线性与非线性积分方程，方法及应用 英文PDF|Epub|txt|kindle电子书版本网盘下载

线性与非线性积分方程，方法及应用 英文PDF格式电子书版下载

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PartⅠ Linear Integral Equations3

1 Preliminaries3

1.1 Taylor Series4

1.2 Ordinary Differential Equations7

1.2.1 First Order Linear Differential Equations7

1.2.2 Second Order Linear Differential Equations9

1.2.3 The Series Solution Method13

1.3 Leibnitz Rule for Differentiation of Integrals17

1.4 Reducing Multiple Integrals to Single Integrals19

1.5 Laplace Transform22

1.5.1 Properties of Laplace Transforms23

1.6 Infinite Geometric Series28

References30

2 Introductory Concepts of Integral Equations33

2.1 Classification of Integral Equations34

2.1.1 Fredholm Integral Equations34

2.1.2 Volterra Integral Equations35

2.1.3 Volterra-Fredholm Integral Equations35

2.1.4 Singular Integral Equations36

2.2 Classification of Integro-Differential Equations37

2.2.1 Fredholm Integro-Differential Equations38

2.2.2 Volterra Integro-Differential Equations38

2.2.3 Volterra-Fredholm Integro-Differential Equations39

2.3 Linearity and Homogeneity40

2.3.1 Linearity Concept40

2.3.2 Homogeneity Concept41

2.4 Origins of Integral Equations42

2.5 Converting IVP to Volterra Integral Equation42

2.5.1 Converting Volterra Integral Equation to IVP47

2.6 Converting BVP to Fredholm Integral Equation49

2.6.1 Converting Fredholm Integral Equation to BVP54

2.7 Solution of an Integral Equation59

References63

3 Volterra Integral Equations65

3.1 Introduction65

3.2 Volterra Integral Equations of the Second Kind66

3.2.1 The Adomian Decomposition Method66

3.2.2 The Modified Decomposition Method73

3.2.3 The Noise Terms Phenomenon78

3.2.4 The Variational Iteration Method82

3.2.5 The Successive Approximations Method95

3.2.6 The Laplace Transform Method99

3.2.7 The Series Solution Method103

3.3 Volterra Integral Equations of the First Kind108

3.3.1 The Series Solution Method108

3.3.2 The Laplace Transform Method111

3.3.3 Conversion to a Volterra Equation of the Second Kind114

References118

4 Fredholm Integral Equations119

4.1 Introduction119

4.2 Fredholm Integral Equations of the Second Kind121

4.2.1 The Adomian Decomposition Method121

4.2.2 The Modified Decomposition Method128

4.2.3 The Noise Terms Phenomenon133

4.2.4 The Variational Iteration Method136

4.2.5 The Direct Computation Method141

4.2.6 The Successive Approximations Method146

4.2.7 The Series Solution Method151

4.3 Homogeneous Fredholm Integral Equation154

4.3.1 The Direct Computation Method155

4.4 Fredholm Integral Equations of the First Kind159

4.4.1 The Method of Regularization161

4.4.2 The Homotopy Perturbation Method166

References173

5 Volterra Integro-Differential Equations175

5.1 Introduction175

5.2 Volterra Integro-Differential Equations of the Second Kind176

5.2.1 The Adomian Decomposition Method176

5.2.2 The Variational Iteration Method181

5.2.3 The Laplace Transform Method186

5.2.4 The Series Solution Method190

5.2.5 Converting Volterra Integro-Differential Equations to Initial Value Problems195

5.2.6 Converting Volterra Integro-Differential Equation to Volterra Integral Equation199

5.3 Volterra Integro-Differential Equations of the First Kind203

5.3.1 Laplace Transform Method204

5.3.2 The Variational Iteration Method206

References211

6 Fredholm Integro-Differential Equations213

6.1 Introduction213

6.2 Fredholm Integro-Differential Equations of the Second Kind214

6.2.1 The Direct Computation Method214

6.2.2 The Variational Iteration Method218

6.2.3 The Adomian Decomposition Method223

6.2.4 The Series Solution Method230

References234

7 Abel's Integral Equation and Singular Integral Equations237

7.1 Introduction237

7.2 Abel's Integral Equation238

7.2.1 The Laplace Transform Method239

7.3 The Generalized Abel's Integral Equation242

7.3.1 The Laplace Transform Method243

7.3.2 The Main Generalized Abel Equation245

7.4 The Weakly Singular Volterra Equations247

7.4.1 The Adomian Decomposition Method248

7.4.2 The Successive Approximations Method253

7.4.3 The Laplace Transform Method257

Reterences260

8 Volterra-Fredholm Integral Equations261

8.1 Introduction261

8.2 The Volterra-Fredholm Integral Equations262

8.2.1 The Series Solution Method262

8.2.2 The Adomian Decomposition Method266

8.3 The Mixed Volterra-Fredholm Integral Equations269

8.3.1 The Series Solution Method270

8.3.2 The Adomian Decomposition Method273

8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables277

8.4.1 The Modified Decomposition Method278

References283

9 Volterra-Fredholm Integro-Differential Equations285

9.1 Introduction285

9.2 The Volterra-Fredholm Integro-Differential Equation285

9.2.1 The Series Solution Method285

9.2.2 The Variational Iteration Method289

9.3 The Mixed Volterra-Fredholm Integro-Differential Equations296

9.3.1 The Direct Computation Method296

9.3.2 The Series Solution Method300

9.4 The Mixed Volterra-Fredholm Integro-Differential Equations in Two Variables303

9.4.1 The Modified Decomposition Method304

References309

10 Systems of Volterra Integral Equations311

