthecharacteristicsdx=dt=u-cand

dx=dt=u+crespectively;thenthesystemtransformstothecanonical

form

dtdtdxdxdudddudd

==1,=u-c,=u+c,

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d-c=0andd+c=0.

d.dad.dad.d.dada

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2.4SystemsofEquations

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Chapter3

SecondOrderLinearandSemilinearEquationsinTwoVariables

Contents

3.1Classi

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cationandStandardFormReduction............373.2ExtensionsoftheTheory........................443.1Classi

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cationandStandardFormReductionConsiderageneralsecondorderlinearequationintwoindependentvariables

@2u@2u@2u@u@u

a(x,

y)+2b(x,y)+c(x,y)+d(x,y)+e(x,y)+f(x,y)u=g(x,

y);

@x2@x@y@y2@x@y

inthecaseof

asemilinearequation,thecoe cientsd,e,fandg

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couldbefunctionsof@xu,@yuanduaswell.Recall,fora

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rstorderlinearandsemilinearequation,a@u=@x+b@u=@y=c,wecouldde

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nenewindependentvariables, (x,y)and (x,y)withJ=@( , )=@(x,y)=6f0,1g,toreducetheequationtothesimplerform,

@u=@.= ( , ).Forthesecondorderequation,canwealsotransform

thevariablesfrom(x,y)to( , )toputtheequationinto

asimplerform?

So,considerthecoordinatetransform

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(x,y).( , )where.andsaresuchthattheJacobian,

J

=

@( , )

=

@.@.@x@y@f@f@x@y

=f0,1g.

@(x,y)

Thenby

inversetheoremthereisanopenneighbourhoodof(x,y)

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andanotherneighbourhoodof( , )suchthatthetransformationisinvertibleandone-to-oneontheseneighbourhoods.Asbeforewecomputechainrulederivations

@u@u@.@u@s@u@u@.@u@s

=+,=+;

@x@.@x@s@x@y@.@y

@s@y

37

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3.1Classi

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cationandStandardFormReduction@2u@2u @. 2@2u@.@s@2u @s 2@u@2.@u@2s

=+2+++

;

@x2@ 2@x@ @s@x@x@ 2@x@.@x2@s

@x2 2 2

@2u@2u @.@2u@.@s@2u @s@u@2.

@u@2s

=+2+++;

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@y2@ 2@y@ @s@y@y@ 2@y@.@y2@s@y2

@2u@2u@.@.@2u.@.@s@.@s.@2u@s

@s@u@2.@u@2s

=+++++.

@x@y

@ 2@x@y@ @s@x@y@y@x@ 2@x@y@.@x@y

@s@x@y

Theequationbecomes

@2@2@2

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uuu

A+2B+C+F(u.;uf,u, , )=0,(3.1)

@ 2@ @s@ 2

where

2 2

.@.@.@. @.

A=a+2b+c;

@x@x@y@y@.@s

.@.@s@.@s.@.@s

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B=a+b++c;

@x@x@x@y@y@x@y@y

2 2

@s@s@s

@s

C=a+2b+c.

@x@x

@y@y

Wewriteexplicitlyonlytheprincipalpartofthe

PDE,involvingthehighest-orderderivativesofu(termsofsecond

8/10/2019 m3414_1


order).

Itiseasytoverifythat

.@.@s@.@s 2

(B2-AC)=(b2-ac)

-

@x@y@y@x

where(@x @ys-@y

@x )2isjusttheJacobiansquared.So,providedJ6=0we

seethatthesignofthediscriminantb2-ac

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isinvariantundercoordinatetransformations.Wecanusethisinvariancepropertiestoclassifytheequation.

Equation(3.1)canbesimpli

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edifwecanchoose.andssothatsomeofthecoe cientsA,BorCarezero.Letusde

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ne,

@ =@x@ =@x

D.=andDf=;

@ =@y@ =@y

thenwecanwrite

.. @.

2

A=aD.2+2bD.+c;

@y@.

@s

B=(aD.Df+b(D.+Df)+c)

;

@y@y

.. @s 2

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C=aDf2+2bDf+c.

@y

Nowconsiderthequadraticequation

aD2+2bD+c=0,

(3.2)

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Chapter3 SecondOrderLinearandSemilinearEquationsinTwoVariables

whosesolutionisgivenby

=..b

±b2-ac

D=.

a

Ifthediscriminantb2-ac6

=0,equation(3.2)

hastwodistinctroots;so,wecanmakebothcoe cients

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AandCzeroifwearbitrarilytaketherootwiththenegativesignforD.andtheonewiththepositivesignfor

Df,

=@ [email protected]

D.==vA=0,(3.3)

@ =@ya

=@ =@x

..b+b2-ac

Df==v

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C=0.

@ =@ya

Then,usingD.Df=c=aandD.+Df=..2b=awehave

2@s

.ac-b2.@.

B=

vB=06.

a@y@y

Furthermore,if

thediscriminantb2-ac>0thenD.and

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Dfaswellas.andsarereal.So,wecande

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netwofamiliesofone-parametercharacteristicsofthePDEasthecurvesdescribedbytheequation (x,y)=constantandtheequation (x,

y)=constant.Di erentiate.alongthecharacteristiccurvesgivenby.=

constant,

@.@.

d.=dx+dy=0;

@x

@y

andmakeuseof(3.3)to

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ndthatthischaracteristicssatisfy

=dyb+b2-ac

=.(3.4)

dxa

Similarlywe

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ndthatthecharacteristiccurvesdescribedby (x,y)=constantsatisfy

=dyb-b2-ac

=.

(3.5)

dxa

Ifthediscriminantb2-ac=0,

equation(3.2)hasoneuniquerootandifwetakethisrootfor

D.say,wecanmakethecoe cientAzero,

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@ =@xb

D.==..vA=0.

@ =@ya

Togetsindependentof ,Dfhas

tobedi erentfromD.,soC6=0inthis

case,butBisnowgivenby

..b2

bb.@.@s.@.@s

B

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=..aDf+b-+Df+c=-+c,

aa@y@ya@y@y

sothatB=0.Whenb2-ac=0thePDE

hasonlyonefamilyofcharacteristiccurves,for

(x,y)=

constant,whoseequationisnowdyb

=.

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(3.6)

dxa

Thuswehavetoconsiderthreedi erentcases.

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3.1Classi

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cationandStandardFormReduction1.Ifb2>acwecanapplythechangeofvariable(x,y).( , )totransform

theoriginalPDEto@2

u

+(lowerorderterms)

=0.@ @s

Inthiscasetheequationissaidto

behyperbolicandhastwofamiliesofcharacteristicsgivenby

8/10/2019 m3414_1


equation(3.4)andequation(3.5).

2.Ifb2=ac,asuitablechoicefor.stillsimpli

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esthePDE,butnowwecanchoosesarbitrarily providedsand.areindependent andtheequationreducestothe

form@2

u

+(lowerorderterms)=0.@ 2

Theequationissaidtobeparabolicandhasonlyone

familyofcharacteristicsgivenbyequation(3.6).

3.

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)via.=.+s=2<( ),a=i(.- )=2=( ),

@2@2@2

uuu

i.e.,

=+(viathechainrule);

@ @s@ 2@

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2

so,theequationcanbereducedto

@2@2

uu

++(lowerorderterms)=0.

@ 2@

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2

Inthiscasetheequationissaidtobeellipticandhasnorealcharacteristics.

Theaboveformsare

calledthecanonical(orstandard)formsofthesecondorderlinearorsemilinear

equations(intwovariables).

Summary:

b2-ac>

0=0<0Canonicalform@2u@ @f+

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.

.

.=0@2u@ 2+...=0@2u@ 2+@2u@

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2+...=0TypeHyperbolicParabolicElliptic

E.g. Thewaveequation,@2@2

uu

-

c2=0;

w

@t2@x22

ishyperbolic(b2

-ac=c>0)andthetwofamiliesofcharacteristicsaredescribed

w

bydx=dt= cwi.e.

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.=x-cwtands=x+cwt.So,theequationtransformsintoitscanonicalform@2u=@ @s=0whosesolutions

arewavestravellinginoppositedirectionatspeedcw.

8/10/2019 m3414_1



Thedi usion(heatconduction)equation,@2u1@u

-=0;

@x2t@t

isparabolic(b2-ac

=0).Thecharacteristicsaregivenbydt=dx=0i.e..=

t=constant.

Laplace'sequation,@2@2

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uu

+=0;@x2@y2iselliptic(b2-ac=..1<0).ThetypeofaPDEisalocal

property.So,anequationcanchangeitsformindi erentregionsofthe

planeorasaparameterischanged.E.g.Tricomi'sequation

@2@2

uu

y+=0,(b2-ac

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=0-y=..y)

@x2@y2isellipticiny>0,parabolicfory=0andhyperboliciny<0,and

forsmalldisturbancesinincompressible(inviscid)ow1@2u@2u1

+

=0,(b2-ac=-)2

1-m@x2@y2

1-m2isellipticifm<1andhyperbolic

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ifm>1.

Example1:Reducetothecanonicalform

2@2u@2u2@2u13@u3@u

y-2xy+x=y+x.

@x2

@x@y@y2xy@x@y2.

a=y

.

2

Hereb=..xysob2

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-ac=(xy)2-xy2=0vparabolicequation.

2;

c=xOn.=constant,

=

dyb+b2-acbx2

===..v

.=x+y2.

dxaayWe

canchoosesarbitrarilyprovided.andsareindependent.

8/10/2019 m3414_1


Wechooses=y.(Exercise,tryitwiths=x.)Then@u@u@u@u@u@2u@u2@2u

=2x,=2y+,=2+4x;

@x@.@y@.@s@x2

@.@ 2@2u@2u@2u@2u@u2@2u@2u@2u

=4xy+2x,=2+4y+4y

+;

@x@y@ 2@ @s@y2@.@ 2@ @s

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@ 2

andtheequationbecomes

2@u2@2u2@2u@2u2@u2@2u

2222

2y+4xy-8xy-4xy+2x+4xy

@.@ 2@ 2

@ @s@.@ 2

@2u2@2u13@u@u3

@u

+4x2y+x=2xy+2x

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3y+x;@ @s@ 2xy@.@.@s

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3.1Classi

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cationandStandardFormReduction@2u1@u

i.e.-=0.(canonicalform)@ 2s@s

Thishassolutionu

=f( )+ 2g( ),

wherefandgarearbitraryfunctions

(viaintegratingfactormethod),i.e.

