Linear partial differential equations for scientists and engineers

Copyright
......Page 4
Table of Contents
......Page 10
Preface to the Fourth Edition......Page 16
Preface to the Third Edition......Page 20
1.1 Brief Historical Comments......Page 24
1.2 Basic Concepts and Definitions......Page 35
1.3 Mathematical Problems......Page 38
1.4 Linear Operators......Page 39
1.5 Superposition Principle......Page 43
1.6 Exercises......Page 45
2.2 Classification of First-Order Equations......Page 50
2.3 Construction of a First-Order Equation......Page 52
2.4 Geometrical Interpretation of a First-Order Equation
......Page 56
2.5 Method of Characteristics and General Solutions......Page 58
2.6 Canonical Forms of First-Order Linear Equations......Page 72
2.7 Method of Separation of Variables......Page 74
2.8 Exercises......Page 78
3.1 Classical Equations......Page 86
3.2 The Vibrating String......Page 88
3.3 The Vibrating Membrane......Page 90
3.4 Waves in an Elastic Medium......Page 92
3.5 Conduction of Heat in Solids......Page 98
3.6 The Gravitational Potential......Page 99
3.7 Conservation Laws and The Burgers Equation......Page 102
3.8 The Schrödinger and the Korteweg–de Vries Equations
......Page 104
3.9 Exercises......Page 106
4.1 Second-Order Equations in Two Independent Variables
......Page 114
4.2 Canonical Forms......Page 116
(A) Hyperbolic Type......Page 117
(C) Elliptic Type......Page 118
4.3 Equations with Constant Coefficients......Page 122
(A) Hyperbolic Type......Page 123
(B) Parabolic Type......Page 124
(C) Elliptic Type......Page 125
4.4 General Solutions......Page 130
4.5 Summary and Further Simplification......Page 134
4.6 Exercises......Page 136
5.1 The Cauchy Problem......Page 140
5.2 The Cauchy–Kowalewskaya Theorem......Page 143
5.3 Homogeneous Wave Equations......Page 144
(A) Semi-infinite String with a Fixed End......Page 153
(B) Semi-infinite String with a Free End......Page 156
5.5 Equations with Nonhomogeneous Boundary Conditions
......Page 157
5.6 Vibration of Finite String with Fixed Ends......Page 159
5.7 Nonhomogeneous Wave Equations......Page 162
5.8 The Riemann Method......Page 165
5.9 Solution of the Goursat Problem......Page 172
5.10 Spherical Wave Equation......Page 176
5.11 Cylindrical Wave Equation......Page 178
5.12 Exercises......Page 181
6.1 Introduction......Page 190
6.2 Piecewise Continuous Functions and Periodic Functions
......Page 191
6.3 Systems of Orthogonal Functions......Page 193
6.4 Fourier Series......Page 194
6.5 Convergence of Fourier Series......Page 196
6.6 Examples and Applications of Fourier Series......Page 200
6.7 Examples and Applications of Cosine and Sine Fourier Series
......Page 206
6.8 Complex Fourier Series......Page 217
6.9 Fourier Series on an Arbitrary Interval......Page 219
6.10 The Riemann–Lebesgue Lemma and Pointwise Convergence Theorem
......Page 224
6.11 Uniform Convergence, Differentiation, and Integration
......Page 231
6.12 Double Fourier Series......Page 235
6.13 Fourier Integrals......Page 237
6.14 Exercises......Page 243
7.1 Introduction......Page 254
7.2 Separation of Variables......Page 255
7.3 The Vibrating String Problem......Page 258
7.4 Existence and Uniqueness of Solution of the Vibrating String Problem
......Page 266
7.5 The Heat Conduction Problem......Page 271
7.6 Existence and Uniqueness of Solution of the Heat Conduction Problem
......Page 274
7.7 The Laplace and Beam Equations......Page 277
7.8 Nonhomogeneous Problems......Page 281
7.9 Exercises......Page 288
8.1 Sturm–Liouville Systems......Page 296
8.2 Eigenvalues and Eigenfunctions......Page 300
8.3 Eigenfunction Expansions......Page 306
8.4 Convergence in the Mean......Page 307
8.5 Completeness and Parseval’s Equality......Page 309
8.6 Bessel’s Equation and Bessel’s Function......Page 312
8.7 Adjoint Forms and Lagrange Identity......Page 318
8.8 Singular Sturm–Liouville Systems......Page 320
8.9 Legendre’s Equation and Legendre’s Function......Page 325
8.10 Boundary-Value Problems Involving Ordinary Differential Equations
......Page 331
8.11 Green’s Functions for Ordinary Differential Equations
......Page 333
8.12 Construction of Green’s Functions......Page 338
8.13 The Schr¨odinger Equation and Linear Harmonic Oscillator
......Page 340
8.14 Exercises......Page 344
9.1 Boundary-Value Problems......Page 352
1. The First Boundary-Value Problem......Page 353
4. The Fourth Boundary-Value Problem......Page 354
9.2 Maximum and Minimum Principles......Page 355
9.3 Uniqueness and Continuity Theorems......Page 356
1. Interior Problem......Page 357
