线性与非线性积分方程，方法及应用 英文

PartⅠ Linear Integral Equations 3

1 Preliminaries 3

1.1 Taylor Series 4

1.2 Ordinary Differential Equations 7

1.2.1 First Order Linear Differential Equations 7

1.2.2 Second Order Linear Differential Equations 9

1.2.3 The Series Solution Method 13

1.3 Leibnitz Rule for Differentiation of Integrals 17

1.4 Reducing Multiple Integrals to Single Integrals 19

1.5 Laplace Transform 22

1.5.1 Properties of Laplace Transforms 23

1.6 Infinite Geometric Series 28

References 30

2 Introductory Concepts of Integral Equations 33

2.1 Classification of Integral Equations 34

2.1.1 Fredholm Integral Equations 34

2.1.2 Volterra Integral Equations 35

2.1.3 Volterra-Fredholm Integral Equations 35

2.1.4 Singular Integral Equations 36

2.2 Classification of Integro-Differential Equations 37

2.2.1 Fredholm Integro-Differential Equations 38

2.2.2 Volterra Integro-Differential Equations 38

2.2.3 Volterra-Fredholm Integro-Differential Equations 39

2.3 Linearity and Homogeneity 40

2.3.1 Linearity Concept 40

2.3.2 Homogeneity Concept 41

2.4 Origins of Integral Equations 42

2.5 Converting IVP to Volterra Integral Equation 42

2.5.1 Converting Volterra Integral Equation to IVP 47

2.6 Converting BVP to Fredholm Integral Equation 49

2.6.1 Converting Fredholm Integral Equation to BVP 54

2.7 Solution of an Integral Equation 59

References 63

3 Volterra Integral Equations 65

3.1 Introduction 65

3.2 Volterra Integral Equations of the Second Kind 66

3.2.1 The Adomian Decomposition Method 66

3.2.2 The Modified Decomposition Method 73

3.2.3 The Noise Terms Phenomenon 78

3.2.4 The Variational Iteration Method 82

3.2.5 The Successive Approximations Method 95

3.2.6 The Laplace Transform Method 99

3.2.7 The Series Solution Method 103

3.3 Volterra Integral Equations of the First Kind 108

3.3.1 The Series Solution Method 108

3.3.2 The Laplace Transform Method 111

3.3.3 Conversion to a Volterra Equation of the Second Kind 114

References 118

4 Fredholm Integral Equations 119

4.1 Introduction 119

4.2 Fredholm Integral Equations of the Second Kind 121

4.2.1 The Adomian Decomposition Method 121

4.2.2 The Modified Decomposition Method 128

4.2.3 The Noise Terms Phenomenon 133

4.2.4 The Variational Iteration Method 136

4.2.5 The Direct Computation Method 141

4.2.6 The Successive Approximations Method 146

4.2.7 The Series Solution Method 151

4.3 Homogeneous Fredholm Integral Equation 154

4.3.1 The Direct Computation Method 155

4.4 Fredholm Integral Equations of the First Kind 159

4.4.1 The Method of Regularization 161

4.4.2 The Homotopy Perturbation Method 166

References 173

5 Volterra Integro-Differential Equations 175

5.1 Introduction 175

5.2 Volterra Integro-Differential Equations of the Second Kind 176

5.2.1 The Adomian Decomposition Method 176

5.2.2 The Variational Iteration Method 181

5.2.3 The Laplace Transform Method 186

5.2.4 The Series Solution Method 190

5.2.5 Converting Volterra Integro-Differential Equations to Initial Value Problems 195

5.2.6 Converting Volterra Integro-Differential Equation to Volterra Integral Equation 199

5.3 Volterra Integro-Differential Equations of the First Kind 203

5.3.1 Laplace Transform Method 204

5.3.2 The Variational Iteration Method 206

References 211

6 Fredholm Integro-Differential Equations 213

6.1 Introduction 213

6.2 Fredholm Integro-Differential Equations of the Second Kind 214

6.2.1 The Direct Computation Method 214

6.2.2 The Variational Iteration Method 218

6.2.3 The Adomian Decomposition Method 223

6.2.4 The Series Solution Method 230

References 234

7 Abel's Integral Equation and Singular Integral Equations 237

7.1 Introduction 237

7.2 Abel's Integral Equation 238

7.2.1 The Laplace Transform Method 239

7.3 The Generalized Abel's Integral Equation 242

7.3.1 The Laplace Transform Method 243

7.3.2 The Main Generalized Abel Equation 245