222

u=

f(x+y2)+yg(x+y2).

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Weneedtoimposetwoconditionsonuoritspartialderivativestodeterminethefunctionsfandgi.e.to

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ndaparticularsolution.

Example2:Reducetocanonicalformandthensolve

@2u@2u@2u@u

+-2

+1=0in0.x.1;y>0,withu==xon

y=0.@x2@x@y@y2@y

9..

a

=1

Hereb=1=2sob2-ac

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=9=4(>0)vequationishyperbolic.

c=..2

Characteristics:

dy13

@ =@x@ =@x

±

=..1or2=..

-

=

or

.

dx22Twomethods

ofsolving:1.directly:dydx=2vx

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-12y=constantand@ =@y@ =@ydydx=..1vx+y=constant.2.simultaneousequations:

.

>>.

@.@.

-=2

@x@y

.>.

1

y

x

=

(s+2 )

.=x-

8/10/2019 m3414_1


3

2,

v

.

2

@s@s

-=-@x@ySo,

s=x+y

>>.

>.

y=

(s- )

3

@u@u@u@u1@u@u@2u@2u@u

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@2u

=+,=-

+,=+2+

@x@.@s@y2@.@s@x2

@ 2@ @s@ 2@2u1@2u1@2u@2u@2u1@2u@2u@2u

=

-++,=

-+;

@x@y2

@ 22@ @s@ 2@y24@ 2@ @s@ 2and

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theequationbecomes

@2u@u@2u1@2u1@2u@2u1@2u@2u@2u

+2+-+

+

-+2-2+1=0;@ 2@ @s@ 22@ 22@ @s@ 2

2@ 2@ @s@ 29@2u

v

+1=0,

canonicalform.2@ @s

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So@2u=@ @s=..2=9andgeneralsolutionisgiven

by

2

vu( , )=-

s

+f( )+g( );

9

wherefandgare

arbitraryfunctions;now,weneedtoapplytwoconditionsto

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determinethesefunctions.

When,y=0,.=s=xsotheconditionu=xaty=0

gives

u(.=x,s=x)=-2x2

+f(x)+g(x)=x(vf(x)+g(x)=x+2x2.

(3.7)

99Also,usingtherelation@u1@u

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@u112

=-+=s-f0( )-.+g0( );

@y2@.@s929the

condition@u=@y=xaty=0gives@u112110

f0(x)..(.=x,s=x)=

x

-x+g0(x)=x(vg0(x)-

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f0(x)=x;@y92929

andafterintegration,g(x)-1f(x)=5x2+k,(3.8)

29wherek

isaconstant.Solvingequation(3.7)andequation(3.8)simultaneouslygives,

222142

2..f(x)=kandg(x)=2

x

-x

x+x+k,393

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393222142

2..or,intermsof.and f( )=

.-kandg( )=s+ 2

+k.393393So,fullsolutionis

22214

u( , )=-

s+.- 2

+s+ 2;9393912

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=

(2.+ )+(s- )(2s+ ).39

2

y

vu(x,y)=x

+xy+.(checkthissolution.)

2

Example

3:Reducetocanonicalform

@2@2@2

uuu

+

+=0.

@x2@x@y@y2

.

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a=1

.

Hereb=1=2sob2-ac=..3=4(<0)vequationiselliptic.

;

c=1Find.andsvia

p=

.=constantondy=dx=(1+i3)=2.=

y-12(1+i3)xpv=

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.s=constantondy=dx=(1-i3)=2s=y-21(1-i3)x

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3.2ExtensionsoftheTheoryToobtainarealtransformation,put

=.=s+.=2y-xand

a=i(.- )=x3.

So,

@u@u

=@u@u@u@2u@2u=@2@2

uu

=-+3,=2,=-

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23+3;

@x@.@a@y@.@x2@ 2@ @a@

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2

@2@2=@2@2@2

uu

uuu

=..2+23,=4;

@x@y@ 2@ @a

@y2@ 2

andtheequationtransformsto

@2=

@2@2@2=@2@2u

uuu

uu

-23+3-2+23+4=0.

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@ 2@ @a@

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2@ 2@ @a@ 2

@2@2

uu

v+=0,canonicalform.

@ 2@

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2

3.2ExtensionsoftheTheory3.2.1LinearsecondorderequationsinnvariablesTherearetwoobviouswaysinwhichwe

mightwishtoextendthetheory.

Toconsiderquasilinearsecondorder

equations(stillintwoindependentvariables.)Suchequationscanbeclassi

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edinaninvariantwayaccordingtoruleanalogoustothosedevelopedaboveforlinearequations.However,sincea,bandcarenow

functionsof@xu,@yuanduitstypeturnsouttodepend

ingeneralontheparticularsolutionsearchedandnotjustonthevalues

oftheindependentvariables.

Toconsiderlinearsecond

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orderequationsinmorethantwoindependentvariables.Insuchcasesitisnotusuallypossibletoreducetheequationtoasimplecanonical

form.However,forthecaseofanequationwithconstantcoe cientssucha

reductionispossible.Althoughthisseemsaratherrestrictiveclassofequations,we

canregardtheclassi

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cationobtainedasalocalone,ataparticularpoint.ConsiderthelinearPDE

nn

@2

@u

u

aij+bi+cu=d.

@xi@xj@xi

i;j=1i=1

Withoutlossofgeneralitywecantakethe

matrixA=(aij),i;j=1 ·n,tobe

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symmetric(assumingderivativescommute).ForanyrealsymmetricmatrixA,thereisanassociateorthogonalmatrixPsuchthatPTAP=F

whereFisadiagonalmatrixwhoseelementaretheeigenvalues, i,of

AandthecolumnsofPthelinearlyindependenteigenvectorsofA,ei

=(e1i;e2i, ·;eni).So

P=(eij)and

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=( i ij),i;j=1, ·,n.

P..1PT

Nowconsiderthetransformationx=P ,i.e..=

x=x(Porthogonal)wherex=(x1;x2, ·;xn)and.=( 1; 2,

·; n);thiscanbewrittenas

nn

xi

=eij jand j=eijxi.j=1i=1

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So,

nn

@u

@u

@2

u

@u

=eik

and=eikejr.

@xi@ k@xi@xj@ k@ r

k=1k;r=1

Theoriginalequationbecomes,

nn

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@u

aijeikejr+(lowerorderterms)=0.

@ k@ r

i;j=1k;r=1

Butbyde

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nitionoftheeigenvectorsofA,

n

T

kAer

eikaijejr. r

rk:

e

=

i;j=1

Thenequation

simpli

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esto

n

k=1

@u

k+(lowerorderterms)=0.@ k2

Wearenowinapositiontoclassifytheequation.

Equationisellipticifandonlyifall karenon-zeroandhave

thesamesign.E.g.Laplace'sequation@2@2@2

uuu

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++=0.

@x2@y2@z2

Whenallthe karenon-zeroandhavethesamesignexceptfor

preciselyoneofthem,theequationishyperbolic.E.g.thewaveequation@2 @2@2@2

uuuu

2

-c++

=0.

@t2

@x2

@y2

8/10/2019 m3414_1


@z2

Whenallthe karenon-zeroandthereareatleasttwoofeachsign,theequationultra-hyperbolic.E.g.@2@2@2@2

uuuu

+=+;@x2@x2@x2@x2

1234

suchequationdonotoftenariseinmathematical

physics.

Ifanyofthe kvanish

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theequationisparabolic.E.g.heatequation @2

@uu@2u@2u

-t++=0.

@t@x2@y2@z2

3.2.2TheCauchyProblemConsidertheproblemof

8/10/2019 m3414_1


ndingthesolutionoftheequation

@2@2@2

uuu

a+2b+c+F(@xu,@yu,u,

x,y)=0

@x2@x@y@y2

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3.2ExtensionsoftheTheorywhichtakesprescribedvaluesonagivencurve.whichweassumeisrepresentedparametricallyintheform

x= ( );y= ( ),

fore=I,

whereIisaninterval,

.e.

say.(Usuallyconsider

piecewisesmoothcurves.)WespecifyCauchydataon..:u,

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@u=@xand@u=@yaregiven8e=I,butnotethatwecannotspecifyallthesequantitiesarbitrarily.Toshowthis,supposeuis

givenon.byu=f( );thenthederivativetangentto..,

du=d ,canbecalculatedfromdu=de=f0( )butalso

du@udx@udy

=+;

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de@xde@yde

@u@u

= 0( )+ 0( )=f0( );

@x@yso,on..,thepartialderivatives

@u=@x,@[email protected]

to.anducanbeprescribedindependently.So,theCauchyproblemconsists

in

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ndingthesolutionu(x,y)whichsatis

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esthefollowing

conditions

u( ( ); ( ))=f( )

.>.>.

;

@u

and

( ( ); ( ))=g( )

@n

wheree=Iand@=@n=

n .denotesanormalderivativeto.(e.g.n=[ =,.. 0]T

);thepartialderivatives@u=@xand@u=@yareuniquelydetermined

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on.bytheseconditions.Set,p=@u=@xandq=@u=@ysothaton..,pandqareknown;then

dp@2udx@2udydq@2udx@2udy

=+and=+

.

ds@x2ds@x@ydsds@x@yds@y2ds

CombiningthesetwoequationswiththeoriginalPDE

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givesthefollowingsystemofequationsfor@2u=@x2,@2u=@x@yand@2u=@y2on.(inmatrixform),

.

@2

u

.

0

..F

a2bc

@x2

@2

u

@x@y

@2

u

BBBBBB.

CCCCCC.

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=

BBBBB.

CCCCC.

whereM=

BBBB.

dp

ds

dq

CCCC.

dxdy

0

M

.

dsds

dxdy

0

ds

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dsds

@y2

So,ifdet(M)60wecansolvetheequationsuniquelyand

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nd@2u=@x2

=,@2u=@x@yand@[email protected] erentiationsoftheseequationsitcanbeshownthatthederivativesofu

ofallordersareuniquelydeterminedateachpointon.forwhich

det(M)6

=0.Thevaluesofuatneighbouringpointscan

beobtainedusingTaylor'stheorem.

So,weconclude

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thattheequationcanbesolveduniquelyinthevicinityof.provideddet(M)6

=0(Cauchy-Kowaleskitheoremprovidesamajorantseries

ensuringconvergenceofTaylor'sexpansion).