9.5 Dirichlet Problem for a Circular Annulus......Page 363
9.6 Neumann Problem for a Circle......Page 364
9.7 Dirichlet Problem for a Rectangle......Page 366
9.8 Dirichlet Problem Involving the Poisson Equation......Page 369
9.9 The Neumann Problem for a Rectangle......Page 371
9.10 Exercises......Page 374
10.2 Dirichlet Problem for a Cube......Page 384
10.3 Dirichlet Problem for a Cylinder......Page 386
10.4 Dirichlet Problem for a Sphere......Page 390
10.6 Vibrating Membrane......Page 395
10.7 Heat Flow in a Rectangular Plate......Page 398
10.8 Waves in Three Dimensions......Page 402
10.9 Heat Conduction in a Rectangular Volume......Page 404
10.10 The Schr¨odinger Equation and the Hydrogen Atom
......Page 405
10.11 Method of Eigenfunctions and Vibration of Membrane
......Page 415
10.12 Time-Dependent Boundary-Value Problems......Page 418
10.13 Exercises......Page 421
11.1 Introduction......Page 430
11.2 The Dirac Delta Function......Page 432
11.3 Properties of Green’s Functions......Page 435
(1) Laplace Operator......Page 437
(2) Helmholtz Operator......Page 438
11.5 Dirichlet’s Problem for the Laplace Operator......Page 439
11.6 Dirichlet’s Problem for the Helmholtz Operator......Page 441
11.7 Method of Images......Page 443
11.8 Method of Eigenfunctions......Page 446
11.9 Higher-Dimensional Problems......Page 448
11.10 Neumann Problem......Page 453
11.11 Exercises......Page 456
12.1 Introduction......Page 462
12.2 Fourier Transforms......Page 463
12.3 Properties of Fourier Transforms......Page 467
12.4 Convolution Theorem of the Fourier Transform......Page 471
12.5 The Fourier Transforms of Step and Impulse Functions
......Page 476
12.6 Fourier Sine and Cosine Transforms......Page 479
12.7 Asymptotic Approximation of Integrals by Stationary Phase Method
......Page 481
12.8 Laplace Transforms......Page 483
12.9 Properties of Laplace Transforms......Page 486
12.10 Convolution Theorem of the Laplace Transform......Page 490
12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions
......Page 493
12.12 Hankel Transforms......Page 511
12.13 Properties of Hankel Transforms and Applications
......Page 514
12.14 Mellin Transforms and their Operational Properties
......Page 518
12.15 Finite Fourier Transforms and Applications......Page 522
12.16 Finite Hankel Transforms and Applications......Page 527
12.17 Solution of Fractional Partial Differential Equations
......Page 533
12.18 Exercises......Page 544
13.1 Introduction
......Page 558
13.2 One-Dimensional Wave Equation and Method of Characteristics
......Page 559
13.3 Linear Dispersive Waves......Page 563
13.4 Nonlinear Dispersive Waves and Whitham’s Equations
......Page 568
13.5 Nonlinear Instability......Page 571
13.6 The Traffic Flow Model......Page 572
13.7 Flood Waves in Rivers......Page 575
13.8 Riemann’s Simple Waves of Finite Amplitude......Page 576
13.9 Discontinuous Solutions and Shock Waves......Page 584
13.10 Structure of Shock Waves and Burgers’ Equation......Page 586
13.11 The Korteweg–de Vries Equation and Solitons......Page 596
13.12 The Nonlinear Schr¨odinger Equation and Solitary Waves
......Page 604
13.13 The Lax Pair and the Zakharov and Shabat Scheme
......Page 613
13.14 Exercises......Page 618
14.1 Introduction......Page 624
14.2 Finite Difference Approximations, Convergence, and Stability
......Page 625
14.3 Lax–Wendroff Explicit Method......Page 628
(A) Wave Equation and the Courant–Friedrichs–Lewy Convergence Criterion
......Page 631
(B) Parabolic Equations......Page 635
(C) Elliptic Equations......Page 641
(D) Simultaneous First-Order Equations......Page 645
(A) Parabolic Equations......Page 647
(B) Hyperbolic Equations......Page 650
14.6 Variational Methods and the Euler–Lagrange Equations
......Page 652
(A) Hamilton Principle......Page 659
(B) The Generalized Coordinates, Lagrange Equation, and Hamilton Equation
......Page 661
14.7 The Rayleigh–Ritz Approximation Method......Page 670
14.8 The Galerkin Approximation Method......Page 678
14.9 The Kantorovich Method......Page 682
14.10 The Finite Element Method......Page 686
14.11 Exercises......Page 691
15.1 Fourier Transforms......Page 704
15.2 Fourier Sine Transforms......Page 706
15.3 Fourier Cosine Transforms......Page 708
15.4 Laplace Transforms......Page 710
15.5 Hankel Transforms......Page 714
15.6 Finite Hankel Transforms......Page 718
Answers and Hints to Selected Exercises......Page 720
A-1 Gamma, Beta, Error, and Airy Functions......Page 772
Legendre Duplication Formula......Page 774
A-2 Hermite Polynomials and Weber–Hermite Functions
......Page 780
Bibliography......Page 784
Index......Page 794