Partial Differential Equations and Boundary-value Problems with Applications

Cover......Page 1
Title: Partial DifferentialEquations andBoundary-Value Problems withApplications, Third Edition......Page 2
ISBN 978-0-8218-6889-8......Page 4
Preface......Page 6
Contents......Page 10
0.1.1. What is a partial differential equation?......Page 16
EXERCISES 0.1.1......Page 17
0.1.2. Superposition principle and subtraction principle......Page 18
0.1.3. Sources of PDEs in classical physics......Page 19
0.1.4. The one-dimensional heat equation......Page 20
0.1.5. Classification of second-order PDEs.......Page 24
0.2.1. What is a separated solution?......Page 25
0.2.2. 稴eparated solutions of Laplace's equation......Page 26
0.2.3. Real and complex separated solutions.......Page 28
EXERCISES 0.2.3......Page 32
0.2.4. Separated solutions with boundary conditions......Page 33
EXERCISES 0.2.4......Page 35
0.3.1. Inner product space of functions......Page 36
0.3.2. Projection of a function onto an orthogonal set.......Page 39
0.3.3. Orthonormal sets of functions.......Page 43
0.3.4. Parseval's equality, completeness, and mean square convergence......Page 44
0.3.5. Weighted inner product......Page 45
0.3.6. Gram-Schmidt orthogonalization......Page 46
0.3.7. Complex inner product......Page 47
EXERCISES 0.3......Page 48
1.1.1. Orthogonality relations......Page 50
1.1.2. Definition of Fourier coefficients......Page 51
1.1.3. Even functions and odd functions......Page 52
1.1.5. Implementation with Mathematica......Page 56
1.1.6. Fourier sine and cosine series......Page 57
EXERCISES 1.1......Page 59
1.2. Convergence of Fourier Series!......Page 61
1.2.1. Piecewise smooth func......Page 62
1.2.2. Dirichlet kernel.......Page 66
1.2.3. Proof of convergence......Page 67
EXERCISES 1.2......Page 69
1.3.1. Example of Gibbs overshoot......Page 73
1.3.2. Implementation with Mathematica......Page 76
1.3.4. Two criteria for uniform convergence......Page 79
1.3.5. Differentiation of Fourier series.......Page 80
1.3.6. Integration of Fourier series.......Page 81
1.3.7. A continuous function with a divergent Fourier series......Page 82
EXERCISES 1.3......Page 84
1.4.1. Statement and proof of Parseval's theorem.......Page 86
1.4.2. Application to mean square error......Page 87
1.4.3. Application to the isoperimetric theorem......Page 89
EXERCISES 1.4......Page 90
1.5.1. Fourier series and Fourier coefficients......Page 93
1.5.3. Applications and examples......Page 94
1.5.4. Fourier series of mass distributions.......Page 96
EXERCISES 1.5......Page 98
1.6. Sturm-Liouville Eigenvalue Problems......Page 99
1.6.1. Examples of Sturm-Liouville eigenvalue problems......Page 100
1.6.2. Some general properties of S-L eigenvalue problems......Page 101
1.6.3. Example of transcendental eigenvalues......Page 102
1.6.4. Further properties: completeness and positivity......Page 104
1.6.5. General Sturm-Liouville problems......Page 107
1.6.6. Complex-valued eigenfunctions and eigenvalues.......Page 110
EXERCISES 1.6......Page 111
2.1.1. Fourier's law of heat conduction......Page 114
2.1.2. Derivation of the heat equation.......Page 115
2.1.3. Boundary conditions......Page 116
2.1.4. Steady-state solutions in a slab.......Page 117
2.1.5. Time-periodic solutions......Page 118
2.1.6. Applications to geophysics......Page 120
2.1.7. Implementation with Mathematica......Page 121
EXERCISES 2.1......Page 123
2.2.1. Separated solutions with boundary conditions......Page 125
2.2.2. Solution of the initial-value problem in a slab.......Page 127
2.2.3. Asymptotic behavior and relaxation time......Page 128
