Nonlinear Partial Differential Equations for Scientists and Engineers, Second Edition

Cover......Page 1
Nonlinear Partial Differential Equations for Scientists and Engineers, Second Edition......Page 4
Copyright - ISBN: 0817643230......Page 5
Contents......Page 8
Preface to the Second Edition......Page 14
Preface to the First Edition......Page 16
1.1 Introduction......Page 22
1.2 Basic Concepts and Definitions......Page 23
1.3 The Linear Superposition Principle......Page 26
1.4 Some Important Classical Linear Model Equations......Page 29
1.5 Second-Order Linear Equations and Method of Characteristics......Page 32
1.6 The Method of Separation of Variables......Page 43
1.7 Fourier Transforms and Initial Boundary-Value Problems......Page 56
1.8 Multiple Fourier Transforms and Partial Differential Equations......Page 69
1.9 Laplace Transforms and Initial Boundary-Value Problems......Page 74
1.10 Hankel Transforms and Initial Boundary-Value Problems......Page 84
1.11 Green’s Functions and Boundary-Value Problems......Page 92
1.12 Sturm–Liouville Systems and Some General Results......Page 105
1.13 Energy Integrals and Higher Dimensional Equations......Page 124
1.14 Fractional Partial Differential Equations......Page 138
1.15 Exercises......Page 149
2.1 Introduction......Page 170
2.3 Some Nonlinear Model Equations......Page 171
2.4 Variational Principles and the Euler–Lagrange Equations......Page 177
2.5 The Variational Principle for Nonlinear Klein–Gordon Equations......Page 190
2.6 The Variational Principle for Nonlinear Water Waves......Page 191
2.7 The Euler Equation of Motion and Water Wave Problems......Page 193
2.8 Exercises......Page 206
3.1 Introduction......Page 214
3.2 The Classification of First-Order Equations......Page 215
3.3 The Construction of a First-Order Equation......Page 216
3.4 The Geometrical Interpretation of a First-Order Equation......Page 219
3.5 The Method of Characteristics and General Solutions......Page 222
3.6 Exercises......Page 236
4.1 Introduction......Page 242
4.2 The Generalized Method of Characteristics......Page 243
4.3 Complete Integrals of Certain Special Nonlinear Equations......Page 246
4.4 The Hamilton–Jacobi Equation and Its Applications......Page 253
4.5 Applications to Nonlinear Optics......Page 261
4.6 Exercises......Page 269
5.1 Introduction......Page 272
5.2 Conservation Laws......Page 273
5.3 Discontinuous Solutions and Shock Waves......Page 285
5.4 Weak or Generalized Solutions......Page 287
5.5 Exercises......Page 293
6.1 Introduction......Page 298
6.2 Kinematic Waves......Page 299
6.3 Traffic Flow Problems......Page 301
6.4 Flood Waves in Long Rivers......Page 313
6.5 Chromatographic Models and Sediment Transport in Rivers......Page 315
6.6 Glacier Flow......Page 321
6.7 RollWaves and Their Stability Analysis......Page 323
6.8 Simple Waves and Riemann’s Invariants......Page 329
6.9 The Nonlinear Hyperbolic System and Riemann’s Invariants......Page 346
6.10 Generalized Simple Waves and Generalized Riemann’s Invariants......Page 356
6.11 Exercises......Page 360
7.1 Introduction......Page 368
7.2 Linear Dispersive Waves......Page 369
7.3 Initial-Value Problems and Asymptotic Solutions......Page 372
7.4 Nonlinear Dispersive Waves and Whitham’s Equations......Page 375
7.5 Whitham’s Theory of Nonlinear Dispersive Waves......Page 377
7.6 Whitham’s Averaged Variational Principle......Page 380
7.7 Whitham’s Instability Analysis of Water Waves......Page 382
7.8 Whitham’s Equation: Peaking and Breaking of Waves......Page 384
7.9 Exercises......Page 390
8.1 Introduction......Page 394
8.2 Burgers’ Equation and the Plane Wave Solution......Page 395
8.3 Traveling Wave Solutions and Shock-Wave Structure......Page 397
8.4 The Exact Solution of the Burgers Equation......Page 399
8.5 The Asymptotic Behavior of the Burgers Solution......Page 404
8.6 The Wave Solution......Page 406