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Considerwhathappenswhendet(M)=0,sothat

Missingularandwecannotsolveuniquelyforthesecondorderderivatives

on...Inthiscasethedeterminantdet(M)=0gives,

2 2

dydxdy dx

a

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-2b+c=0.

dsdsdsds

But,

dydy=ds

=

dx

dx=dsandso(dividingthroughbydx=ds),dy=dxsatis

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estheequation,

p

2

dydydyb±b2-acdyba-2b+c=0,i.e.=

or=.

dxdxdxadxa

Theexceptionalcurves

..,onwhich,ifuanditsnormalderivativeareprescribed,nounique

solutioncanbefoundsatisfyingtheseconditions,arethecharacteristics

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curves.

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3.2ExtensionsoftheTheory

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Chapter4

EllipticEquations

Contents

4.1De

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nitions.................................494.2PropertiesofLaplace'sandPoisson'sEquations..........504.3SolvingPoissonEquation

UsingGreen'sFunctions........544.4

ExtensionsofTheory:..........................684.1De

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nitionsEllipticequationsaretypicallyassociatedwithsteady-statebehavior.ThearchetypalellipticequationisLaplace'sequation

@2@2

uu

r2

u=0,e.g.+=0in2-D;@x2@y2

anddescribes

steady,irrotationalows, electrostaticpotentialintheabsenceofcharge,

equilibriumtemperaturedistributioninamedium.Becauseoftheirphysical

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origin,ellipticequationstypicallyariseasboundaryvalueproblems(BVPs).SolvingaBVPforthegeneralellipticequation

nn

.

@2u.@u

L[u]=aij+bi+cu=F

@xi@xj@xi

i;j=1i=1

(recall:alltheeigenvalues

ofthematrixA=(aij),i,j=1 ·

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n,arenon-zeroandhavethesamesign)isto

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ndasolutionuinsomeopenregionGofspace,withconditionsimposedon@G(theboundaryof

)oratin

8/10/2019 m3414_1


nity.E.g.inviscidowpastasphereisdeterminedbyboundaryconditionsonthesphere(u·n=0)andatin

8/10/2019 m3414_1


nity(u=Const).Therearethreetypesofboundaryconditionsforwell-posedBVPs,

1.Dirichletcondition utakesprescribedvalues

ontheboundary@G(

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rstBVP).49

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4.2PropertiesofLaplace'sandPoisson'sEquations2.Neumannconditions thenormalderivative,@u=@n=n ruisprescribedontheboundary

@G(secondBVP).Inthiscasewehavecompatibilityconditions(i.e.globalconstraints):

E.g.,supposeusatis

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esr2u=FonGandn ru=@[email protected],

r2udV=r ru

dV=ru·ndS=@udS(divergencetheorem);

@n

.@.@.

.

.vFdV

=fdSfortheproblemtobewell-de

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ned..@.

3.Robinconditions acombinationofuanditsnormalderivativesuchas@u=@n+ uisprescribed

ontheboundary@G(thirdBVP).y

@nx

Sometimeswemayhaveamixedproblem,inwhichuis

givenonpartof@Gand@u=@ngivenonthe

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[email protected]

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niteregion,wehaveaninteriorproblem;if,however,Gisunbounded,wehaveanexteriorproblem,andwemustimposeconditions'atin

8/10/2019 m3414_1


nity'.NotethatinitialconditionsareirrelevantfortheseBVPsandtheCauchyproblemforellipticequationsisnotalwayswell-posed(evenifCauchy-Kowaleski

theoremstatesthatthesolutionexistandisunique).

Asa

generalrule,itishardtodealwithellipticequationssincethesolution

isglobal,a ectedbyallpartsofthedomain.(Hyperbolic

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equations,posedasinitialvalueorCauchyproblem,aremorelocalised.)

Fromnow,weshalldealmainlywiththeHelmholtzequationr2u

+Pu=F,wherePandFarefunctionsofx,

andparticularlywiththespecialoneifP=0,Poisson'sequation,or

Laplace'sequation,ifF=0too.Thisisnot

8/10/2019 m3414_1


toosevererestriction;recallthatanylinearellipticequationcanbeputintothecanonicalform

n

@2

.

u

+ ·=0@x2k

k=1

and

thatthelowerorderderivativesdonotaltertheoverallpropertiesofthe

solution.

4.2PropertiesofLaplace'sandPoisson'sEquations

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De

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nition:AcontinuousfunctionsatisfyingLaplace'sequationinanopenregion

,withcontinuous

8/10/2019 m3414_1


rstandsecondorderderivatives,iscalledanharmonicfunction.Functionsu

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Chapter4 EllipticEquations

inC2()withr2u~0(respectivelyr2u.0)arecallsubharmonic(respectivelysuperharmonic).

4.2.1MeanValuePropertyDe

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nition:Letx0beapointinGandletBR(x0)denotetheopenballhavingcentrex0andradiusR.Let R(x0)denote

theboundaryofBR(x0)andletA(R)bethesurfaceareaof R(x0).

Thenafunctionuhasthemeanvaluepropertyatapointx0

=Gif

1

u(x0)=u(x)dS

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A(R)

R

foreveryR>0suchthatBR(x0)iscontainedin

.Ifinsteadu(x0)satis

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es

1

u(x0)=u(x)dV,

V(R)

BR

whereV(R)isthevolumeof

theopenballBR(x0),wesaythatu(x0)hasthesecondmeanvalue

propertyatapointx0=

.Thetwomeanvalueproperties

areequivalent.

xy@

Rx0BR RTheorem:Ifu

8/10/2019 m3414_1


isharmonicinGanopenregionofRn,thenuhasthemeanvaluepropertyon

.

Proof:Weneed

tomakeuseofGreen'stheoremwhichsays,

@u@v

v-udS=vr2u-ur2v

dV.(4.1)

@n@n

SV

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beBr(x0)-B"(x0),0<"<R.Then,inRn-x0,

1... 1 .

r22

v

=

r

=0

2

r@r@r

r

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4.2PropertiesofLaplace'sandPoisson'sEquationssovisharmonictooandequation(4.1)becomes

@v@v@v@v

u

dS+u

dS=u

dS-u

dS=0

@n@n@r

@r

r . r .

@v

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@v11

vu

dS=u

dSi.eudS=

udS.

@r@r"2r2

. r . rSinceuis

continuous,thenasp.0theLHSconvergesto4 u(x0;y0;z0)(withn

=3,say),so

1

u(x0)=

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udS.

A(r)

r

Recoveringthesecondmeanvalueproperty(withn=3,say)isstraightforward

.r3.rZZ

r11

2u(x0)dd

=

u(x0)=

udSdd=

u

dV.

34.4

00 .Br

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Theinverseofthistheoremholdstoo,butishardertoprove.Ifuhasthemeanvaluepropertythenuisharmonic.

4.2.2Maximum-MinimumPrincipleOneofthemostimportantfeaturesofelliptic

equationsisthatitispossibletoprovetheoremsconcerningtheboundednessof

thesolutions.

Theorem:Supposethatthesubharmonicfunction

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usatis

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es

r2u=Fin

,withF>0in

.Thenu(x,y)attainshismaximumon@

.

Proof:(Theoremstatedin2-Dbutholdsinhigherdimensions.)

Supposeforacontradictionthatuattainsitsmaximumataninteriorpoint

(x0;y0)of

.Thenat(x0;y0),

@u

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@u@2u@2u

=0,=0,.0and.0;

@x@y@x2@y2

sinceitisamaximum.So,

@2@2

uu

+.0,whichcontradictsF>

0in

.@x2@y2

Henceumustattainits

maximumon@,i.e.ifu.Mon

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@,u<Min

.

Theorem:TheweakMaximum-MinimumPrincipleforLaplace'sequation.Supposethatusatis

8/10/2019 m3414_1


es

r2u=0inaboundedregion

;ifm.u.Mon@,thenm

.u

.Min

.

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Proof:(Theoremstatedin2-Dbutholdsinhigherdimensions.)Considerthefunction

2

2

v=u+"(x+y2),forany">0.Then

r2v=4">0inG(sincer2(x+y2)=4),andusingthe

previoustheorem,

v.M+"R2in

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,

whereu.Mon@GandRistheradiusofthecirclecontaining

.Asthisholds

forany",letp.0toobtain

u.

Min

,

i.e.,ifusatis

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esr2u=0in

,thenucannotexceedM,[email protected],ifuis

asolutionofr2u=0,sois..u.Thus,wecanapply

alloftheaboveto..utogetaminimumprinciple:ifu

~mon@,thenu~min

8/10/2019 m3414_1


.

ThistheoremdoesnotsaythatharmonicfunctioncannotalsoattainmandMinsideGthough.Weshallnow

progressintothestrongMaximum-MinimumPrinciple.

Theorem:Supposethatuhas

themeanvaluepropertyinaboundedregionGandthatu

iscontinuousinG=G.@.

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ifuattainsitsmax.ataninteriorpointof

,thenuisconstantin

¯

.

Supposeu(x0)=Mandletx.besomeotherpointof

.

Jointhesepointswithapathcoveredbyasequenceofoverlappingballs,

Br.

y

xx0x1x?@

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Considertheballwithx0atitscenter.Sinceuhasthemeanvaluepropertythen

1

M=u(x0)=

udS.M.

A(r)

r

This

equalitymustholdthroughoutthisstatementandu=Mthroughoutthesphere

surroundingx0.Sincetheballsoverlap,thereisx1,centre

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ofthenextballsuchthatu(x1)=M;themeanvaluepropertyimpliesthatu=Minthisspherealso.Continuinglikethis

gives

?

u(x?)=M.Sincexisarbitrary,weconcludethat

u=Mthroughout

,andbycontinuity

throughout

.ThusifuisnotaconstantinG

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itcanattainitsmaximumvalueonlyontheboundary@.

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4.3SolvingPoissonEquationUsingGreen'sFunctionsCorollary:Applyingtheabovetheoremto..uestablishesthatifuisnonconstantitcan

[email protected],we

canstatethefollowingtheorem.(Theprooffollowsimmediatelytheprevioustheoremand

theweakMaximum-MinimumPrinciple.)

Theorem:ThestrongMaximum-Minimum

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PrincipleforLaplace'sequation.

Letubeharmonicin

,i.e.solutionofr2u=0inGand

continuousin

,withMandmthemaximumandminimumvalues

[email protected],either

m<u<MinGorelsem=u=

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Min

.