2.2.4. Uniqueness of solutions.......Page 129
2.2.5. Examples of transcendental eigenvalues......Page 131
EXERCISES 2.2......Page 135
2.3. Nonhomogeneous Boundary Conditions......Page 136
2.3.2. Five-stage method of solution......Page 137
2.3.3. Temporally nonhomogeneous problems......Page 145
EXERCISES 2.3......Page 148
2.4.1: Derivation of the equation.......Page 149
2.4.2. Linearized model......Page 152
2.4.3. Motion of the plucked string......Page 153
2.4.4. Acoustic interpretation......Page 155
2.4.5. Explicit (d' Alembert) representation......Page 156
2.4.6. Motion of the struck string......Page 160
2.4.7. d'Alembert's general solution......Page 161
2.4.8. Vibrating string with external forcing......Page 163
EXERCISES 2.4......Page 165
2.5. Applications of Multiple Fourier Series......Page 167
2.5.1. The heat equation (homogeneous boundary conditions)......Page 168
2.5.2. Laplace's equation......Page 170
2.5.3. The heat equation (nonhomogeneous boundary conditions)......Page 172
2.5.4. The wave equation (nodal lines)......Page 174
2.5.5. Multiplicities of the eigenvalues......Page 177
2.5.6. Implementation with Mathematica......Page 179
2.5.7. Application to Poisson '8 equation......Page 180
EXERCISES 2.5......Page 183
3.1.1. Laplacian in cylindrical coordinates.......Page 186
3.1.2. Separated solutions of Laplace's equation in $\rho$, $\phi$.......Page 188
3.1.3. Application to boundary-value problems......Page 189
3.1.5. Uniqueness of solutions......Page 192
3.1.7. Wedge domains.......Page 193
3.1.9. Explicit representation by Poisson's formula......Page 194
EXERCISES 3.1......Page 196
3.2.1. Bessel's equation......Page 198
3.2.2. The power series solution of Bessel's equation......Page 199
3.2.3. Integral representation of Bessel functions......Page 203
3.2.4. The second solution of Bessel's equation......Page 206
3.2.5. Zeros of the Bessel function Jo......Page 207
3.2.6. Asymptotic behavior and zeros of Bessel functions......Page 208
3.2.7. Fourier-Bessel series......Page 212
3.2.8. Implementation with Mathematica......Page 217
EXERCISES 3.2......Page 222
3.3.1. Wave equation in polar coordinates.......Page 224
3.3.2. Solution of initial-value problems......Page 226
3.3.3. Implementation with Mathematica......Page 229
3.4. Heat Flow in the Infinite Cylinder......Page 231
3.4.2. Initial-value problems in a cylinder.......Page 232
3.4.3. Initial-value problems between two cylinders......Page 236
3.4.5. Time-periodic heat flow in the cylinder......Page 239
EXERCISES 3.4......Page 241
3.5.1. Separated solutions......Page 242
3.5.2. Solution of Laplace's equation......Page 243
3.5.3. Solutions of the heat equation with zero boundary conditions.......Page 246
3.5.4. General initial-value problems for the heat equation......Page 247
EXERCISES 3.5......Page 248
4.1. Spherically Symmetric Solutions......Page 250
4.1.1. Laplacian in spherical coordinates......Page 251
4.1.2. Time-periodic heat flow: Applications to geophysics......Page 252
4.1.3. Initial-value problem for heat flow in a sphere......Page 255
4.1.4. The three-dimensional wave equation......Page 262
4.1.5. Convergence of series in three dimensions......Page 264
EXERCISES 4.1......Page 265
4.2.1. Separated solutions in spherical coordinates......Page 266
4.2.2. Legendre polynomials......Page 268
4.2.3. Legendre polynomial expansions......Page 273
4.2.4. Implementation with Mathematica......Page 274
4.2.5. Associated Legendre functions......Page 276
4.2.6. Spherical Bessel functions......Page 278
EXERCISES 4.2......Page 281
