偏微分方程 第4版PDF电子书下载

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Chapter 1 The Single First-Order Equation 1

1.Introduction 1

2.Examples 2

3.Analytic Solution and Approximation Methods in a Simple Example 4

Problems 8

4.Quasi-linear Equations 9

5.The Cauchy Problem for the Quasi-linear Equation 11

6.Examples 15

Problems 18

7.The General First-Order Equation for a Function of Two Variables 19

8.The Cauchy Problem 24

9.Solutions Generated as Envelopes 29

Problems 31

Chapter 2 Second-Order Equations：Hyperbolic Equations for Functions of Two Independent Variables 33

1.Characteristics for Linear and Quasi-linear Second-order Equations 33

2.Propagation of Singularities 35

3.The Linear Second-Order Equation 37

Problems 39

4.The One-Dimensional Wave Equation 40

Problems 45

5.Systems of First-Order Equations 46

6.A Quasi-linear System and Simple Waves 52

Problem 53

Chapter 3 Characteristic Manifolds and the Cauchy Problem 54

1.Notation of Laurent Schwartz 54

Problems 55

2.The Cauchy Problem 56

Problems 61

3.Real Analytic Functions and the Cauchy-Kowalevski Theorem 61

（a） Multiple infinite series 62

Problems 63

（b） Real analytic functions 64

Problems 69

（c） Analytic and real analytic functions 70

Problems 72

（d） The proof of the Cauchy-Kowalevski theorem 73

Problems 78

4.The Lagrange-Green Identity 79

5.The Uniqueness Theorem of Holmgren 80

Problems 88

6.Distribution Solutions 89

Problems 92

Chapter 4 The Laplace Equation 94

1.Green’s Identity，Fundamental Solutions，and Poisson’s Equation 94

Problems 101

2.The Maximum Principle 103

Problems 105

3.The Dirichlet Problem，Green’s Function，and Poisson’s Formula 106

Problems 110

4.Proof of Existence of Solutions for the Dirichlet Problem Using Subharmonic Functions（“Perron’s Method”） 111

Problems 116

5.Solution of the Dirichlet Problem by Hilbert-Space Methods 117

Problems 125

Chapter 5 Hyperbolic Equations in Higher Dimensions 126

1.The Wave Equation in n-Dimensional Space 126

（a） The method of spherical means 126

Problems 132

（b） Hadamard’s method of descent 133

Problems 134

（c） Duhamel’s principle and the general Cauchy problem 135

Problem 139

（d） Initial-boundary-value problems（“Mixed” problems） 139

Problems 142

2.Higher-Order Hyperbolic Equations with Constant Coefficients 143

（a） Standard form of the initial-value problem 143

Problem 145

（b） Solution by Fourier transformation 145

Problems 156

（c） Solution of a mixed problem by Fourier transformation 157

（d） The method of plane waves 158

Problems 161

3.Symmetric Hyperbolic Systems 163

（a） The basic energy inequality 163

Problems 169

（b） Existence of solutions by the method of finite differences 172

Problems 181

（c） Existence of solutions by the method of approximation by analytic functions（Method of Schauder） 182

Chapter 6 Higher-Order Elliptic Equations with Constant Coefficients 185

1.The Fundamental Solution for Odd n 186

Problems 188

2.The Dirichlet Problem 190

Problems 195

3.More on the Hilbert Space Hμ and the Assumption of Boundary Values in the Dirichlet Problem 198

Problems 201

Chapter 7 Parabolic Equations 206

1.The Heat Equation 206

（a） The initial-value problem 206

Problems 213

（b） Maximum principle，uniqueness，and regularity 215

Problem 220

（c） A mixed problem 220

Problems 221

（d） Non-negative solutions 222

Problems 226

2.The Initial-Value Problem for General Second-Order Linear Parabolic Equations 227

（a） The method of finite differences and the maximum principle 227

（b） Existence of solutions of the initial-value problem 231

Problems 233

Chapter 8 H.Lewy’s Example of a Linear Equation without Solutions 235

Problems 239

Bibliography 241

Glossary 243

Index 245

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