NotethatitisimportantthatGbeboundedforthetheoremtohold.E.g.,consideru(x,y)=

exsinywithG=f(x,y)j...<x<+1,0<y<2 g.

Thenr2u=0andontheboundaryofGwehaveu

=0,sothatm=M=0.But

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ofcourseuisnotidenticallyzeroin

.

Corollary:Ifu=Cisconstanton@,thenu

=Cisconstantin

.Armedwiththeabovetheoremswe

areinpositiontoprovetheuniquenessandthestabilityofthesolution

ofDirichletproblemforPoisson'sequation.ConsidertheDirichletBVP

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r2u=FinGwithu=fon@Gandsupposeu1;u2twosolutionstotheproblem.Thenv

=u1-u2satis

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es

r2v=r2(u1-u2)=0in,withv=0on@.

Thus,v.0in

,i.e.u1=u2;thesolutionisunique.Toestablishthecontinuous

dependenceofthesolutionontheprescribeddata(i.e.thestabilityofthe

solution)letu1andu2satisfy

r2uf1;2}

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=FinGwithuf1;2}=ff1;2}on@,withmaxjf1-f2|=".Thenv=u1-u2

[email protected],

vmusthaveitsmaximumandminimumvalueson@;

henceju1-u2j.pin

.So,the

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solutionisstable smallchangesintheboundarydataleadtosmallchangesinthesolution.

WemayusetheMaximum-Minimum

Principletoputboundsonthesolutionofanequationwithoutsolvingit.

ThestrongMaximum-MinimumPrinciplemaybeextendedtomoregenerallinear

ellipticequations

nn

.@2u.

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@u

L[u]=aij+bi+cu=F,

@xi@xj@xi

i;j=1i=1

and,asforPoisson's

equationitispossiblethentoprovethatthesolutiontotheDirichlet

BVPisuniqueandstable.

4.3SolvingPoissonEquationUsingGreen's

FunctionsWeshalldevelopaformalrepresentationforsolutionsto

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boundaryvalueproblemsforPoisson'sequation.

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4.3.1De

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nitionofGreen'sFunctionsConsideragenerallinearPDEintheform

L(x)u(x)=F(x)in

,

whereL(x)

isalinear(self-adjoint)di erentialoperator,u(x)istheunknownandF(x)

istheknownhomogeneousterm.(Recall:Lisself-adjointifL=L?,

whereL.isde

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nedbyhvjLu.=hL?vju.andwhere

hvju.=v(x)w(x)u(x)dx((w(x)istheweightfunction).)Thesolutiontotheequationcanbe

writtenformally

u(x)=L..1F(x),

whereL..1,theinverse

ofL,issomeintegraloperator.(WecanexpecttohaveLL..1=

LL..1=I,identity.)Wede

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netheinverseL..1usingaGreen'sfunction:let

u(x)=L..1F(x)=-G(x, )F( )d ,(4.2)

whereG(x,.

)istheGreen'sfunctionassociatedwithL(Gisthekernel).Note

thatGdependsonboththeindependentvariablesxandthenewindependent

variables ,overwhichweintegrate.

Recallthe

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Dirac -function(morepreciselydistributionorgeneralisedfunction) (x)whichhastheproperties,

(x)dx=1and (x- )h( )d.=

h(x).

RnRn

Now,applyingLtoequation(4.2)

weget

Lu(x)=F(x)=..LG(x, )F( )d ;

hence,

theGreen'sfunctionG(x, )satis

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es

u(x)=-G(x,.)F( )d.withLG(x, )=.. (x- )andx,.=

.

4.3.2Green'sfunctionforLaplaceOperatorConsiderPoisson'sequationintheopen

boundedregionVwithboundaryS,

r2u=Fin

V.(4.3)

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4.3SolvingPoissonEquationUsingGreen'sFunctionsxyVSnThen,Green'stheorem(nisnormaltoSoutwardfromV),whichstates

@v@u

.ur2.

v-vr2

udV=u-vdS;

@n@n

VS

foranyfunctionsuandv,with@h=@n

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=n rh,becomes

.ZZ.

@v@u

ur2vdV=vFdV+u-v

dS;

@n@n

VVS

so,ifwe

choosev.v(x, ),singularatx= ,suchthatr2v

=.. (x- ),thenuissolutionofthe

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equation

@v@u

u( )=-vFdV-u-vdS(4.4)

@n@n

VS

whichisanintegralequationsinceuappearsintheintegrand.To

addressthisweconsideranotherfunction,w.w(x, ),regularatx

= ,suchthatr2w=0inV.

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Hence,applyGreen'stheoremtothefunctionuandw

Z .Z

@w@u

.ur2w-wr2

.

u-wdS=udV=-wF

dV.

@n@n

SVV

Combiningthis

equationwithequation(4.4)we

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nd

.@u

u( )=-(v+w)FdV-u(v+w)-(v+w)dS;

@n@n

VS

so,ifweconsiderthefundamentalsolutionof

Laplace'sequation,G=v+w,suchthatr2G=.. (x-

)inV,

@G@u

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u( )=-GFdV-u-GdS.(4.5)

@n@n

VS

Notethatif,F,f

andthesolutionuaresu cientlywell-behavedatin

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nitythisintegralequationisalsovalidforunboundedregions(i.e.forexteriorBVPforPoisson'sequation).

Thewaytoremoveu

or@u=@nfromtheRHSoftheaboveequationdependsonthechoice

ofboundaryconditions.

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DirichletBoundaryConditions

Here,thesolutiontoequation(4.3)satis

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estheconditionu=fonS.So,wechoosewsuchthatw=..vonS,i.e.G=0on

S,inordertoeliminate@u=@nformtheRHSofequation(4.5).Then,

thesolutionoftheDirichletBVPforPoisson'sequation

r2u

=FinVwithu=fonS

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is

@G

u( )=-GFdV-fdS;

@n

VS

whereG=v

+w(wregularatx= )withr2v=.. (x.. )andr2w

=0inVandv+w=0onS.So,theGreen's

functionGissolutionoftheDirichletBVP

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r2G=.. (x- )inV,

withG=0onS.

NeumannBoundaryConditions

Here,the

solutiontoequation(4.3)satis

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notsatisfyacompatibilityequation,hasnosolution.RecallthattheNeumannBVPr2u=FinV,with@u=@n=fon

S,isill-posedif

FdV6fdS:

=

VS

WeneedtoaltertheGreen'sfunctionalittleto

satisfythecompatibilityequation;put

r2G=...

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+C,whereCisaconstant,thenthecompatibilityequationfortheNeumannBVPforGis1

(...+C)dV

=0dS=0vC=;

V

VS

whereVisthevolumeofV.Now,applyingGreen'stheoremto

Gandu:

@u@G

..Gr2

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u-ur2G.dV=G-udS

@n@n

VS

weget

1

u( )=-GFdV+GfdS+udV.

V

VSV

u

Thisshowsthat,whereasthe

solutionofPoisson'sequationwithDirichletboundaryconditionsisunique,

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thesolutionoftheNeumannproblemisuniqueuptoanadditiveconstant uwhichisthemeanvalueofuover

.

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4.3SolvingPoissonEquationUsingGreen'sFunctionsThus,thesolutionoftheNeumannBVPforPoisson'sequation

@u

r2

u=FinVwith=fonS@n

is

u( )=u -GFdV+GfdS,

VS

whereG=v+w(wregularatx

8/10/2019 m3414_1


= )withr2v=.. (x- ),r2w=1=VinVand@w=@n=..@[email protected],theGreen'sfunctionGis

solutionoftheNeumannBVP

r2G=.. (x- )+1

inV,

V

@G

with=0onS.

@n

RobinBoundaryConditions

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Here,thesolutiontoequation(4.3)satis

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esthecondition@u=@n+ u=fonS.So,wechoosewsuchthat@w=@n+ w=..@v=@n- vonS,i.e.

@G=@n+ G=0onS.Then,

Z. Z. Z

@G@u@G

u-GdS=u+G( u

-f)dS=-GfdS.

@n@n

8/10/2019 m3414_1


@n

SSS

Hence,thesolutionoftheRobinBVPforPoisson'sequation

@u

r2u

=FinVwith+ u=fonS

@n

is

u( )=-GFdV+GfdS,

VSwhereG=v+w(w

8/10/2019 m3414_1


regularatx= )withr2v=.. (x- )andr2w=0inVand@w=@n+ w=..@v=@n-

vonS.So,theGreen'sfunctionGissolutionoftheRobin

BVP

r2G=.. (x- )inV,

@G

with+ G=0onS.

8/10/2019 m3414_1


@n

SymmetryofGreen'sFunctions

TheGreen'sfunctionissymmetric(i.e.,G(x, )=G( ,x)).Toshowthis,consider

twoGreen'sfunctions,G1(x).G(x,.1)andG2(x).G(x,.

2),andapplyGreen'stheoremtothese,

@G2@G1

..G1r2G2

-G2r2G1.dV=G1-G2dS.

8/10/2019 m3414_1


@n@n

VS

Now,since,G1andG2arebyde

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nitionGreen'sfunctions,G1=G2=0onSforDirichletboundaryconditions,@G1=@n=@G2=@n=0onSforNeumannboundary

conditionsorG2@G1=@n=G1@G2=@nonSforRobinboundaryconditions,

soinanycasetheright-handsideisequaltozero.Also,r2G1

=.. (x...1),r2G2=.. (x...2)andtheequation

8/10/2019 m3414_1


becomes

G(x,.1) (x- 2)dV=G(x,.2) (x- 1)dV,

VV

G( 2, 1)=G( 1,

2).

Nevertheless,notethatforNeumannBVPs,theterm1=Vwhich

providestheadditiveconstanttothesolutiontoPoisson'sequationbreaksthesymmetry

ofG.

8/10/2019 m3414_1



Example:

Considerthe2-dimensionalDirichletproblemforLaplace'sequation,

r2u=

0inV,withu=fonS(boundaryofV

).SinceuisharmonicinV(i.e.r2u=0)andu

=fonS,thenGreen'stheoremgives@v@u

8/10/2019 m3414_1


ur2vdV=f-vdS.

@n@n

VS

Notethatwehaveno

informationabout@[email protected],

1

v=-ln(x- )2+(y- )2.