4.3. Laplace's Equation in Spherical Coordinates......Page 282
4.3.1. Boundary-value problems in a sphere......Page 283
4.3.2. Boundary-value problems exterior to a sphere......Page 284
4.3.3. Applications to potential theory.......Page 287
EXERCISES 4.3......Page 290
5.1.1. Passage from Fourier series to Fourier integrals......Page 292
5.1.2. Definition and properties of the Fourier transform......Page 294
5.1.3. Fourier sine and cosine transforms......Page 300
5.1.4. Generalized h-transform......Page 302
5.1.5. Fourier transforms in several variables......Page 303
5.1.6. The uncertainty principle......Page 304
5.1.7. Proof of convergence......Page 306
EXERCISES 5.1......Page 307
5.2.1. First method: Fourier series and passage to the limit......Page 309
5.2.2. Second method: Direct solution by Fourier transform......Page 310
5.2.3. Verification of the solution.......Page 311
5.2.4. Explicit representation by the Gauss-Weierstrass kernel......Page 312
5.2.5. Some explicit formulas......Page 315
5.2.6. Solutions on a half-line: The method of images......Page 318
EXERCISES 5.2.6......Page 323
5.2.6.1. Dirichlet boundary condition at x = O.......Page 319
5.2.6.2. Neumann boundary condition at x = O......Page 320
5.2.6.3. Mixed boundary condition at x = O......Page 321
5.2.7. The Black-Scholes model......Page 325
5.2.7.1. Transformation to the heat equation.......Page 326
5.2.7.2. Solution of the Black-Scholes equation.......Page 328
5.2.8. Hermite polynomials.......Page 329
EXERCISES 5.2.8......Page 332
5.3.1. One-dimensional wave equation and d'Alembert's formula......Page 333
5.3.2. General solution of the wave equation.......Page 336
5.3.3. Three-dimensional wave equation and Huygens' principle......Page 338
5.3.4. Extended validity of t he explicit representation......Page 342
5.3.5. Application to one- and two-dimensional wave equations......Page 344
5.3.6. Laplace's equation in a half-space: Poisson's formula......Page 347
EXERCISES 5.3......Page 349
5.4. Solution of the Telegraph Equation......Page 350
5.4.1. Fourier representation of the solution.......Page 351
5.4.2. Uniqueness of the solution......Page 353
5.4.3. Time-periodic solutions of the telegraph equation......Page 354
EXERCISES 5.4......Page 356
Appendix 5A: Derivation of the telegraph equation......Page 357
6.1. Asymptotic Analysis of the Factorial Function......Page 360
6.1.1. Geometric mean approximation: Analysis by logarithms......Page 361
6.1.2. Refined method using functional equations......Page 362
6.1.3. Stirling's formula via an integral representation......Page 363
EXERCISES 6.1......Page 364
6.2. Integration by Parts......Page 365
6.2.1. Two applications......Page 366
6.3.1. Statement and proof of the result......Page 369
6.3.2. Three applications to integral......Page 372
6.3.3. Applications to the heat equation......Page 373
6.3.4. Improved error with gaussian approximation......Page 374
EXERCISES 6.3......Page 376
6.4. The Method of Stationary Phase......Page 377
6.4.1. Statement of the result......Page 378
6.4.3. Proof of the method of stationary phase......Page 379
EXERCISES 6.4......Page 382
6.5.1. Extension of integration by parts.......Page 383
6.5.2. Extension of Laplace's method......Page 384
6.6. Asymptotic Analysis of the Telegraph Equation......Page 386
6.6.2. Asymptotic behavior in case a > o.......Page 387
6.6.3. Asymptotic behavior in case a. < O......Page 389
EXERCISES 6.6......Page 391
7.1. Numerical Analysis of Ordinary Differential Equations......Page 393
7.1.1. The Euler method.......Page 394
EXERCISES 7.1.1......Page 396