7.4 The Weakly Singular Volterra Equations 247

7.4.1 The Adomian Decomposition Method 248

7.4.2 The Successive Approximations Method 253

7.4.3 The Laplace Transform Method 257

Reterences 260

8 Volterra-Fredholm Integral Equations 261

8.1 Introduction 261

8.2 The Volterra-Fredholm Integral Equations 262

8.2.1 The Series Solution Method 262

8.2.2 The Adomian Decomposition Method 266

8.3 The Mixed Volterra-Fredholm Integral Equations 269

8.3.1 The Series Solution Method 270

8.3.2 The Adomian Decomposition Method 273

8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables 277

8.4.1 The Modified Decomposition Method 278

References 283

9 Volterra-Fredholm Integro-Differential Equations 285

9.1 Introduction 285

9.2 The Volterra-Fredholm Integro-Differential Equation 285

9.2.1 The Series Solution Method 285

9.2.2 The Variational Iteration Method 289

9.3 The Mixed Volterra-Fredholm Integro-Differential Equations 296

9.3.1 The Direct Computation Method 296

9.3.2 The Series Solution Method 300

9.4 The Mixed Volterra-Fredholm Integro-Differential Equations in Two Variables 303

9.4.1 The Modified Decomposition Method 304

References 309

10 Systems of Volterra Integral Equations 311

10.1 Introduction 311

10.2 Systems of Volterra Integral Equations of the Second Kind 312

10.2.1 The Adomian Decomposition Method 312

10.2.2 The Laplace Transform Method 318

10.3 Systems of Volterra Integral Equations of the First Kind 323

10.3.1 The Laplace Transform Method 323

10.3.2 Conversion to a Volterra System of the Second Kind 327

10.4 Systems of Volterra Integro-Differential Equations 328

10.4.1 The Variational Iteration Method 329

10.4.2 The Laplace Transform Method 335

References 339

11 Systems of Fredholm Integral Equations 341

11.1 Introduction 341

11.2 Systems of Fredholm Integral Equations 342

11.2.1 The Adomian Decomposition Method 342

11.2.2 The Direct Computation Method 347

11.3 Systems of Fredholm Integro-Differential Equations 352

11.3.1 The Direct Computation Method 353

11.3.2 The Variational Iteration Method 358

References 364

12 Systems of Singular Integral Equations 365

12.1 Introduction 365

12.2 Systems of Generalized Abel Integral Equations 366

12.2.1 Systems of Generalized Abel Integral Equations in Two Unknowns 366

12.2.2 Systems of Generalized Abel Integral Equations in Three Unknowns 370

12.3 Systems of the Weakly Singular Volterra Integral Equations 374

12.3.1 The Laplace Transform Method 374

12.3.2 The Adomian Decomposition Method 378

References 383

PartII Nonlinear Integral Equations 387

13 Nonlinear Volterra Integral Equations 387

13.1 Introduction 387

13.2 Existence of the Solution for Nonlinear Volterra Integral Equations 388

13.3 Nonlinear Volterra Integral Equations of the Second Kind 388

13.3.1 The Successive Approximations Method 389

13.3.2 The Series Solution Method 393

13.3.3 The Adomian Decomposition Method 397

13.4 Nonlinear Volterra Integral Equations of the First Kind 404

13.4.1 The Laplace Transform Method 405

13.4.2 Conversion to a Volterra Equation of the Second Kind 408

13.5 Systems of Nonlinear Volterra Integral Equations 411

13.5.1 Systems of Nonlinear Volterra Integral Equations of the Second Kind 412

13.5.2 Systems of Nonlinear Volterra Integral Equations of the First Kind 417

References 423

14 Nonlinear Volterra Integro-Differential Equations 425

14.1 Introduction 425

14.2 Nonlinear Volterra Integro-Differential Equations of the Second Kind 426

14.2.1 The Combined Laplace Transform-Adomian Decomposition Method 426

14.2.2 The Variational Iteration Method 432

14.2.3 The Series Solution Method 436

14.3 Nonlinear Volterra Integro-Differential Equations of the First Kind 440

14.3.1 The Combined Laplace Transform-Adomian Decomposition Method 440

14.3.2 Conversion to Nonlinear Volterra Equation of the Second Kind 446

14.4 Systems of Nonlinear Volterra Integro-Differential Equations 450