;

4.

thenr2v=0onV

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forallpointsexceptP.(x= ,y= ),whereitisunde

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ned.Toeliminatethissingularity,we'cutthispointPout i.e,surroundPbyasmallcircleofradiusp=p(x

- )2+(y- )2anddenotethecircleby ,whoseparametric

forminpolarcoordinatesis

:fx-.=p

cos ,y-s=psinßwith

8/10/2019 m3414_1


">0andß=(0,2 )g.

xy" V?S Hence,v=..1=2.lnpanddv=dp=..1=2 pandapplyingGreen'stheorem

touandvinthisnewregionV.(withboundariesS

and ),weget

Z. Z.

@v@u@v

@u

f-vdS+u-

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vdS=0.(4.6)

@n@n@n@n

S.

sincer2u=r2v=0forallpoint

inV..Bytransformingtopolarcoordinates,dS="dßand

@u=@n=..@u=@p(unitnormalisinthedirection")onto ;then

Z.2

@uplnp@u

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vdS=dß.0asp.0;

@n2.@p

0

andalso

Z.2a.2a.2

@v@v111

udS=-updß=up

dß=

udß.u( , )asp.0;@n

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@p2 p2

000andso,inthelimitp.0,equation(4.6)gives

@u@v1

u( , )=v-fdS,wherev=-ln(x-

)2+(y- )2..

@n@n4

S

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4.3SolvingPoissonEquationUsingGreen'sFunctionsnow,considerw,suchthatr2w=0inVbutwithwregularat(x=

,y= ),andwithw=..vonS.ThenGreen's

theoremgives

.Z. Z.

@w@u@w@u

.ur2w-wr2u.dV

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=u-wdS8f+vdS=0

@n@n@n@n

VSS

since

r2u=r2w=0inVandw=..vonS.

Then,subtractthisequationfromequationabovetoget

Z. Z.

Z

@u@v@w@u.

u( ,

8/10/2019 m3414_1


)=v-fdS-f+vdS=-f(v+w)dS.

@n@n@n@n@n

SSS

SettingG(x,y; , )=v+w,

then

@G

u( , )=-fdS.

@n

S

SuchafunctionG

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thenhastheproperties,r2G=.. (x- )inV,withG=0onS.

4.3.3FreeSpaceGreen'sFunction

WeseekaGreen'sfunctionGsuchthat,G(x, )=v(x, )+w(x,

)wherer2v=.. (x- )inV.Howdowe

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ndthefreespaceGreen'sfunctionvde

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nedsuchthatr2v=.. (x- )inV?Notethatitdoesnotdependontheformoftheboundary.(The

functionvisa'sourceterm andforLaplace'sequationisthepotential

duetoapointsourceatthepointx= .)Asan

illustrationofthemethod,wecanderivethat,intwo

8/10/2019 m3414_1


dimensions,1

v=-ln(x- )2+(y- )2.;

4.aswehavealreadyseen.Wemove

topolarcoordinatearound( , ),x-.=rcosß

&y-s=rsin ,andlookforasolution

ofLaplace'sequationwhichisindependentofßandwhich

8/10/2019 m3414_1


issingularasr.0.

y

Cr

r Dr.x

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Laplace'sequationinpolarcoordinatesis

1.

@v

@2

v1@v

r=+

=0

r@r@r@r2r@r

whichhassolution

v=Blnr+AwithAand

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Bconstant.PutA=0and,todeterminetheconstantB,applyGreen'stheoremtovand1inasmalldiscDr

(withboundaryCr),ofradiusraroundtheorigin( , ),

@v

dS=r2vdV=- (x- )dV

=..1;

@n

CrDrDr

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sowechooseBtomake

@v

dS=..1.

@n

Cr

Now,

inpolarcoordinates,@v=@n=@v=@r=B=randdS=rdß(going

aroundcircleCr).So,

.2a.2

B1

rdß=Bdß=..1vB

8/10/2019 m3414_1


=-.

r2

00

Hence,

111

(x- )2+(y- )2.

.

2

v=-lnr

=-lnr

=-ln

2.4.

4.(Wedonotusetheboundaryconditionin

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ndingv.)

Similar(butmorecomplicated)methodsleadtothefree-spaceGreen'sfunctionvfortheLaplaceequationinndimensions.In

particular,

.>>>>>>.

1

-

jx- j;n=1;

2

1

jx-

j2.

4.

1

-

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ln

n=2,

v(x,.)=

,

>>>>>>.

jx- j2..n

-

;n~3,

(2-n)An(1)

wherex

and.aredistinctpointsandAn(1)denotestheareaoftheunit

n-sphere.Weshallrestrictourselvestotwodimensionsforthis

8/10/2019 m3414_1


course.

NotethatPoisson'sequation,r2u=F,issolvedinunboundedRnby

u(x)=-v(x,.)

F( )d.

Rn

wherefromequation(4.2)thefree

spaceGreen'sfunctionv,de

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nedabove,servesasGreen'sfunctionforthedi erentialoperatorr2whennoboundariesarepresent.

4.3.4MethodofImagesInorder

tosolveBVPsforPoisson'sequation,suchasr2u=Finan

openregionVwithsomeconditionsontheboundaryS,weseeka

Green'sfunctionGsuchthat,inV

G(x,

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)=v(x, )+w(x,.)wherer2v=.. (x- )andr2w=0or1=V(V).

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4.3SolvingPoissonEquationUsingGreen'sFunctionsHavingfoundthefreespaceGreen'sfunctionv whichdoesnotdependontheboundary

conditions,andsoisthesameforallproblems westillneed

to

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ndthefunctionw,solutionofLaplace'sequationandregularinx= ,which

8/10/2019 m3414_1


xestheboundaryconditions(vdoesnotsatis

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estheboundaryconditionsrequiredforGbyitself).So,welookforthefunctionwhichsatis

8/10/2019 m3414_1


es

r2w=0or1=V(V)inV,(ensuringwisregularat( , )),withw=..v(i.e.

G=0)onSforDirichletboundaryconditions,@w@v@G

or=-(i.e.=0)onSforNeumannboundaryconditions.

@n@n@n

Toobtainsuch

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afunctionwesuperposefunctionswithsingularitiesattheimagepointsof( , )).(Thismayberegardedasaddingappropriatepointsourcesand

sinkstosatisfytheboundaryconditions.)Notealsothat,sinceGandv

aresymmetricthenwmustbesymmetrictoo(i.e.w(x, )=w( ,x)).

Example1

Supposewewishto

8/10/2019 m3414_1


solvetheDirichletBVPforLaplace'sequation

@2@2

uu

r2u=+=0iny>0withu=

f(x)ony=0.@x2@y2

Weknowthatin2-D

thefreespacefunctionis1

v=-ln(x

- )2+(y- )2..

4.If

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wesuperposetovthefunction1

w=+ln(x- )2+(y+ )2.;

4.solutionofr2w=0

inVandregularat(x= ,y= ),then

1 (x- )2+(y- )2

G(x,y, ,

)=v+w=-ln.

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4.(x- )2+(y+ )2

xyS( ; )V( ;.. )+..G=v+wwvy=.. y= yx= Notethat,settingy=0inthisgives,

1 (x-

)2+ 2

G(x,0, , )=-ln=0,as

required.

4.(x- )2+ 2

8/10/2019 m3414_1



Thesolutionisthengivenby

@G

u( , )=-fdS.

@n

S

Now,wewant@G=@nfor

theboundaryy=0,whichis

@G@G1s

.=-.=-

8/10/2019 m3414_1


(exercise,checkthis).

@n

8/10/2019 m3414_1


@y..(x- )2+ 2

Sy=0

Thus,

s.+.f(x)

u( , )=

dx;

.(x- )2+ 2

...

andwecanrelabeltogetintheoriginalvariablesy

.+.f( )

u(x,y)=2

8/10/2019 m3414_1


d .

.(.-x)2+y

...

Example2

FindGreen'sfunctionfortheDirichletBVP

@2@2

uu

r2u=+=Finthe

quadrantx>0;y>0.@x2@y2

Weusethesametechnique

butnowwehavethreeimages.

x( ; )( ;.. )+..VSy(.. ;.. )+(.. ; )..

8/10/2019 m3414_1


Then,theGreen'sfunctionGis11

G(x,y, , )=-ln(x- )2+(y- )2.+ln(x-

)2+(y+ )2.

4.4.11

-ln

(x+ )2+(y+ )2.+ln(x+ )2+(y- )2.

.

4.4.

So,"..(x-

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8/10/2019 m3414_1


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4.3SolvingPoissonEquationUsingGreen'sFunctionsExample3

ConsidertheNeumannBVPforLaplace'sequationintheupperhalf-plane,

@2u@2u@u@u

r2u=+=0iny>0

with=-=f(x)ony=0.@x2@y2@n@y

xyS( ; )V( ;.. )....x= y= y=.. vG=v+wywAddanimagetomake@G=@y=0

8/10/2019 m3414_1


ontheboundary:

1

1

-ln

.. 22(- )+(- )xy

(x-

)2+(y+ )2

G(x,y, , )=-ln

.

4.

4.

Notethat,

@G1

2(y- )2(y+ )

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=-

+

,

(x- )2+(y- )2

(x- )2+(y

+ )2

@y4.

andasrequiredforNeumannBVP,

@G

@n

@G

1

..2s2s

=-

=

8/10/2019 m3414_1


+=0.

4.(x- )2+ 2(x- )2+ 2(x- )2

@y

S

y=0

+ 2.,

Then,sinceG(x,0, ,

)=..1=2.ln

.+1

1

u( , )=

-f(x)ln

2.

...

8/10/2019 m3414_1


(x- )2+ 2.

dx,

.+1

1

i.e.u(x,y)=-f( )ln2.

...

(x- )2+y2.

d ,

RemindthatallthetheoryonGreen'sfunctionhasbeen

developedinthecasewhentheequationisgivenin

8/10/2019 m3414_1


aboundedopendomain.Inanin

8/10/2019 m3414_1


nitedomain(i.e.forexternalproblems)wehavetobeabitcarefulsincewehavenotgivenconditionsonGand@G=@nat

in

8/10/2019 m3414_1


nity.Forinstance,wecanthinkoftheboundaryoftheupperhalf-planeasasemi-circlewithR.+1.

yS1S2..Rx+R

8/10/2019 m3414_1



Green'stheoreminthehalf-disc,foruandG,is

@u@G

..Gr2u-ur2G.dV=G-udS.

