Sachdev P.L. Self-similarity and beyond.. exact solutions of nonlinear problems (CRC, 2000)(ISBN 1584882115)(315s)_PNc_.pdf

Preface

Ihaveenjoyedﬁndingexactsolutionsofnonlinearproblemsforseveraldecades.

Ihavealsohadthepleasureofassociationofalargenumberofstudents,postdoc-

toralfellows,andothercolleagues,frombothIndiaandabroad,inthispursuit.

Thepresentmonographisanattempttoputdownsomeofthisexperience.

Nonlinearproblemsposeachallengethatisoftendiculttoresist;eachnew

exactsolutionisathingofjoy.

InthewritingofthisbookIhavebeenmuchhelpedbymycolleagueProfessor

V.Philip,andformerstudentsDr.B.MayilVagananandDr.Ch.Srinivasa

Rao. IamparticularlyindebtedtoDr.Raoforhisunstintinghelpinthe

preparationofthemanuscript.Mr.Renugopal,withconsiderablepatienceand

care,putitinLaTexform.

Mywife,Rita,providedinvaluablesupport,care,andcomfortasshehas

doneinmyearlierendeavours.Oursons,DeepakandAnurag,eachcontributed

intheirownways.

IamgratefultotheCouncilofScientiﬁcandIndustrialResearch,India

forﬁnancialsupport. IalsowishtothankDr.SunilNair,Commissioning

Editor,Chapman&Hall,CRCPress,forhispromptactioninseeingthisproject

through.

P.L.Sachdev

Contents

1Introduction

2First-OrderPartialDierentialEquations

2.1LinearPartialDierentialEquationsofFirstOrder

2.2QuasilinearPartialDierentialEquationsofFirstOrder

2.3Reductionofu t +u n u x +H(x,t,u)=0

2.4InitialValueProblemforu t +g(u)u x +h(u)=0

2.5InitialValueProblemforu t +u u x +u =0

3ExactSimilaritySolutionsofNonlinearPDEs

3.1ReductionofPDEsbyInﬁnitesimalTransformations

3.2SystemsofPartialDierentialEquations

3.3Self-SimilarSolutionsoftheSecondKind

3.4Introduction

3.5ANonlinearHeatEquationinThreeDimensions

3.6SimilaritySolutionofBurgersEquationbytheDirectMethod

3.7ExactFreeSurfaceFlowsforShallow-WaterEquations

3.8AnExamplefromGasdynamics

4ExactTravellingWaveSolutions

4.1TravellingWaveSolutions

4.2SimpleWavesin1-DGasdynamics

4.3ElementaryNonlinearDiusiveTravellingWaves

4.4TravellingWavesforHigher-OrderDiusiveSystems

4.5MultidimensionalHomogeneousPartialDierentialEquations

4.6SystemsofNonhomogeneousPartialDierentialEquations

4.7ExactHydromagneticTravellingWaves

4.8ExactSimpleWavesonShearFlows

5ExactLinearisationofNonlinearPDEs

5.1Introduction

5.2CommentsontheSolutionofLinearPDEs

5.3BurgersEquationinOneandHigherDimensions

5.4NonlinearDegenerateDiusionEquationu t =[f(u)u − x ] x

5.5MotionofCompressibleIsentropicGasintheHodographPlane

5.6TheBorn-InfeldEquation

5.7WaterWavesupaUniformlySlopingBeach

5.8SimpleWavesonShearFlows

5.9C-IntegrableNonlinearPDEs

6NonlinearisationandEmbeddingofSpecialSolutions

6.1Introduction

6.2GeneralisedBurgersEquations

6.3BurgersEquationinCylindricalCoordinateswithAxisym-

metry

6.4NonplanarBurgersEquation–ACompositeSolution

6.5ModiﬁedBurgersEquation

6.6EmbeddingofSimilaritySolutioninaLargerClass

7AsymptoticSolutionsbyBalancingArguments

7.1AsymptoticSolutionbyBalancingArguments

7.2NonplanarBurgersEquation

7.3One-DimensionalContaminantTransportthroughPorous

Media

8SeriesSolutionsofNonlinearPDEs

8.1Introduction

8.2AnalysisofExpansionofaGasSphere(Cylinder)intoVacuum

8.3CollapseofaSphericalorCylindricalCavity

8.4ConvergingShockWavefromaSphericalorCylindricalPiston

R eferences

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: