偏微分方程与孤波理论 英文版PDF|Epub|txt|kindle电子书版本网盘下载

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Part Ⅰ　Partial Differential Equations1

1 Basic Concepts3

1.1　Introduction3

1.2　Definitions4

1.2.1　Definition of a PDE4

1.2.2 Order of a PDE5

1.2.3　Linear and Nonlinear PDEs6

1.2.4　Some Linear Partial Differential Equations7

1.2.5 Some Nonlinear Partial Differential Equations7

1.2.6　Homogeneous and Inhomogeneous PDEs9

1.2.7 Solution of a PDE9

1.2.8 Boundary Conditions11

1.2.9 Initial Conditions12

1.2.10　Well-posed PDEs12

1.3 Classifications of a Second-order PDE14

References17

2 First-order Partial Differential Equations19

2.1　Introduction19

2.2　Adomian Decomposition Method19

2.3　The Noise Terms Phenomenon36

2.4　The Modified Decomposition Method41

2.5　The Variational Iteration Method47

2.6　Method of Characteristics54

2.7 Systems of Linear PDEs by Adomian Method59

2.8 Systems of Linear PDEs by Variational Iteration Method63

References68

3 One Dimensional Heat Flow69

3.1 Introduction69

3.2　The Adomian Decomposition Method70

3.2.1　Homogeneous Heat Equations73

3.2.2　Inhomogeneous Heat Equations80

3.3　The Variational Iteration Method83

3.3.1　Homogeneous Heat Equations84

3.3.2　Inhomogeneous Heat Equations87

3.4　Method of Separation of Variables89

3.4.1 Analysis of the Method89

3.4.2　Inhomogeneous Boundary Conditions99

3.4.3 Equations with Lateral Heat Loss102

References106

4　Higher Dimensional Heat Flow107

4.1　Introduction107

4.2　Adomian Decomposition Method108

4.2.1　Two Dimensional Heat Flow108

4.2.2　Three Dimensional Heat Flow116

4.3　Method of Separation of Variables124

4.3.1　Two Dimensional Heat Flow124

4.3.2　Three Dimensional Heat Flow134

References140

5 One Dimensional Wave Equation143

5.1 Introduction143

5.2　Adornian Decomposition Method144

5.2.1　Homogeneous Wave Equations146

5.2.2 Inhomogeneous Wave Equations152

5.2.3 Wave Equation in an Infinite Domain157

5.3　The Variational Iteration Method162

5.3.1　Homogeneous Wave Equations162

5.3.2 Inhomogeneous Wave Equations168

5.3.3　Wave Equation in an Infinite Domain170

5.4 Method of Separation of Variables174

5.4.1 Analysis of the Method174

5.4.2 Inhomogeneous Boundary Conditions184

5.5　Wave Equation in an Infinite Domain:D'Alembert Solution190

References194

6　Higher Dimensional Wave Equation195

6.1　Introduction195

6.2 Adomian Decomposition Method195

6.2.1 Two Dimensional Wave Equation196

6.2.2 Three Dimensional Wave Equation210

6.3　Method of Separation of Variables220

6.3.1　Two Dimensional Wave Equation221

6.3.2　Three Dimensional Wave Equation231

References236

7　Laplace's Equation237

7.1　Introduction237

7.2　Adomian Decomposition Method238

7.2.1　Two Dimensional Laplace's Equation238

7.3　The Variational Iteration Method247

7.4　Method of Separation of Variables251

7.4.1　Laplace's Equation in Two Dimensions251

7.4.2　Laplace's Equation in Three Dimensions259

7.5　Laplace's Equation in Polar Coordinates267

7.5.1　Laplace's Equation for a Disc268

7.5.2　Laplace's Equation for an Annulus275

References283

8 Nonlinear Partial Differential Equations285

8.1　Introduction285

8.2　Adomian Decomposition Method287

8.2.1 Calculation of Adomian Polynomials288

8.2.2　Alternative Algorithm for Calculating Adomian Polynomials292

8.3 Nonlinear ODEs by Adomian Method301

8.4 Nonlinear ODEs by VIM312

8.5 Nonlinear PDEs by Adomian Method319

8.6　Nonlinear PDEs by VIM334

8.7　Nonlinear PDEs Systems by Adomian Method341

8.8 Systems of Nonlinear PDEs by VIM347

References351

9　Linear and Nonlinear Physical Models353

9.1 Introduction353

9.2　The Nonlinear Advection Problem354

9.3　The Goursat Problem360

9.4 The Klein-Gordon Equation370

9.4.1　Linear Klein-Gordon Equation371

9.4.2 Nonlinear Klein-Gordon Equation375

9.4.3 The Sine-Gordon Equation378

9.5　The Burgers Equation381

