Table Of ContentPartial Differential Equations
Thesubjectofpartialdifferentialequations(PDE)hasundergonegreatchangeduringthelast
70yearsorso,afterthedevelopmentofmodernfunctionalanalysis;inparticular,distribution
theoryandSobolevspaces.Inthemodernconcept,thePDEisvisualizedinamoregeneralsetup
of functional analysis, where we look for solutions in a sense weaker than the usual classical
sensetoaddressthemorephysicallyrelevantsolutions.Thoughtheaimofthepresentbookis
tointroducethefundamentaltopicsinaclassicalwayasinanyfirstbookonPDE,theauthors
havedemonstratedthebasictopicsinawaythatopensthedoorstothemoderntheory.Readers
can immediately and naturally sense the importance of studying these topics in a modern
approach. As a lead, after introducing method of characteristics for first order equations, the
authorshaveimmediatelydiscussedtheimportanceofintroducingthenotionofweaksolutions
totwoimportantclassesoffirstorderequations,namelyconservationlawsandHamilton–Jacobi
equations. The implication is that physically relevant solutions cannot be obtained within the
realmofclassicalsolutions.Almostallthechapterscoversomethingaboutmoderntopics.Also
includedaremanyexercisesinmostchapters,whichhelpstudentsgetbetterinsight.Hintsor
answersareprovidedtosomeselectedexercises.
A. K. Nandakumaran is Professor, Department of Mathematics, Indian Institute of
Science, Bangalore. His areas of interest are analysis of partial differential equations, in
particularhomogenizationofPDE,controlandcontrollabilityproblems,inverseproblemsand
computations. He received the Sir C. V. Raman Young Scientist State Award in Mathematics
in 2003. He and P. S. Datti are also co-authors of the book ‘Ordinary Differential Equations:
PrinciplesandApplications’,publishedbytheCambridgeUniversityPressin2017.
P. S. Datti is former faculty at Tata Institute of Fundamental Research (TIFR) Centre for
ApplicableMathematics,Bangalore.Hismainareasofresearchincludenon-linearhyperbolic
equations, hyperbolic conservation laws, ordinary differential equations, evolution equations
andboundarylayerphenomena.HehaswrittenTIFRLectureNotesforthelecturesdelivered
byG.B.Whitham(CalTech)andCathleenMorawetz(CourantInstitute).
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Cambridge-IISc Series
Partial Differential
Equations
Classical Theory with a Modern Touch
A. K. Nandakumaran
P. S. Datti
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©A.K.NandakumaranandP.S.Datti2020
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ToOurTeachers
Contents
ListofIllustrations xi
Preface xiii
Acknowledgments xv
Notations xvii
1 Introduction 1
1.1 GeneralNatureofPDE 1
1.2 TwoExamples 2
1.3 DescriptionoftheContents 4
2 Preliminaries 7
2.1 MultivariableCalculus 7
2.1.1 Introduction 7
2.1.2 Partial,DirectionalandFrechétDerivatives 9
2.1.3 InverseFunctionTheorem 13
2.1.4 ImplicitFunctionTheorem 15
2.2 MultipleIntegralsandDivergenceTheorem 17
2.2.1 MultipleIntegrals 17
2.2.2 Green’sTheorem 30
2.3 SystemsofFirst-OrderOrdinaryDifferentialEquations:Existenceand
UniquenessResults 34
2.4 FourierTransform,ConvolutionandMollifiers 43