7.1.2. The Heun method......Page 398
EXERCISES 7.1.2......Page 399
7.1.3. Error analysis......Page 401
EXERCISES 7.1.3......Page 404
7.2.1. Formulation of a difference equation.......Page 407
7.2.2. Computational molec......Page 408
7.2.3. Examples and comparison with the Fourier method......Page 409
7.2.4. Stability analysis......Page 412
7.2.5. Other boundary conditions......Page 413
EXERCISES 7.2......Page 416
7.3. Equations in Several Dimensions......Page 417
7.3.1. Heat equation in a triangular region......Page 418
7.3.2. Laplace's equation in a triangular region......Page 419
EXERCISES 7.3......Page 422
7.4.1. Variational formulation of Poisson's equation......Page 423
7.4.3. Variational formulation of eigenvalue probems......Page 425
7.4.4. Variational problems, minimization, and critical points.......Page 426
EXERCISES 7.4......Page 427
7.5. Approximate Methods of Ritz and Kantorovich......Page 429
7.5.1. The Ritz method: Rectangular regions......Page 430
7.5.2. The Kantorovich method: Rectangular regions......Page 431
EXERCISES 7.5......Page 433
7.6.1. The Galerkin method: Rectangular regions.......Page 434
7.6.2. Nonrectangular regions.......Page 436
7.6.3. The finite element method.......Page 439
8.1.1. An example......Page 442
8.1.2. The generic case......Page 444
8.1.3. The exceptional case: Modified Green's function......Page 446
8.1.4. The Fredholm alternative.......Page 447
8.2. The Three-Dimensional Poisson Equation......Page 448
8.2.1. Newtonian potential kernel......Page 449
8.2.2. Single- and double-layer potentials......Page 451
8.2.3. Green '8 function of a bounded region......Page 452
8.2.4. Solution of the Dirichlet problem.......Page 455
EXERCISES 8.2......Page 456
8.3.1. The logarithmic potential......Page 458
8.3.2. Green's function of a bounded plane region......Page 459
8.3.3. Solution of the Dirichlet problem......Page 460
8.3.4. Green's functions and separation of variables......Page 461
EXERCISES 8.3......Page 463
8.4.1. Nonhomogeneous heat equation......Page 465
8.4.2. The one-dimensional heat kernel and the method of images.......Page 467
8.5.1. Derivation of the retarded potential......Page 469
8.5.2. Green's function for the Helmholtz equation......Page 474
8.5.3. Application to the telegraph equation......Page 476
EXERCISES 8.5......Page 477
A.l.1. First-order linear equations......Page 480
A.1.2. Second-order linear equations......Page 481
A.I.3. Second-order linear equations with constant coefficients......Page 483
A.1.4. Euler's equidimensional equation.......Page 485
A.I.5. Power series solutions......Page 486
A.1.6. Steady state and relaxation time......Page 489
EXERCISES A.l......Page 490
A.2.1. Numerical series.......Page 491
A.2.2. Taylor's theorem......Page 493
A.2.3. Series of functions: Uniform convergence.......Page 495
A.2.4. Abel's lemma......Page 498
A.2.5. Double series......Page 499
A.2.6. Big-O notation.......Page 500
EXERCISES A.2......Page 501
A.3. Review of Vector Integral Calculus......Page 504
A.3.1. Implementation with Mathematica.......Page 506
A.4.2. The notebook front end.......Page 507
A.4.2.4. Copying input and output from above......Page 508
A.4.2.9. Graphics......Page 509
A.4.4.2. Defining functions......Page 510
A.4.4.5. Lists.......Page 511
A.4.4. 7. Differentiation......Page 512
A.4.4.9. Integration......Page 513
A.4.4.10. Simplifying expressions......Page 514
EXERCISES A.4......Page 515
ANSWERS TO SELECTED EXERCISES......Page 518
INDEX......Page 536
List of Titles Published in This Series......Page 542
Back Cover......Page 543