9.6　The Telegraph Equation388

9.7 Schrodinger Equation394

9.7.1 The Linear Schrodinger Equation394

9.7.2　The Nonlinear Schrodinger Equation397

9.8　Korteweg-de Vries Equation401

9.9　Fourth-order Parabolic Equation405

9.9.1　Equations with Constant Coefficients405

9.9.2　Equations with Variable Coefficients408

References413

10 Numerical Applications and Padé Approximants415

10.1　Introduction415

10.2　Ordinary Differential Equations416

10.2.1　Perturbation Problems416

10.2.2　Nonperturbed Problems421

10.3　Partial Differential Equations427

10.4　The Padé Approximants430

10.5　Padé Approximants and Boundary Value Problems439

References455

11 Solitons and Compactons457

11.1　Introduction457

11.2　Solitons459

11.2.1　The KdV Equation460

11.2.2　The Modified KdV Equation462

11.2.3　The Generalized KdV Equation464

11.2.4　The Sine-Gordon Equation464

11.2.5　The Boussinesq Equation465

11.2.6　The Kadomtsev-Petviashvili Equation467

11.3　Compactons469

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

References475

Part Ⅱ　Solitray Waves Theory479

12 Solitary Waves Theory479

12.1 Introduction479

12.2　Definitions480

12.2.1　Dispersion and Dissipation482

12.2.2　Types of Travelling Wave Solutions484

12.2.3 Nonanalytic Solitary Wave Solutions490

12.3 Analysis of the Methods491

12.3.1　The Tanh-coth Method491

12.3.2　The Sine-cosine Method493

12.3.3　Hirota's Bilinear Method494

12.4　Conservation Laws496

References502

13　The Family of the KdV Equations503

13.1　Introduction503

13.2　The Family of the KdV Equations505

13.2.1　Third-order KdV Equations505

13.2.2　The K(n,n) Equation507

13.3　The KdV Equation507

13.3.1 Using the Tanh-coth Method508

13.3.2　Using the Sine-cosine Method510

13.3.3 Multiple-soliton Solutions of the KdV Equation510

13.4　The Modified KdV Equation518

13.4.1 Using the Tanh-coth Method519

13.4.2　Using the Sine-cosine Method520

13.4.3 Multiple-soliton Solutions of the mKdV Equation521

13.5 Singular Soliton Solutions523

13.6　The Generalized KdV Equation526

13.6.1 Using the Tanh-coth Method526

13.6.2　Using the Sine-cosine Method528

13.7　The Potential KdV Equation528

13.7.1 Using the Tanh-coth Method529

13.7.2 Multiple-soliton Solutions of the Potential KdV Equation……531 13.8　The Gardner Equation533

13.8.1　The Kink Solution533

13.8.2　The Soliton Solution534

13.8.3　N-soliton Solutions of the Positive Gardner Equation535

13.8.4　Singular Soliton Solutions537

13.9 Generalized KdV Equation with Two Power Nonlinearities542

13.9.1 Using the Tanh Method543

13.9.2　Using the Sine-cosine Method544

13.10　Compactons:Solitons with Compact Support544

13.10.1 The K(n,n)Equation546

13.11 Variants of the K(n,n) Equation547

13.11.1 First Variant548

13.11.2 Second Variant549

13.11.3　Third Variant551

13.12 Compacton-like Solutions553

13.12.1 The Modified KdV Equation553

13.12.2 The Gardner Equation554

13.12.3 The Modified Equal Width Equation554

References555

14 KdV and mKdV Equations of Higher-orders557

14.1　Introduction557

14.2　Family of Higher-order KdV Equations558

14.2.1　Fifth-order KdV Equations558

14.2.2　Seventh-order KdV Equations561

14.2.3　Ninth-order KdV Equations562

14.3 Fifth-order KdV Equations562

14.3.1　Using the Tanh-coth Method563

14.3.2　The First Condition564

14.3.3　The Second Condition566

14.3.4　N-soliton Solutions of the Fifth-order KdV Equations567

14.4　Seventh-order KdV Equations576

14.4.1　Using the Tanh-coth Method576

14.4.2 N-soliton Solutions of the Seventh-order KdV Equations578

14.5　Ninth-order KdV Equations582

14.5.1 Using the Tanh-coth Method583

14.5.2　The Soliton Solutions584

14.6　Family of Higher-order mKdV Equations585

14.6.1 N-soliton Solutions for Fifth-order mKdV Equation586

14.6.2 Singular Soliton Solutions for Fifth-order mKdV Equation587