2.4.1 Convolution 46
2.4.2 Mollifiers 47
3 First-OrderPartialDifferentialEquations:MethodofCharacteristics 48
3.1 Introduction 48
3.2 LinearEquations 50
3.3 QuasilinearEquations 53
3.4 GeneralFirst-OrderEquationinTwoVariables 63
3.5 First-OrderEquationinSeveralVariables 70
3.5.1 LinearFirst-OrderEquationinSeveralVariables 71
3.5.2 QuasilinearEquationinSeveralVariables 72
3.5.3 GeneralNon-linearEquationinSeveralVariables 74
vii
viii CONTENTS
3.6 Hamilton–JacobiEquation 77
3.7 Notes 84
3.8 Exercises 84
4 Hamilton–JacobiEquation 87
4.1 Hamilton–JacobiEquation 88
4.2 Hopf–LaxFormula 89
4.3 Euler–LagrangeEquations 93
4.4 LegendreTransformation 98
4.5 Notes 107
4.6 Exercises 108
5 ConservationLaws 111
5.1 Introduction 111
5.2 GeneralizedSolutionandRankine–Hugoniot(R–H)Condition 114
5.3 Lax–OleinikFormula 123
5.4 GeneralizedSolutionandUniqueness 128
5.5 RiemannProblem 137
5.6 Notes 139
5.7 Exercises 143
6 ClassificationofSecond-OrderEquations 145
6.1 Introduction 145
6.2 CauchyProblem 145
6.2.1 Non-characteristicCauchyProblem 150
6.3 ClassificationofLinearEquations 152
6.3.1 Second-OrderEquationsinTwoVariables 157
6.4 Higher-OrderLinearEquations 161
6.5 Notes 163
6.6 Exercises 164
7 LaplaceandPoissonEquations 166
7.1 Introduction 166
7.1.1 PhysicalInterpretation 167
7.2 FundamentalSolution,MeanValueFormulaandMaximumPrinciples 168
7.2.1 MeanValueFormula 172
7.2.2 MaximumandMinimumPrinciples 175
7.2.3 UniquenessandRegularityoftheDirichletProblem 178
7.2.4 Green’sFunctionandRepresentationFormula 183
7.2.5 MVPImpliesHarmonicity 188
7.3 ExistenceofSolutionofDirichletProblem(Perron’sMethod) 189
7.4 PoissonEquationandNewtonianPotential 193
7.4.1 HölderContinuousFunctions 197
CONTENTS ix
7.5 HilbertSpaceMethod:WeakSolutions 202
7.5.1 FourierMethod 206
7.6 Notes 209
7.7 Exercises 211
8 HeatEquation 216
8.1 Introduction 216
8.1.1 DerivationofOne-DimensionalHeatEquation 216
8.2 HeatTransferinanUnboundedRod 218
8.2.1 SolutioninHigherDimensions 223
8.2.2 Uniqueness 228
8.2.3 InhomogeneousEquation 230
8.3 MaximumandMinimumPrinciples 231
8.4 HeatEquationonaFiniteInterval:FourierMethod 241
8.4.1 PrescribedNon-zeroBoundaryConditions 243
8.4.2 FreeExchangeofHeatattheEnds 243
8.5 Notes 246
8.6 Exercises 250
9 One-DimensionalWaveEquation 252
9.1 Introduction 252
9.2 CauchyProblemontheLine 253
9.2.1 InhomogeneousEquation:Duhamel’sPrinciple 259
9.2.2 CharacteristicParallelogram 262
9.3 CauchyProbleminaQuadrant(Semi-infiniteString) 263
9.4 WaveEquationinaFiniteInterval 267
9.5 NotionofaWeakSolution 268
9.6 GeneralSecond-OrderEquations 269
9.6.1 AnExample 277
9.7 Notes 279
9.8 Exercises 279
10 WaveEquationinHigherDimensions 281
10.1 Introduction 281
10.2 Three-DimensionalWaveEquation:MethodofSphericalMeans 282
10.2.1 CharacteristicCone:SecondMethod 288
10.3 Two-DimensionalWaveEquation:MethodofDescent 294
10.3.1 TelegraphEquation 296
10.4 WaveEquationforGeneraln 297
10.4.1 SolutionFormulaviaEuler–Poisson–DarbouxEquation 298
10.4.2 AnInversionMethod 303
10.5 MixedorInitialBoundaryValueProblem 308
10.6 GeneralHyperbolicEquationsandSystems 311