14.4.1 The Variational Iteration Method 451

14.4.2 The Combined Laplace Transform-Adomian Decomposition Method 456

References 465

15 Nonlinear Fredholm Integral Equations 467

15.1 Introduction 467

15.2 Existence of the Solution for Nonlinear Fredholm Integral Equations 468

15.2.1 Bifurcation Points and Singular Points 469

15.3 Nonlinear Fredholm Integral Equations of the Second Kind 469

15.3.1 The Direct Computation Method 470

15.3.2 The Series Solution Method 476

15.3.3 The Adomian Decomposition Method 480

15.3.4 The Successive Approximations Method 485

15.4 Homogeneous Nonlinear Fredholm Integral Equations 490

15.4.1 The Direct Computation Method 490

15.5 Nonlinear Fredholm Integral Equations of the First Kind 494

15.5.1 The Method of Regularization 495

15.5.2 The Homotopy Perturbation Method 500

15.6 Systems of Nonlinear Fredholm Integral Equations 505

15.6.1 The Direct Computation Method 506

15.6.2 The Modified Adomian Decomposition Method 510

References 515

16 Nonlinear Fredholm Integro-Differential Equations 517

16.1 Introduction 517

16.2 Nonlinear Fredholm Integro-Differential Equations 518

16.2.1 The Direct Computation Method 518

16.2.2 The Variational Iteration Method 522

16.2.3 The Series Solution Method 526

16.3 Homogeneous Nonlinear Fredholm Integro-Differential Equations 530

16.3.1 The Direct Computation Method 530

16.4 Systems of Nonlinear Fredholm Integro-Differential Equations 535

16.4.1 The Direct Computation Method 535

16.4.2 The Variational Iteration Method 540

References 545

17 Nonlinear Singular Integral Equations 547

17.1 Introduction 547

17.2 Nonlinear Abel's Integral Equation 548

17.2.1 The Laplace Transform Method 549

17.3 The Generalized Nonlinear Abel Equation 552

17.3.1 The Laplace Transform Method 553

17.3.2 The Main Generalized Nonlinear Abel Equation 556

17.4 The Nonlinear Weakly-Singular Volterra Equations 559

17.4.1 The Adomian Decomposition Method 559

17.5 Systems of Nonlinear Weakly-Singular Volterra Integral Equations 562

17.5.1 The Modified Adomian Decomposition Method 563

References 567

18 Applications of Integral Equations 569

18.1 Introduction 569

18.2 Volterra's Population Model 570

18.2.1 The Variational Iteration Method 571

18.2.2 The Series Solution Method 572

18.2.3 The PadéApproximants 573

18.3 Integral Equations with Logarithmic Kernels 574

18.3.1 Second Kind Fredholm Integral Equation with a Logarithmic Kernel 577

18.3.2 First Kind Fredholm Integral Equation with a Logarithmic Kernel 580

18.3.3 Another First Kind Fredholm Integral Equation with a Logarithmic Kernel 583

18.4 The Fresnel Integrals 584

18.5 The Thomas-Fermi Equation 587

18.6 Heat Transfer and Heat Radiation 590

18.6.1 Heat Transfer:Lighthill Singular Integral Equation 590

18.6.2 Heat Radiation in a Semi-Infinite Solid 592

References 594

Appendix A Table of Indefinite Integrals 597

A.1 Basic Forms 597

A.2 Trigonometric Forms 597

A.3 Inverse Trigonometric Forms 598

A.4 Exponential and Logarithmic Forms 598

A.5 Hyperbolic Forms 599

A.6 Other Forms 599

Appendix B Integrals Involving Irrational Algebraic Functions 600

B.1 Integrals Involving？,n is an integer,n≥0 600

B.2 Integrals Involving？,n is an odd integer,n≥1 600

Appendix C Series Representations 601

C.1 Exponential Functions Series 601

C.2 Trigonometric Functions 601

C.3 Inverse Trigonometric Functions 602

C.4 Hyperbolic Functions 602

C.5 Inverse Hyperbolic Functions 602

C.6 Logarithmic Functions 602

Appendix D The Error and the Complementary Error Functions 603

D.1 The Error Function 603

D.2 The Complementary Error Function 603

Appendix E Gamma Function 604

Appendix F Infinite Series 605

F.1 Numerical Series 605

F.2 Trigonometric Series 605

Appendix G The Fresnel Integrals 607

G.1 The Fresnel Cosine Integral 607

G.2 The Fresnel Sine Integral 607

Answers 609

Index 637