14.6.3 N-soliton Solutions for the Seventh-order mKdV Equation589

14.7 Complex Solution for the Seventh-order mKdV Equations591

14.8　The Hirota-Satsuma Equations592

14.8.1 Using the Tanh-coth Method593

14.8.2　N-soliton Solutions of the Hirota-Satsuma System594

14.8.3 N-soliton Solutions by an Alternative Method596

14.9　Generalized Short Wave Equation597

References602

15　Family of KdV-type Equations605

15.1 Introduction605

15.2　The Complex Modified KdV Equation606

15.2.1 Using the Sine-cosine Method607

15.2.2　Using the Tanh-coth Method608

15.3 The Benjamin-Bona-Mahony Equation612

15.3.1 Using the Sine-cosine Method612

15.3.2　Using the Tanh-coth Method613

15.4　The Medium Equal Width(MEW)Equation615

15.4.1 Using the Sine-cosine Method615

15.4.2　Using the Tanh-coth Method616

15.5　The Kawahara and the Modified Kawahara Equations617

15.5.1　The Kawahara Equation618

15.5.2　The Modified Kawahara Equation619

15.6　The Kadomtsev-Petviashvili(KP)Equation620

15.6.1 Using the Tanh-coth Method621

15.6.2 Multiple-soliton Solutions of the KP Equation622

15.7　The Zakharov-Kuznetsov(ZK)Equation626

15.8 The Benjamin-Ono Equation629

15.9　The KdV-Burgers Equation630

15.10 Seventh-order KdV Equation632

15.10.1　The Sech Method632

15.11 Ninth-order KdV Equation634

15.11.1　The Sech Method634

References637

16 Boussinesq,Klein-Gordon and Liouville Equations639

16.1 Introduction639

16.2　The Boussinesq Equation641

16.2.1 Using the Tanh-coth Method641

16.2.2 Multiple-soliton Solutions of the Boussinesq Equation643

16.3　The Improved Boussinesq Equation646

16.4　The Klein-Gordon Equation648

16.5 The Liouville Equation649

16.6　The Sine-Gordon Equation651

16.6.1 Using the Tanh-coth Method651

16.6.2　Using the B？cklund Transformation654

16.6.3 Multiple-soliton Solutions for Sine-Gordon Equation655

16.7　The Sinh-Gordon Equation657

16.8 The Dodd-Bullough-Mikhailov Equation658

16.9　The Tzitzeica-Dodd-Bullough Equation659

16.10 The Zhiber-Shabat Equation661

References662

17 Burgers,Fisher and Related Equations665

17.1　Introduction665

17.2　The Burgers Equation666

17.2.1　Using the Tanh-coth Method667

17.2.2　Using the Cole-Hopf Transformation668

17.3　The Fisher Equation670

17.4　The Huxley Equation671

17.5　The Burgers-Fisher Equation673

17.6　The Burgers-Huxley Equation673

17.7　The FitzHugh-Nagumo Equation675

17.8　Parabolic Equation with Exponential Nonlinearity676

17.9　The Coupled Burgers Equation678

17.10　The Kuramoto-Sivashinsky(KS)Equation680

References681

18 Families of Camassa-Holm and Schrodinger Equations683

18.1　Introduction683

18.2　The Family of Camassa-Holm Equations686

18.2.1　Using the Tanh-coth Method686

18.2.2　Using an Exponential Algorithm688

18.3 Schrodinger Equation of Cubic Nonlinearity689

18.4 Schrodinger Equation with Power Law Nonlinearity690

18.5　The Ginzburg-Landau Equation692

18.5.1 The Cubic Ginzburg-Landau Equation693

18.5.2　The Generalized Cubic Ginzburg-Landau Equation694

18.5.3　The Generalized Quintic Ginzburg-Landau Equation695

References696

Appendix699

A Indefinite Integrals699

A.1 Fundamental Forms699

A.2　Trigonometric Forms700

A.3 Inverse Trigonometric Forms700

A.4 Exponential and Logarithmic Forms701

A.5　Hyperbolic Forms701

A.6 Other Forms702

B Series703

B.1 Exponential Functions703

B.2 Trigonometric Functions703

B.3 Inverse Trigonometric Functions704

B.4　Hyperbolic Functions704

B.5 Inverse Hyperbolic Functions704

C　Exact Solutions of Burgers' Equation705

D Padé Approximants for Well-Known Functions707

D.1 Exponential Functions707

D.2 Trigonometric Functions707

D.3　Hyperbolic Functions709

D.4　Logarithmic Functions709

E　The Error and Gamma Functions711

E.1　The Error function711

E.2 The Gamma function Г（x）711

F Infinite Series712

F.1 Numerical Series712

F.2 Trigonometric Series713

Answers715

Index739