@n

@n

VS

SplitSintoS1,theportionalong

thex-axisandS2,thesemi-circulararc.Then,inthe

8/10/2019 m3414_1


aboveequationwehavetoconsiderthebehaviouroftheintegrals

Z.aZ.

@u@u@G@G

(1)GdS=GRdßand(2)udS=uRdß

@n@R@n@R

S20S20

as

R.+1.Green'sfunctionGisO(lnR)on

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S2,sofromintegral(1)weneed@u=@Rtofalloffsu cientlyrapidlywiththedistance:fasterthan1=(RlnR)i.e.umust

fallofffasterthanln(ln(R)).Inintegral(2),@G=@R=O(1=R)onS2

providesamorestringentconstraintsinceumustfalloffmorerapidlythat

O(1)atlargeR.IfbothintegralsoverS2vanish

8/10/2019 m3414_1


asR.+.thenwerecoverthepreviouslystatedresultsonGreen'sfunction.

Example4

SolvetheDirichletproblem

forLaplace'sequationinadiscofradiusa, .@2

1

.@u1u

r2

u=

r+=0inr<awithu=f( )on

8/10/2019 m3414_1


r=a.

r@r@rr2@ 2

xyrSV(x;y) Q +..( ; )PConsiderimageofpointPatinversepointQP=(dcos

,dsin ),Q=(qcos ,qsin ),

2

with q=a(i.e.OP·OQ=a2).1

G(x,y, , )=-ln(x- )2

8/10/2019 m3414_1


+(y- )2.

4.

.22.

+1ln(x-acos )2+(y-asin )2+

h(x,y, , )(with 2+ 2= 2).4 ddWe

needtoconsiderthefunctionh(x,y, , )tomakeGsymmetric

andzeroontheboundary.Wecanexpressthisin

8/10/2019 m3414_1


polarcoordinates,x=rcos ,y=rsin ,

1 (rcosß-a2=dcos )2+(rsin

ß-a2=dsin )2

G(r, , , )=ln+

h;

4.(rcosß-dcos )2+(rsinß

-dsin )2

.2

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1r+a4= 2-2a2r=dcos(ß- )

=ln+h.

4 r2+ 2-2rdcos(ß- )

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4.3SolvingPoissonEquationUsingGreen'sFunctionsChoosehsuchthatG=0onr=a,

2

+

a4= 2-2a3=dcos(ß- )1

a

Gjr=a

=ln

4.

+h,a

2

+ 2-2adcos(ß- )

2

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2

2+a2-2adcos(ß- )

1

1

a

=

ln

+h=0vh=ln.

2 2+a2-2adcos(ß- )

2

4.

4 a

Notethat,

8/10/2019 m3414_1


2

42

1

1

a

a

r

2

w(r,

, , )=ln

4.

+-2cos(ß

- )+lnr

2

2

d

4 a

r

2

lna

8/10/2019 m3414_1


+

2 2

1

-2rdcos(ß- )

=

2

4 a

is

symmetric,regularandsolutionofr2w=0inV.So,

2

+r2 2=a2-2rdcos(ß- )1

a

G(r, , , )=v+w

8/10/2019 m3414_1


=ln

;

2+ 2-2rdcos(ß- )

4 r

Gissymmetricandzeroon

theboundary.ThisenableustogettheresultforDirichlet

problemforacircle,

.2a

@G

ad ,

u( , )=-f( )

8/10/2019 m3414_1


0

@r

r=a

where

so

2r 2=a2-2dcos(ß- )

@G1

2r-2dcos(ß- )

..

=

,

2+r2 2=a2-2rdcos(ß- )

2+ 2-2rdcos(ß- )

8/10/2019 m3414_1


@r4 a

r

2=a-dcos(ß- )

@G

@n

@G

1

a-dcos(ß- )

..

=

=

,

2+ 2-2adcos(ß

- )

2+ 2-2adcos(ß

8/10/2019 m3414_1


- )

@r

2 a

a

S

r=a

1 2-a2

=.

2 aa2+ 2-2adcos(ß- )

Then

1.2aa2- 2

u( , )=f( )d ,

2 a2+ 2-2ad

8/10/2019 m3414_1


cos(ß- )

0

andrelabelling,

2.2

a2-rf( )

u(r, )=d .

2 a2+r2-2arcos(ß- )

0

Notethat,fromtheintegralformofu(r, )above,wecanrecover

theMeanValueTheorem.Ifweputr=0

8/10/2019 m3414_1


(centreofthecircle)then,

.2

1

u(0)=f( )d ;

2.

0

i.e.

theaverageofanharmonicfunctionoftwovariablesoveracircleis

equaltoitsvalueatthecentre.

8/10/2019 m3414_1



Furthermorewemayintroducemoresubtleinequalitieswithintheclassofpositiveharmonicfunctionsu~

0.Since..1.cos(ß.. ).1then(a..r)2.a2..2ar

cos(ß.. )+r2.(a+r)2.Thus,thekerneloftheintegrandinthe

integralformofthesolutionu(r, )canbebounded

8/10/2019 m3414_1


1a-r1a2-r21a+r

.:

2

2 a+r

2.(a-r)2.a2-2arcos(ß- )+r2 a-

r

Forpositiveharmonicfunctionsu,wemayusetheseinequalities

toboundthesolutionofDirichletproblemforLaplace'sequation

8/10/2019 m3414_1


inadisc

.2a.2

1a-r1a+r

f( )dß.u(r, )

.f( )d ,

2 a+r2 a-r

00

i.e.usingtheMeanValueTheoremweobtainHarnack'sinequalitiesa-

ra+ra+ru(0).u(r,

8/10/2019 m3414_1


).a-ru(0).Example5

InteriorNeumannproblemforLaplace'sequationinadisc,

1.

@u

1

@2

u

r2

r+=0inr<a,

u=r@r

@rr2@ 2

@u

=

8/10/2019 m3414_1


f( )onr=a.

@n

Here,weneed

r2G=.. (x- ) (y- )+1

with@G

=0,

V@r

r=a

whereV= a2isthesurfaceareaofthedisc.In

ordertodealwiththistermwesolvethe

8/10/2019 m3414_1


equation

2

1.

@t

1r

r2 (r)=r=v (r)=+c1

lnr+c2,

r@r@r a24 a2

andtaketheparticularsolutionwithc1=c2=0.Then,add

insourceatinversepointandanarbitraryfunctionh

8/10/2019 m3414_1


to

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xthesymmetryandboundaryconditionofG

1

2

+ 2-2rdcos(ß- )G(r,

, , )=-ln

r

4.

2

22

r

r

2

1

a

2

-ln

8/10/2019 m3414_1


4.

+2-2rdcos(ß- )++h:a

2

4 a2

a

So,

@G12r-2dcos(ß- )12r-2a2=d

cos(ß- )r@h

=..-++;

@r

4 r2+ 2-2rdcos(ß- )4 r2+a4= 2

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-2a2r=dcos(ß- )2 a2@r

@G

=-

1

2=dcos(ß- )

a

-dcos(ß- )a-a

1@h

+

++

@r

2.

a

2+ 2-2adcos(ß-

8/10/2019 m3414_1


)

a

2+a4= 2-2a3=dcos(ß- )

2 a@r

,

r=a

r=a

1a-dcos(ß- )+ 2=a

-dcos(ß- )1@h

=-++

,

2 2+a2-2adcos(ß-

8/10/2019 m3414_1


)

2 a@r

r=a

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4.4ExtensionsofTheory:@G11@h@h@G

.=-++.and.=0implies=0

ontheboundary.

@r.2 a2 a@r.@r.

@r

r=ar=ar=a

Then,puth.1=2.ln(a= )

;so, . 22 .2

.2

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.2

G(r, , , )=-1lnr+ 2-2rdcos(ß- )a+r-2rdcos(ß- )+

r.

4.a24 a2

Onr=a,

1h.1

2 2.

Gjr=a=-ln

a+ 2-2adcos(ß- )+;

8/10/2019 m3414_1


4.4.

.2.

=-1lna+ 2-2adcos(ß- )-1

.

2.2

Then,

.2au( ,

)=u +f( )Gjr=aad ,

0.2a .

=u -alna2+ 2-

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2adcos(ß- )-1f( )d .

2.2

0

Now,recalltheNeumannproblemcompatibilitycondition,

.

2a

f( )dß=0.

0Z..2

@u

Indeed,r2udV=dSfromdivergencetheoremvf( )dß=0.

@n

VS0

.

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2a

Sotheterminvolvingf( )dßinthesolutionu( , )vanishes;hence

0.2a

a

.2.

u( , )=u -lna+ 2

-2adcos(ß- )f( )d ,

2.

0.2a

a

.2.

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oru(r, )=u -lna+r2-2arcos(ß- )f( )d .

2.

0

Exercise:Exterior

NeumannproblemforLaplace'sequationinadisc,

.2a

a

.2.

u(r, )=lna

+r2-2arcos(ß- )f( )d .

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2.

0

4.4ExtensionsofTheory: AlternativetothemethodofimagestodeterminetheGreen'sfunctionG:(a)

eigenfunctionmethodwhenGisexpendedonthebasisoftheeigenfunctionof

theLaplacianoperator;conformalmappingofthecomplexplaneforsolving2-Dproblems.

Green'sfunctionformoregeneraloperators.

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Chapter5

ParabolicEquations

Contents

5.1De

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nitionsandProperties.......................695.2FundamentalSolutionoftheHeatEquation............725.3

SimilaritySolution............................755.4MaximumPrinciplesandComparisonTheorems.....

..

.

.785.1De

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nitionsandPropertiesUnlikeellipticequations,whichdescribesasteadystate,parabolic(andhyperbolic)evolutionequationsdescribeprocessesthatareevolvingintime.For

suchanequationtheinitialstateofthesystemispartofthe

auxiliarydataforawell-posedproblem.

Thearchetypalparabolicevolutionequation

isthe\heatconduction or\di usion equation:

@u

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@2u

=(1-dimensional);

@t@x2ormoregenerally,for >0,

@u

=r·(tru)

@t=tr2u(tconstant),@2

u

=

t(1-D).

@x2

Problemswhicharewell-posedforthe

heatequationwillbewell-posedformoregeneralparabolicequation.

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5.1.1Well-PosedCauchyProblem(InitialValueProblem)Consider >0,

@u= r2uinRn,t>0;

@t

withu=f(x)inRnatt=0,andju|<.

inRn,t>0.

69

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5.1De

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nitygivesruj.=0.Wealsoimposeconditionsonf,

jf(x)j2dx<1vf(x).0asjxj!1.

Rn

Sometimesf(x)hascompactsupport,i.e.f(x)=0outside

some

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niteregion.(E.g.,in1-D,seegraphhereafter.)

u

...+.x

5.1.2Well-PosedInitial-BoundaryValueProblemConsider

anopenboundedregionGofRnand >0;

@u

=tr2uin

,t>0;

@twithu

=f(x)att=0in

,@u

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and u(x;t)+a(x;t)=g(x;t)ontheboundary@.

@nThen,a=0givestheDirichletproblem,.=

0givestheNeumannproblem(@u=@n=0ontheboundaryisthezero-ux

condition)and.60,a=0givestheRobinorradiation

problem.(Theproblemcanalsohavemixedboundaryconditions.)

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IfGisnotbounded(e.g.half-plane),thenadditionalbehavior-at-in

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nityconditionmaybeneeded.

5.1.3TimeIrreversibilityoftheHeatEquationIftheinitialconditionsinawell-posedinitialvalueor

initial-boundaryvalueproblemforanevolutionequationarereplacedbyconditionsonthe

solutionatotherthaninitialtime,theresultingproblemmaynotbewell-posed

(evenwhenthetotalnumberofauxiliaryconditionsisunchanged).

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E.g.thebackwardheatequationin1-Disill-posed;thisproblem,

@u@2u

=tin0<x<

l,0<t<T,

@t@x2withu=

f(x)att=T,x=(0;l),andu(0;t)=u(l,t)=0for

t=(0;T),

whichisto

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ndpreviousstatesu(x,t),(t<T)whichwillhaveevolvedintothestatef(x),hasnosolutionforarbitraryf(x).Evenwhena

thesolutionexists,itdoesnotdependcontinuouslyonthedata.

Theheatequationisirreversibleinthemathematicalsensethatforwardtimeis

distinguishablefrombackwardtime(i.e.itmodelsphysicalprocessesirreversible

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inthesenseoftheSecondLawofThermodynamics).

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Chapter5 ParabolicEquations

5.1.4UniquenessofSolutionforCauchyProblem:The1-Dinitialvalueproblem

@u@2u

=;x=R,t>0;

@t@x2

..

withu=f(x)att=0(x=R),

suchthatjf(x)j2dx<1.

...

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hasauniquesolution.

Proof:

WecanprovetheuniquenessofthesolutionofCauchyproblemusingtheenergymethod.

Supposethatu1andu2aretwoboundedsolutions.Considerw=u1

-u2;thenwsatis

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es

@w@2w

=(...<x<1,t>0);

@t@x2

@w

withw=0at

t=0(...<x<1)and.=0,8t.

@x

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.

Considerthefunctionoftime

.

.

I(t)=

1w2(x,t)dx,such

thatI(0)=0andI(t)~08t(asw2~0);2

...

whichrepresentstheenergyofthefunctionw.

Then,

.

.

.

.

.1

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dI1@w2@w@2w

=

dx=wdx=wdx(fromtheheatequation);

dt

2@t@t@x2

..1...

... ... 2

@w @w

=w-dx(integrationbyparts);

@x@x

...

...

.

. 2.

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@w@w.

=-dx.0since.=0.

@x@x

...

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.

Then,0.I(t).I(0)=0,8t>0,

sincedI=dt<0.So,I(t)=0andw.0

i.e.u1=u2,8t>0.

5.1.5UniquenessofSolutionfor

Initial-BoundaryValueProblem:Similarlywecanmakeuseoftheenergymethodto

provetheuniquenessofthesolutionofthe1-DDirichlet

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orNeumannproblem

@u@2u

=in0<x<l,t>0;

@t@x2withu=f(x)att=0;x

=(0;l),

u(0;t)=g0(t)andu(l,t)=gl(t),8t>0(Dirichlet),

@u@u

or(0;t)=g0(t)and(l,t)=gl(t),8t>0

(Neumann).

@x@x

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5.2FundamentalSolutionoftheHeatEquationSupposethatu1andu2aretwosolutionsandconsiderw=u1-u2;then

wsatis

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es

@w@2w

=(0<x<l,t>0);

@t@x2withw=0att=0(0<x<l),and

w(0;t)=w(l,t)=0,8t>0(Dirichlet),@w@w

or(0;t)=(l,

t)=0,8t>0(Neumann).

@x@x

Considerthefunction

oftime

.l

I(t)=

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1w2(x,t)dx,suchthatI(0)=0andI(t)~08t(asw2~0);2

0

whichrepresentstheenergyofthefunctionw.Then,

.l

.l

dI1@w2@2w

=dx=w;

dt20@t0@x2 l.l 2

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.l 2

@w @w @w

=w-dx=-dx.0.

@x@x@x

000

Then,0.I(t).I(0)=0,

8t>0,

sincedI=dt<0.SoI(t)=08t>andw.

0andu1=u2.

5.2FundamentalSolution

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oftheHeatEquationConsiderthe1-DCauchyproblem,

@u@2u

=on...<x<1,t>0;

@t

@x2withu=f(x)att=0(...<x<1),

..

suchthatjf(x)j2dx<1.

...

Example:Toillustratethetypicalbehaviourofthesolution

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ofthisCauchyproblem,considerthespeci

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ccasewhereu(x,0)=f(x)=exp(..x2);thesolutionis

.2

1x

u(x,t)=

exp-(exercise:checkthis).

(1+4t)1=21+4t

=

Startingwithu(x,0)=exp(..x2)att=

0,thesolutionbecomesu(x,t)=1=2texp(..x2=4t),

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=

=

fortlarge,i.e.theamplitudeofthesolutionscalesas1=tanditswidth

scalesast.

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t=0t=1ut=10x

SpreadingoftheSolution:ThesolutionoftheCauchyproblemforthe

heatequationspreadssuchthatitsintegralremainsconstant:

.

.

Q(t)=udx=constant.

...

Proof:

ConsiderdQdt=.....@u@tdx

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=.....@2u@x2dx(fromequation), @u .=@x...=0(fromconditionsonu).So,Q=constant.

5.2.1IntegralFormoftheGeneralSolutionTo

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ndthegeneralsolutionoftheCauchyproblemwede

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netheFouriertransformofu(x,t)anditsinverseby

.+.

U(k,t)==1u(x,t)e..ikxdx,

2.....+.

u(x,t)==1U(k,

t)eikxdk.

2....

So,theheatequationgives,

.+.

1 @U(k,t)=

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+k2U(k,t)eikxdk=08x,

2....@t

whichimpliesthattheFouriertransformU(k,t)satis

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estheequation

@U(k,t)

+k2U(k,t)=0.@t

Thesolutionofthislinearequationis

U(k,t)=F(k)e..k2t,

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5.2FundamentalSolutionoftheHeatEquationwhereF(k)istheFouriertransformoftheinitialdata,u(x,t=0),

.+.

F(k)==1f(x)e..ikxdx.

2....

.+1

(Thisrequiresjf(x)j2dx<1.)Then,

webacksubstituteU(k,t)intheintegralform

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...

ofu(x,t)to

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nd,

.+..+. .+.

1ikxdk=1..k2tikxdk,

u(x,t)==

F(k)e..k2tef( )e..ik.d.ee

2

2......1...

.+.

.+.

..k2t

=1f( )ee

ik(x.. )dkd .

2.

...

...

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Nowconsider

.+..+. 2

x-.(x- )2

..k2tik(x.. )dk

=

H(x,t, )=eeexp..tk-i-dk,

2t4t

...

...

sincetheexponentsatis

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es

. ". 2.

x-.x-.(x- )2

..k2t+ik(x-

)=..tk2-ik=..tk-i+,

t

2t4t2

=

andsetk-i(x

- )=2t=s/t,withdk=ds,suchthat

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.+..r

2(x- )2.ds

..s

..(x.. )2=4t

H(x,t, )=eexp..==e

;

4t

...tt

.+.

since

e..s2ds==.(seeappendixA).

...

So,

.+.. .+1

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1(x- )2

u(x,t)==f( )exp..d.=K(x- ,t)f( )d.4 t...4t...

Wherethefunction

.2

1x

K(x,

t)==exp-

4 t4tiscalledthefundamentalsolution

orsourcefunction,Green'sfunction,propagator,di usionkernel

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oftheheatequation.

5.2.2PropertiesoftheFundamentalSolutionThefunctionK(x,t)issolution(positive)oftheheatequationfor

t>0(checkthis)andhasasingularityonlyatx=0;t=

0:

=

1.K(x,t).0as

t.0+withx6=0(K=O(1=texp[..1=t])),=

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2.K(x,t).+.ast.0+withx=0(K=O(1=t)),=3.K(x,t).0ast.+.

(K=O(1=t)),..

4.K(x- ,t)d.=1

...

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Atanytimet>0(nomatterhowsmall),thesolutiontotheinitialvalueproblem

fortheheatequationatanarbitrarypointxdependsonallof

theinitialdata,i.e.thedatapropagatewithanin

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nitespeed.(Asaconsequence,theproblemiswellposedonlyifbehaviour-atin

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nityconditionsareimposed.)However,theinuenceoftheinitialstatediesoutveryrapidlywiththedistance(asexp(..r2)).

5.2.3Behaviour

atlargetSupposethattheinitialdatahaveacompactsupport

ordecaystozerosu cientlyquicklyasjxj!.andthatwelookat

thesolutionoftheheatequationonspatialscales,x,

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largecomparedtothespatialscaleofthedata.andattlarge.Thus,weassumetheorderingx2=t=O(1)and 2=t

=O(")wherep.1(sothat,x =t=O("1=2)).Then,the

solution

.+1..x2=4t.+1

1e..x =2td ,

u(x,

t)==f( )e..(x.. )2=4td.==f( )e.. 2=4te4 t...

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4 t...

2=4t.+.2

e..xF(0).x.

'=f( )d.'=exp-

,

4 t...2t4t

whereF(0)istheFourier

transformoffatk=0,i.e.

.+..

+.

p

1f(x)e..ikxdx)

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F(k)==f(x)dx=2 F(0).2.......

=

So,atlarget,onlargespatialscalesxthe

solutionevolvesasu.u0=texp(.. 2)whereu0

=

isaconstantands=x/2tisthedi usionvariable.This

solutionspreadsanddecreasesastincreases.

