偏微分方程与孤波理论 英文版PDF格式文档图书下载

Part Ⅰ　Partial Differential Equations 1

1 Basic Concepts 3

1.1　Introduction 3

1.2　Definitions 4

1.2.1　Definition of a PDE 4

1.2.2 Order of a PDE 5

1.2.3　Linear and Nonlinear PDEs 6

1.2.4　Some Linear Partial Differential Equations 7

1.2.5 Some Nonlinear Partial Differential Equations 7

1.2.6　Homogeneous and Inhomogeneous PDEs 9

1.2.7 Solution of a PDE 9

1.2.8 Boundary Conditions 11

1.2.9 Initial Conditions 12

1.2.10　Well-posed PDEs 12

1.3 Classifications of a Second-order PDE 14

References 17

2 First-order Partial Differential Equations 19

2.1　Introduction 19

2.2　Adomian Decomposition Method 19

2.3　The Noise Terms Phenomenon 36

2.4　The Modified Decomposition Method 41

2.5　The Variational Iteration Method 47

2.6　Method of Characteristics 54

2.7 Systems of Linear PDEs by Adomian Method 59

2.8 Systems of Linear PDEs by Variational Iteration Method 63

References 68

3 One Dimensional Heat Flow 69

3.1 Introduction 69

3.2　The Adomian Decomposition Method 70

3.2.1　Homogeneous Heat Equations 73

3.2.2　Inhomogeneous Heat Equations 80

3.3　The Variational Iteration Method 83

3.3.1　Homogeneous Heat Equations 84

3.3.2　Inhomogeneous Heat Equations 87

3.4　Method of Separation of Variables 89

3.4.1 Analysis of the Method 89

3.4.2　Inhomogeneous Boundary Conditions 99

3.4.3 Equations with Lateral Heat Loss 102

References 106

4　Higher Dimensional Heat Flow 107

4.1　Introduction 107

4.2　Adomian Decomposition Method 108

4.2.1　Two Dimensional Heat Flow 108

4.2.2　Three Dimensional Heat Flow 116

4.3　Method of Separation of Variables 124

4.3.1　Two Dimensional Heat Flow 124

4.3.2　Three Dimensional Heat Flow 134

References 140

5 One Dimensional Wave Equation 143

5.1 Introduction 143

5.2　Adornian Decomposition Method 144

5.2.1　Homogeneous Wave Equations 146

5.2.2 Inhomogeneous Wave Equations 152

5.2.3 Wave Equation in an Infinite Domain 157

5.3　The Variational Iteration Method 162

5.3.1　Homogeneous Wave Equations 162

5.3.2 Inhomogeneous Wave Equations 168

5.3.3　Wave Equation in an Infinite Domain 170

5.4 Method of Separation of Variables 174

5.4.1 Analysis of the Method 174

5.4.2 Inhomogeneous Boundary Conditions 184

5.5　Wave Equation in an Infinite Domain:D'Alembert Solution 190

References 194

6　Higher Dimensional Wave Equation 195

6.1　Introduction 195

6.2 Adomian Decomposition Method 195

6.2.1 Two Dimensional Wave Equation 196

6.2.2 Three Dimensional Wave Equation 210

6.3　Method of Separation of Variables 220

6.3.1　Two Dimensional Wave Equation 221

6.3.2　Three Dimensional Wave Equation 231

References 236

7　Laplace's Equation 237

7.1　Introduction 237

7.2　Adomian Decomposition Method 238

7.2.1　Two Dimensional Laplace's Equation 238

7.3　The Variational Iteration Method 247

7.4　Method of Separation of Variables 251

7.4.1　Laplace's Equation in Two Dimensions 251

7.4.2　Laplace's Equation in Three Dimensions 259

7.5　Laplace's Equation in Polar Coordinates 267

7.5.1　Laplace's Equation for a Disc 268

7.5.2　Laplace's Equation for an Annulus 275

References 283

8 Nonlinear Partial Differential Equations 285

8.1　Introduction 285

8.2　Adomian Decomposition Method 287

8.2.1 Calculation of Adomian Polynomials 288

8.2.2　Alternative Algorithm for Calculating Adomian Polynomials 292

8.3 Nonlinear ODEs by Adomian Method 301

8.4 Nonlinear ODEs by VIM 312

8.5 Nonlinear PDEs by Adomian Method 319

8.6　Nonlinear PDEs by VIM 334

8.7　Nonlinear PDEs Systems by Adomian Method 341

8.8 Systems of Nonlinear PDEs by VIM 347

References 351

9　Linear and Nonlinear Physical Models 353

9.1 Introduction 353

9.2　The Nonlinear Advection Problem 354

9.3　The Goursat Problem 360

9.4 The Klein-Gordon Equation 370

9.4.1　Linear Klein-Gordon Equation 371

9.4.2 Nonlinear Klein-Gordon Equation 375

9.4.3 The Sine-Gordon Equation 378

9.5　The Burgers Equation 381

9.6　The Telegraph Equation 388

