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An Introduction to Partial Differential Equations

Publisher: 
Cambridge University Press
Number of Pages: 
370
Price: 
45.00
ISBN: 
0-521-61323-X
Date Received: 
Friday, September 16, 2005
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Yehuda Pinchover and Jacob Rubinstein
Publication Date: 
2005
Format: 
Paperback
Audience: 
Category: 
Textbook

Preface page xi

1 Introduction 1

1.1 Preliminaries 1

1.2 Classification 3

1.3 Differential operators and the superposition principle 3

1.4 Differential equations as mathematical models 4

1.5 Associated conditions 17

1.6 Simple examples 20

1.7 Exercises 21

2 First-order equations 23

2.1 Introduction 23

2.2 Quasilinear equations 24

2.3 The method of characteristics 25

2.4 Examples of the characteristics method 30

2.5 The existence and uniqueness theorem 36

2.6 The Lagrange method 39

2.7 Conservation laws and shock waves 41

2.8 The eikonal equation 50

2.9 General nonlinear equations 52

2.10 Exercises 58

3 Second-order linear equations in two indenpendent

variables 64

3.1 Introduction 64

3.2 Classification 64

3.3 Canonical form of hyperbolic equations 67

3.4 Canonical form of parabolic equations 69

3.5 Canonical form of elliptic equations 70

3.6 Exercises 73

vii

© Cambridge University Press www.cambridge.org

Cambridge University Press

0521848865 - An Introduction to Partial Differential Equations

Yehuda Pinchover and Jacob Rubinstein

Table of Contents

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viii Contents

4 The one-dimensional wave equation 76

4.1 Introduction 76

4.2 Canonical form and general solution 76

4.3 The Cauchy problem and d’Alembert’s formula 78

4.4 Domain of dependence and region of influence 82

4.5 The Cauchy problem for the nonhomogeneous wave equation 87

4.6 Exercises 93

5 The method of separation of variables 98

5.1 Introduction 98

5.2 Heat equation: homogeneous boundary condition 99

5.3 Separation of variables for the wave equation 109

5.4 Separation of variables for nonhomogeneous equations 114

5.5 The energy method and uniqueness 116

5.6 Further applications of the heat equation 119

5.7 Exercises 124

6 Sturm–Liouville problems and eigenfunction expansions 130

6.1 Introduction 130

6.2 The Sturm–Liouville problem 133

6.3 Inner product spaces and orthonormal systems 136

6.4 The basic properties of Sturm–Liouville eigenfunctions

and eigenvalues 141

6.5 Nonhomogeneous equations 159

6.6 Nonhomogeneous boundary conditions 164

6.7 Exercises 168

7 Elliptic equations 173

7.1 Introduction 173

7.2 Basic properties of elliptic problems 173

7.3 The maximum principle 178

7.4 Applications of the maximum principle 181

7.5 Green’s identities 182

7.6 The maximum principle for the heat equation 184

7.7 Separation of variables for elliptic problems 187

7.8 Poisson’s formula 201

7.9 Exercises 204

8 Green’s functions and integral representations 208

8.1 Introduction 208

8.2 Green’s function for Dirichlet problem in the plane 209

8.3 Neumann’s function in the plane 219

8.4 The heat kernel 221

8.5 Exercises 223

© Cambridge University Press www.cambridge.org

Cambridge University Press

0521848865 - An Introduction to Partial Differential Equations

Yehuda Pinchover and Jacob Rubinstein

Table of Contents

More information

Contents ix

9 Equations in high dimensions 226

9.1 Introduction 226

9.2 First-order equations 226

9.3 Classification of second-order equations 228

9.4 The wave equation in R2 and R3 234

9.5 The eigenvalue problem for the Laplace equation 242

9.6 Separation of variables for the heat equation 258

9.7 Separation of variables for the wave equation 259

9.8 Separation of variables for the Laplace equation 261

9.9 Schr¨odinger equation for the hydrogen atom 263

9.10 Musical instruments 266

9.11 Green’s functions in higher dimensions 269

9.12 Heat kernel in higher dimensions 275

9.13 Exercises 279

10 Variational methods 282

10.1 Calculus of variations 282

10.2 Function spaces and weak formulation 296

10.3 Exercises 306

11 Numerical methods 309

11.1 Introduction 309

11.2 Finite differences 311

11.3 The heat equation: explicit and implicit schemes, stability,

consistency and convergence 312

11.4 Laplace equation 318

11.5 The wave equation 322

11.6 Numerical solutions of large linear algebraic systems 324

11.7 The finite elements method 329

11.8 Exercises 334

12 Solutions of odd-numbered problems 337

A.1 Trigonometric formulas 361

A.2 Integration formulas 362

A.3 Elementary ODEs 362

A.4 Differential operators in polar coordinates 363

A.5 Differential operators in spherical coordinates 363

References 364

Index 366

Publish Book: 
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