Partial differential equations: an introduction

Walter A. Strauss0471548685, 9780471548683

Covers the fundamental properties of partial differential equations (PDEs) and proven techniques useful in analyzing them. Uses a broad approach to illustrate the rich diversity of phenomena such as vibrations of solids, fluid flow, molecular structure, photon and electron interactions, radiation of electromagnetic waves encompassed by this subject as well as the role PDEs play in modern mathematics, especially geometry and analysis.

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Table of contents :
Cover……Page 1
Title……Page 2
Title page……Page 3
Date-line……Page 4
Preface……Page 5
CONTENTS……Page 7
Title……Page 11
1.1* What Is a Partial Differential Equation?……Page 13
1.2* First-Order Linear Equations……Page 18
1.3* Flows, Vibrations, and Diffusions……Page 22
1.4* Initial and Boundary Conditions……Page 31
1.5 Well-Posed Problems……Page 37
1.6 Types of Second-Order Equations……Page 39
2.1* The Wave Equation……Page 44
2.2* Causality and Energy……Page 49
2.3* The Diffusion Equation……Page 53
2.4* Diffusion on the Whole Line……Page 57
2.5* Comparison of Waves and Diffusions……Page 64
3.1 Diffusion on the Half-Line……Page 67
3.2 Reflections of Waves……Page 71
3.3 Diffusion with a Source……Page 77
3.4 Waves with a Source……Page 81
3.5 Diffusion Revisited……Page 90
4.1* Separation of Variables, the Dirichlet Condition……Page 94
4.2* The Neumann Condition……Page 99
4.3* The Robin Condition……Page 102
5.1* The Coefficients……Page 113
5.2* Even, Odd, Periodic, and Complex Functions……Page 121
5.3* Orthogonality and General Fourier Series……Page 126
5.4* Completeness……Page 132
5.5 Completeness and the Gibbs Phenomenon……Page 144
5.6 Inhomogeneous Boundary Conditions……Page 152
6.1 * Laplace’s Equation……Page 158
6.2* Rectangles and Cubes……Page 167
6.3* Poisson’s Formula……Page 171
6.4 Circles, Wedges, and Annuli……Page 176
7.1 Green’s First Identity……Page 181
7.2 Green’s Second Identity……Page 188
7.3 Green’s Functions……Page 190
7.4 Half-Space and Sphere……Page 193
8.1 Opportunities and Dangers……Page 201
8.2 Approximations of Diffusions……Page 205
8.3 Approximations of Waves……Page 213
8.4 Approximations of Laplace’s Equation……Page 220
8.5 Finite Element Method……Page 224
9.1 Energy and Causality……Page 228
9.2 The Wave Equation in Space-Time……Page 234
9.3 Rays, Singularities, and Sources……Page 241
9.4 The Diffusion and Schrodinger Equations……Page 247
9.5 The Hydrogen Atom……Page 253
10.1 Fourier’s Method, Revisited……Page 257
10.2 Vibrations of a Drumhead……Page 263
10.3 Solid Vibrations in a Ball……Page 269
10.4 Nodes……Page 276
10.5 Bessel Functions……Page 280
10.6 Legendre Functions……Page 286
10.7 Angular Momentum in Quantum Mechanics……Page 291
11.1 The Eigenvalues Are Minima of the Potential Energy……Page 295
11.2 Computation of Eigenvalues……Page 300
11.3 Completeness……Page 305
11.4 Symmetric Differential Operators……Page 309
11.5 Completeness and Separation of Variables……Page 313
11.6 Asymptotics of the Eigenvalues……Page 316
12.1 Distributions……Page 326
12.2 Green’s Functions, Revisited……Page 332
12.3 Fourier Transforms……Page 337
12.4 Source Functions……Page 342
12.5 Laplace Transform Techniques……Page 346
13.1 Electromagnetism……Page 351
13.2 Fluids and Acoustics……Page 354
13.3 Scattering……Page 358
13.4 Continuous Spectrum……Page 362
13.5 Equations of Elementary Particles……Page 365
14.1 ShockWaves……Page 371
14.2 Solitons……Page 379
14.3 Calculus of Variations……Page 386
14.4 Bifurcation Theory……Page 391
A.1 Continuous and Differentiate Functions……Page 396
A.2 Infinite Series of Functions……Page 400
A.3 Differentiation and Integration……Page 402
A.4 Differential Equations……Page 405
A.5 The Gamma Function……Page 407
References……Page 409
Answers and Hints to Selected Exercises……Page 412
Index……Page 427