9.7 Schrodinger Equation 394

9.7.1 The Linear Schrodinger Equation 394

9.7.2　The Nonlinear Schrodinger Equation 397

9.8　Korteweg-de Vries Equation 401

9.9　Fourth-order Parabolic Equation 405

9.9.1　Equations with Constant Coefficients 405

9.9.2　Equations with Variable Coefficients 408

References 413

10 Numerical Applications and Padé Approximants 415

10.1　Introduction 415

10.2　Ordinary Differential Equations 416

10.2.1　Perturbation Problems 416

10.2.2　Nonperturbed Problems 421

10.3　Partial Differential Equations 427

10.4　The Padé Approximants 430

10.5　Padé Approximants and Boundary Value Problems 439

References 455

11 Solitons and Compactons 457

11.1　Introduction 457

11.2　Solitons 459

11.2.1　The KdV Equation 460

11.2.2　The Modified KdV Equation 462

11.2.3　The Generalized KdV Equation 464

11.2.4　The Sine-Gordon Equation 464

11.2.5　The Boussinesq Equation 465

11.2.6　The Kadomtsev-Petviashvili Equation 467

11.3　Compactons 469

11.4　The Defocusing Branch of K(n,n) 474

References 475

Part Ⅱ　Solitray Waves Theory 479

12 Solitary Waves Theory 479

12.1 Introduction 479

12.2　Definitions 480

12.2.1　Dispersion and Dissipation 482

12.2.2　Types of Travelling Wave Solutions 484

12.2.3 Nonanalytic Solitary Wave Solutions 490

12.3 Analysis of the Methods 491

12.3.1　The Tanh-coth Method 491

12.3.2　The Sine-cosine Method 493

12.3.3　Hirota's Bilinear Method 494

12.4　Conservation Laws 496

References 502

13　The Family of the KdV Equations 503

13.1　Introduction 503

13.2　The Family of the KdV Equations 505

13.2.1　Third-order KdV Equations 505

13.2.2　The K(n,n) Equation 507

13.3　The KdV Equation 507

13.3.1 Using the Tanh-coth Method 508

13.3.2　Using the Sine-cosine Method 510

13.3.3 Multiple-soliton Solutions of the KdV Equation 510

13.4　The Modified KdV Equation 518

13.4.1 Using the Tanh-coth Method 519

13.4.2　Using the Sine-cosine Method 520

13.4.3 Multiple-soliton Solutions of the mKdV Equation 521

13.5 Singular Soliton Solutions 523

13.6　The Generalized KdV Equation 526

13.6.1 Using the Tanh-coth Method 526

13.6.2　Using the Sine-cosine Method 528

13.7　The Potential KdV Equation 528

13.7.1 Using the Tanh-coth Method 529

13.7.2 Multiple-soliton Solutions of the Potential KdV Equation……531 13.8　The Gardner Equation 533

13.8.1　The Kink Solution 533

13.8.2　The Soliton Solution 534

13.8.3　N-soliton Solutions of the Positive Gardner Equation 535

13.8.4　Singular Soliton Solutions 537

13.9 Generalized KdV Equation with Two Power Nonlinearities 542

13.9.1 Using the Tanh Method 543

13.9.2　Using the Sine-cosine Method 544

13.10　Compactons:Solitons with Compact Support 544

13.10.1 The K(n,n)Equation 546

13.11 Variants of the K(n,n) Equation 547

13.11.1 First Variant 548

13.11.2 Second Variant 549

13.11.3　Third Variant 551

13.12 Compacton-like Solutions 553

13.12.1 The Modified KdV Equation 553

13.12.2 The Gardner Equation 554

13.12.3 The Modified Equal Width Equation 554

References 555

14 KdV and mKdV Equations of Higher-orders 557

14.1　Introduction 557

14.2　Family of Higher-order KdV Equations 558

14.2.1　Fifth-order KdV Equations 558

14.2.2　Seventh-order KdV Equations 561

14.2.3　Ninth-order KdV Equations 562

14.3 Fifth-order KdV Equations 562

14.3.1　Using the Tanh-coth Method 563

14.3.2　The First Condition 564

14.3.3　The Second Condition 566

14.3.4　N-soliton Solutions of the Fifth-order KdV Equations 567

14.4　Seventh-order KdV Equations 576

14.4.1　Using the Tanh-coth Method 576

14.4.2 N-soliton Solutions of the Seventh-order KdV Equations 578

14.5　Ninth-order KdV Equations 582

14.5.1 Using the Tanh-coth Method 583

14.5.2　The Soliton Solutions 584

14.6　Family of Higher-order mKdV Equations 585

14.6.1 N-soliton Solutions for Fifth-order mKdV Equation 586

14.6.2 Singular Soliton Solutions for Fifth-order mKdV Equation 587

14.6.3 N-soliton Solutions for the Seventh-order mKdV Equation 589