5.3

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SimilaritySolutionForsomeequations,liketheheatequation,thesolutiondependsonacertaingroupingoftheindependentvariablesratherthandependingon

eachoftheindependentvariablesindependently.Considertheheatequationin1-D

@u@2u

-D=0;

@t@x2

andintroducethedilatationtransformation

.="a

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x,f="btandw( , )="cu("..a ,"..b );p=R.

Thischangeofvariablesgives

@u@w

@f@w@u@w@.@w

="..c="b..c,

="..c="a..c

@t@f@t@f@x@.@x

@.

@2u@2w@.@2w

="a..c=

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"2a..c

and.

@x2@ 2@x@ 2

So,theheatequationtransformsinto

@2@2

@ww

@ww

"b..c-"2a..cD"b..c-"2a..bD

=0i.e.=0;

@f@ 2@f@ 2

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5.3SimilaritySolutionandisinvariantunderthedilatationtransformation(i.e.8")ifb=2a.Thus,ifusolvestheequationatx,

tthenw="..cusolvetheequationatx="..a ,

t="..b .Notealsothatwecanbuildsomegroupingsof

independentvariableswhichareinvariantunderthistransformation,suchas

8/10/2019 m3414_1


"a

.xx

==

a=ba=bta=b

("bt)

=whichde

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nesthedimensionlesssimilarityvariable (x,t)=x/2Dt,sinceb=2a.(s!.

ifx!.ort.0ands

=0ifx=0.)Also,

"c===v( )

wuu

c=bc=btc=b

("bt)

suggests

thatwelookforasolutionoftheheatequation

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oftheformu=tc=2av( ).Indeed,sincetheheatequationisinvariantunderthedilatationtransformation,thenwealsoexpectthesolution

tobeinvariantunderthattransformation.Hence,thepartialderivativesbecome,

@ucc=2a..1c=2a0( )@s1c=2a..1.c.

=tv( )+tv=tv( )- v0( );

@t

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2a@t2a

=since@ =@t=..x=(2t2Dt)=.. =2t,and

@uc=2a@stc=2a..1=2@2utc=2a..100( ):=tv0( )=

=v0( ),=v

@x@x2D@x22D

Then,theheatequationreducestoanODE

=2..1...

tv00( )+ v0( )-v( )=0.(5.1)

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=

t=2

with.=c=a,suchthatu=vands=x/2Dt.So,wemaybeable

tosolvetheheatequationthrough(5.1)ifwecanwritetheauxiliary

conditionsonu,xandtasconditionsonvand .Note

that,ingeneral,theintegraltransformmethodisableto

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dealwithmoregeneralboundaryconditions;ontheotherhand,lookingforsimilaritysolutionpermitstosolveothertypesofproblems(e.g.weaksolutions).

5.3.1In

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niteRegionConsidertheproblem

@u@2u

=Don...<x<1,t>0;

@t@x2withu

=u0att=0;x=R ..;u=0att=0;x=R.

+,andu.u0asx...1;u.0

asx!1,8t>0.

u

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u0t=0

x

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=

Welookforasolutionoftheformu=t=2v( ),where

(x,t)=x/2Dt,suchthatv( )ist=2

solutionofequation

(5.1).Moreover,sinceu=v( ).u0ass...1,where

u0doesnotdependont,.mustbezero.

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Hence,vissolutionofthelinearsecondorderODE

v00( )+ v0( )=0withv.u0ass.

...andv.0ass.+1.Makinguseofthe

integratingfactormethod,

2.

.

2

=2 2=2 2=2

ev00( )+

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sexpv0( )=ev0( )=0vev0( )= 0;

2@s

*

.f

. /2

..s2

v0( )= 0e.. 2

=2vv( )= 0e..h2=2dh+ 1= 2eds+

1.

...

...

Now,applythe

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initialconditionstodeterminetheconstants 2and 1.Ass...1,we

p=

havev= 1=

u0andass!1,v= 2.+u0=0,

so 2=..u0= .Hence,thesolutiontothisCauchyprobleminthe

in

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niteregionis

pp

!.!

. /2.x2/4Dt

..s22

v( )=

u01-edsi.e.u(x,t)=u01-e..sds.

...

...

5.3.2Semi-In

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niteRegionConsidertheproblem

@u@2u

=Don0<x<1,t>0;

@t@x2withu

=0att=0;x=R.

+,@u

and

=..qatx=0,t>0;u.0asx!1,8t>

0.

@x

=

t=2

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Again,welookforasolutionoftheformu=v( ),where (x,t)=x/2Dt,suchthatv( )issolutionof

equation(5.1).However,theboundaryconditionsarenowdi erent

t(..1)=2.

t(..1)=2

@u@s@s

=2.

=tv

0( )==v0( )v.==v0(0)

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=..q,

@x@[email protected]

x=0

sinceqdoesnotdependont,.-1

mustbezero.Hencefromequation(5.1),thefunctionv,

=

suchthatu=vt,issolutionofthelinearsecond

orderODE

=v00( )+ v0( )-v( )

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=0withv0(0)=..q2Dandv.0ass.+1.

Sincethefunctionv.= s

issolutionoftheaboveODE,weseekforsolutionsoftheform

v( )= ( )suchthat

=

v=.+

=andv0==2 =+ 0=.

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Then,back-substituteintheODE

0=

2+ 22

0=+2 =+ 2 =+ .- .

=0i.e.=-=..- .

= s

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5.4MaximumPrinciplesandComparisonTheoremsAfterintegration(integratingfactormethodoranother),weget

.. 2=2.f..s2=2

21 2ee

lnj 0|=..2lns-+

k=ln-+kv == 0v.=

0ds+ 1.

2 22 2

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s2

Anintegrationbypartgives

.#f!.!

..s2=2

.f.. 2=2.f

ee

..s2=2ds..s2=2ds

( )= 0..-e+ 1.=

2+e+ 3.ss

0

Hence,thesolution

becomes,

.f.

.. 2=2

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..s2=2ds

v( )= 2e+se+ 3 ,0

wheretheconstantsofintegration 2and 3aredeterminedby

theinitialconditions:

.f .f

0.. 2=2

..s.. 2=2

..s

v= 2.. e+e2=2ds+ e+

3= 2e2=2ds+ 3,00=

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sothatv0(0)= 3=..q2D.Also

...

ass.+1;v= 2

e..s2=2ds+ 3=0v 2=.. 32;

0

p

.

.

.

.r

2

..h2

..s2=2ds==

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sincee2edh==.

22

00

Thesolutionoftheequationbecomes

*

!

. /2=

.. 2=2

v( )= 2e+s2e..h2dh+ 3 ,0

!

.+. r

=

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..h2

.. 2=2-

= 2e2edh+ 2+ 3;

*

2

/2

r.!

.+1

p

4D

.. 2=2-

=qe2e..h2dh;

p

.

/2

r.

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!

.+1

4Dtx

..x2=4Dt-..h2

u(x,y)=qe=edh:

*

.

Dtx/4Dt

5.4MaximumPrinciplesandComparison

TheoremsLiketheellipticPDEs,theheatequationorparabolicequationsofmost

generalformsatisfyamaximum-minimumprinciple.ConsidertheCauchyproblem,

8/10/2019 m3414_1


@u@2u

=in...<x<1,0.t.T.

@t@x2

andde

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nethetwosetsVandVTas

V=f(x,t)=(..1,+1)×(0;T)g,

andVT

=f(x,t)=(..1,+1)×(0;T]g.

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Lemma:Suppose@u@2u@t-@x2<0inVandu(x,0).M,

thenu(x,t)<MinVT.

Proof:Supposeu(x,

t)achievesamaximuminV,atthepoint(x0;t0).Then,at

thispoint,

@u@u@2u

=0,

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=0and.0.

@t@x@x2

But,@[email protected]@2u=@x2>@u=@t

=0at(x0;t0).Moreover,ifwenowsupposethatthemaximumoccurs

int=Tthen,atthispoint

@u@u@2u

~0,=0and.0;

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@t@x@x2

whichagainleadstoacontradiction.

5.4.1FirstMaximumPrincipleSuppose

@u@2u

...0inVandu(x,0).M,

@t@x2

thenu(x,t).MinVT.

Proof:

Supposethereissomepoint(x0;t0)inVT(0<t

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.T)atwhichu(x0;t0)=M1>M.Putw(x,t)=u(x,t)-(t-t0)pwherep=(M1-M)=t0<

0.Then,

@w@2w@u@2u

-=..-

"<0(informoflemma);

@t@x2@t@x2|{z.

.{..>0

0

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andbylemma,

w(x,t)<maxfw(x,0)}inVT,<M+"t0,

M1-M

<M

+t0,t0

vw(x,t)<M1inVT.

But,w(x0;t0)=u(x0;t0)-(t0-t0)p=u(x0;t0)=M1;since

(x0;t0)=VTwehaveacontradiction.

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5.4MaximumPrinciplesandComparisonTheorems

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AppendixA

2

Integralofe..xinR

Considertheintegrals

.R.+.

..s2

..s2

I(R)=edsandI=eds

00

suchthatI(R).IasR.+1.Then,

.R.R.R.R.

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..x..y..(x2+y..(x2+y

I2(R)=e2dxe2dy=e2)dxdy=e2)dxdy.

0000

R

Sinceitsintegrandispositive,I2(R)isbounded

bythefollowingintegrals

222

..(x+y..(x+y..(x+y

e2)

dxdy<e2)dxdy<e2)

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dxdy,

-

R

+222

where

-:fx=R+;y=R+jx+y=R2}and

+:fx=R+;y=R+jx+y2=2R2g.

y

R..+RRRRp2xRp2Rp2Hence,afterpolarcoordinatestransformation,(x=

dcos ,y=dsin ),withdx

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dy=ddddß

2

andx2+y= 2,thisrelationbecomes

*

. =2

.R

. =2.R2

.. 2

.. 2

deddd <I2(R)<deddd .

0000Put,s=

2sothatds=2dd ,toget

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.R2.2R2

. ....R2.....2R2.

e..sds<I2(R)<e..s

dsi.e.1-e<I2(R)<1-e.

4444

00

81

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TakethelimitR.+.tostatethat=.4.I2..4i.e.I2=.4and

I=.2(I>0).

So,

p

.+..+.

2.2

e

..sds=ve..sds

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