8.7 Burgers’ Initial- and Boundary-Value Problem......Page 407
8.8 Fisher’s Equation and Diffusion-Reaction Process......Page 410
8.9 Traveling Wave Solutions and Stability Analysis......Page 412
8.10 Perturbation Solutions of the Fisher Equation......Page 416
8.11 Method of Similarity Solutions of Diffusion Equations......Page 418
8.12 Nonlinear Reaction-Diffusion Equations......Page 426
8.13 Brief Summary of Recent Work......Page 430
8.14 Exercises......Page 432
9.1 Introduction......Page 438
9.2 The History of the Solitons and Soliton Interactions......Page 439
9.3 The Boussinesq and Korteweg–de Vries Equations......Page 444
9.4 Solutions of the KdV Equation: Solitons and Cnoidal Waves......Page 467
9.5 The Lie Group Method and Similarity Analysis of the KdV Equation......Page 474
9.6 Conservation Laws and Nonlinear Transformations......Page 478
9.7 The Inverse Scattering Transform (IST) Method......Page 483
9.8 Bäcklund Transformations and the Nonlinear Superposition Principle......Page 504
9.9 The Lax Formulation and the Zakharov and Shabat Scheme......Page 509
9.10 The AKNS Method......Page 517
9.11 Asymptotic Behavior of the Solution of the KdV–Burgers Equation......Page 519
9.12 Strongly Dispersive Nonlinear Equations and Compactons......Page 520
9.13 Exercises......Page 531
10.1 Introduction......Page 536
10.2 The One-Dimensional Linear Schrödinger Equation......Page 537
10.3 The Nonlinear Schrödinger Equation and Solitary Waves......Page 538
10.4 Properties of the Solutions of the Nonlinear Schrödinger Equation......Page 543
10.5 Conservation Laws for the NLS Equation......Page 549
10.6 The Inverse Scattering Method for the Nonlinear Schrödinger Equation......Page 552
10.7 Examples of Physical Applications in Fluid Dynamics and Plasma Physics......Page 554
10.8 Applications to Nonlinear Optics......Page 566
10.9 Exercises......Page 575
11.1 Introduction......Page 578
11.2 The One-Dimensional Linear Klein–Gordon Equation......Page 579
11.3 The Two-Dimensional Linear Klein–Gordon Equation......Page 581
11.4 The Three-Dimensional Linear Klein–Gordon Equation......Page 583
11.5 The Nonlinear Klein–Gordon Equation and Averaging Techniques......Page 584
11.6 The Klein–Gordon Equation and the Whitham Averaged Variational Principle......Page 591
11.7 The Sine-Gordon Equation: Soliton and Antisoliton Solutions......Page 593
11.8 The Solution of the Sine-Gordon Equation by Separation of Variables......Page 597
11.9 Bäcklund Transformations for the Sine-Gordon Equation......Page 605
11.10 The Solution of the Sine-Gordon Equation by the Inverse Scattering Method......Page 608
11.12 Nonlinear Optics and the Sine-Gordon Equation......Page 612
11.13 Exercises......Page 616
12.1 Introduction......Page 620
12.2 The Reductive Perturbation Method and Quasi-Linear Hyperbolic Systems......Page 622
12.3 Quasi-Linear Dissipative Systems......Page 626
12.4 Weakly Nonlinear Dispersive Systems and the Korteweg–de Vries Equation......Page 627
12.5 Strongly Nonlinear Dispersive Systems and the NLS Equation......Page 639
12.6 The Perturbation Method of Ostrovsky and Pelinovsky......Page 644
12.7 The Method of Multiple Scales......Page 648
12.8 Asymptotic Expansions and Method of Multiple Scales......Page 654
12.9 Derivation of the NLS Equation and Davey–Stewartson Evolution Equations......Page 662
13.1 Fourier Transforms......Page 674
13.2 Fourier Sine Transforms......Page 676
13.3 Fourier Cosine Transforms......Page 678
13.4 Laplace Transforms......Page 680
13.5 Hankel Transforms......Page 684
13.6 Finite Hankel Transforms......Page 688
1.15 Exercises......Page 690
2.8 Exercises......Page 702
3.6 Exercises......Page 703
4.6 Exercises......Page 708
5.5 Exercises......Page 710
6.11 Exercises......Page 713
7.9 Exercises......Page 716
8.14 Exercises......Page 717
11.13 Exercises......Page 718
Bibliography......Page 720
Index......Page 748