14.7 Complex Solution for the Seventh-order mKdV Equations 591

14.8　The Hirota-Satsuma Equations 592

14.8.1 Using the Tanh-coth Method 593

14.8.2　N-soliton Solutions of the Hirota-Satsuma System 594

14.8.3 N-soliton Solutions by an Alternative Method 596

14.9　Generalized Short Wave Equation 597

References 602

15　Family of KdV-type Equations 605

15.1 Introduction 605

15.2　The Complex Modified KdV Equation 606

15.2.1 Using the Sine-cosine Method 607

15.2.2　Using the Tanh-coth Method 608

15.3 The Benjamin-Bona-Mahony Equation 612

15.3.1 Using the Sine-cosine Method 612

15.3.2　Using the Tanh-coth Method 613

15.4　The Medium Equal Width(MEW)Equation 615

15.4.1 Using the Sine-cosine Method 615

15.4.2　Using the Tanh-coth Method 616

15.5　The Kawahara and the Modified Kawahara Equations 617

15.5.1　The Kawahara Equation 618

15.5.2　The Modified Kawahara Equation 619

15.6　The Kadomtsev-Petviashvili(KP)Equation 620

15.6.1 Using the Tanh-coth Method 621

15.6.2 Multiple-soliton Solutions of the KP Equation 622

15.7　The Zakharov-Kuznetsov(ZK)Equation 626

15.8 The Benjamin-Ono Equation 629

15.9　The KdV-Burgers Equation 630

15.10 Seventh-order KdV Equation 632

15.10.1　The Sech Method 632

15.11 Ninth-order KdV Equation 634

15.11.1　The Sech Method 634

References 637

16 Boussinesq,Klein-Gordon and Liouville Equations 639

16.1 Introduction 639

16.2　The Boussinesq Equation 641

16.2.1 Using the Tanh-coth Method 641

16.2.2 Multiple-soliton Solutions of the Boussinesq Equation 643

16.3　The Improved Boussinesq Equation 646

16.4　The Klein-Gordon Equation 648

16.5 The Liouville Equation 649

16.6　The Sine-Gordon Equation 651

16.6.1 Using the Tanh-coth Method 651

16.6.2　Using the B？cklund Transformation 654

16.6.3 Multiple-soliton Solutions for Sine-Gordon Equation 655

16.7　The Sinh-Gordon Equation 657

16.8 The Dodd-Bullough-Mikhailov Equation 658

16.9　The Tzitzeica-Dodd-Bullough Equation 659

16.10 The Zhiber-Shabat Equation 661

References 662

17 Burgers,Fisher and Related Equations 665

17.1　Introduction 665

17.2　The Burgers Equation 666

17.2.1　Using the Tanh-coth Method 667

17.2.2　Using the Cole-Hopf Transformation 668

17.3　The Fisher Equation 670

17.4　The Huxley Equation 671

17.5　The Burgers-Fisher Equation 673

17.6　The Burgers-Huxley Equation 673

17.7　The FitzHugh-Nagumo Equation 675

17.8　Parabolic Equation with Exponential Nonlinearity 676

17.9　The Coupled Burgers Equation 678

17.10　The Kuramoto-Sivashinsky(KS)Equation 680

References 681

18 Families of Camassa-Holm and Schrodinger Equations 683

18.1　Introduction 683

18.2　The Family of Camassa-Holm Equations 686

18.2.1　Using the Tanh-coth Method 686

18.2.2　Using an Exponential Algorithm 688

18.3 Schrodinger Equation of Cubic Nonlinearity 689

18.4 Schrodinger Equation with Power Law Nonlinearity 690

18.5　The Ginzburg-Landau Equation 692

18.5.1 The Cubic Ginzburg-Landau Equation 693

18.5.2　The Generalized Cubic Ginzburg-Landau Equation 694

18.5.3　The Generalized Quintic Ginzburg-Landau Equation 695

References 696

Appendix 699

A Indefinite Integrals 699

A.1 Fundamental Forms 699

A.2　Trigonometric Forms 700

A.3 Inverse Trigonometric Forms 700

A.4 Exponential and Logarithmic Forms 701

A.5　Hyperbolic Forms 701

A.6 Other Forms 702

B Series 703

B.1 Exponential Functions 703

B.2 Trigonometric Functions 703

B.3 Inverse Trigonometric Functions 704

B.4　Hyperbolic Functions 704

B.5 Inverse Hyperbolic Functions 704

C　Exact Solutions of Burgers' Equation 705

D Padé Approximants for Well-Known Functions 707

D.1 Exponential Functions 707

D.2 Trigonometric Functions 707

D.3　Hyperbolic Functions 709

D.4　Logarithmic Functions 709

E　The Error and Gamma Functions 711

E.1　The Error function 711

E.2 The Gamma function Г（x） 711

F Infinite Series 712

F.1 Numerical Series 712

F.2 Trigonometric Series 713

Answers 715

Index 739

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