10.1 Introduction311

10.2 Systems of Volterra Integral Equations of the Second Kind312

10.2.1 The Adomian Decomposition Method312

10.2.2 The Laplace Transform Method318

10.3 Systems of Volterra Integral Equations of the First Kind323

10.3.1 The Laplace Transform Method323

10.3.2 Conversion to a Volterra System of the Second Kind327

10.4 Systems of Volterra Integro-Differential Equations328

10.4.1 The Variational Iteration Method329

10.4.2 The Laplace Transform Method335

References339

11 Systems of Fredholm Integral Equations341

11.1 Introduction341

11.2 Systems of Fredholm Integral Equations342

11.2.1 The Adomian Decomposition Method342

11.2.2 The Direct Computation Method347

11.3 Systems of Fredholm Integro-Differential Equations352

11.3.1 The Direct Computation Method353

11.3.2 The Variational Iteration Method358

References364

12 Systems of Singular Integral Equations365

12.1 Introduction365

12.2 Systems of Generalized Abel Integral Equations366

12.2.1 Systems of Generalized Abel Integral Equations in Two Unknowns366

12.2.2 Systems of Generalized Abel Integral Equations in Three Unknowns370

12.3 Systems of the Weakly Singular Volterra Integral Equations374

12.3.1 The Laplace Transform Method374

12.3.2 The Adomian Decomposition Method378

References383

PartII Nonlinear Integral Equations387

13 Nonlinear Volterra Integral Equations387

13.1 Introduction387

13.2 Existence of the Solution for Nonlinear Volterra Integral Equations388

13.3 Nonlinear Volterra Integral Equations of the Second Kind388

13.3.1 The Successive Approximations Method389

13.3.2 The Series Solution Method393

13.3.3 The Adomian Decomposition Method397

13.4 Nonlinear Volterra Integral Equations of the First Kind404

13.4.1 The Laplace Transform Method405

13.4.2 Conversion to a Volterra Equation of the Second Kind408

13.5 Systems of Nonlinear Volterra Integral Equations411

13.5.1 Systems of Nonlinear Volterra Integral Equations of the Second Kind412

13.5.2 Systems of Nonlinear Volterra Integral Equations of the First Kind417

References423

14 Nonlinear Volterra Integro-Differential Equations425

14.1 Introduction425

14.2 Nonlinear Volterra Integro-Differential Equations of the Second Kind426

14.2.1 The Combined Laplace Transform-Adomian Decomposition Method426

14.2.2 The Variational Iteration Method432

14.2.3 The Series Solution Method436

14.3 Nonlinear Volterra Integro-Differential Equations of the First Kind440

14.3.1 The Combined Laplace Transform-Adomian Decomposition Method440

14.3.2 Conversion to Nonlinear Volterra Equation of the Second Kind446

14.4 Systems of Nonlinear Volterra Integro-Differential Equations450

14.4.1 The Variational Iteration Method451

14.4.2 The Combined Laplace Transform-Adomian Decomposition Method456

References465

15 Nonlinear Fredholm Integral Equations467

15.1 Introduction467

15.2 Existence of the Solution for Nonlinear Fredholm Integral Equations468

15.2.1 Bifurcation Points and Singular Points469

15.3 Nonlinear Fredholm Integral Equations of the Second Kind469

15.3.1 The Direct Computation Method470

15.3.2 The Series Solution Method476

15.3.3 The Adomian Decomposition Method480

15.3.4 The Successive Approximations Method485

15.4 Homogeneous Nonlinear Fredholm Integral Equations490

15.4.1 The Direct Computation Method490

15.5 Nonlinear Fredholm Integral Equations of the First Kind494

15.5.1 The Method of Regularization495

15.5.2 The Homotopy Perturbation Method500

15.6 Systems of Nonlinear Fredholm Integral Equations505

15.6.1 The Direct Computation Method506

15.6.2 The Modified Adomian Decomposition Method510

References515

16 Nonlinear Fredholm Integro-Differential Equations517

16.1 Introduction517

16.2 Nonlinear Fredholm Integro-Differential Equations518

16.2.1 The Direct Computation Method518

16.2.2 The Variational Iteration Method522

16.2.3 The Series Solution Method526

16.3 Homogeneous Nonlinear Fredholm Integro-Differential Equations530

16.3.1 The Direct Computation Method530

16.4 Systems of Nonlinear Fredholm Integro-Differential Equations535

16.4.1 The Direct Computation Method535

16.4.2 The Variational Iteration Method540

References545

17 Nonlinear Singular Integral Equations547

17.1 Introduction547

17.2 Nonlinear Abel's Integral Equation548

17.2.1 The Laplace Transform Method549

17.3 The Generalized Nonlinear Abel Equation552

17.3.1 The Laplace Transform Method553

17.3.2 The Main Generalized Nonlinear Abel Equation556

17.4 The Nonlinear Weakly-Singular Volterra Equations559

17.4.1 The Adomian Decomposition Method559

17.5 Systems of Nonlinear Weakly-Singular Volterra Integral Equations562

17.5.1 The Modified Adomian Decomposition Method563

References567

18 Applications of Integral Equations569

18.1 Introduction569

18.2 Volterra's Population Model570

18.2.1 The Variational Iteration Method571

18.2.2 The Series Solution Method572

18.2.3 The PadéApproximants573

18.3 Integral Equations with Logarithmic Kernels574

18.3.1 Second Kind Fredholm Integral Equation with a Logarithmic Kernel577

18.3.2 First Kind Fredholm Integral Equation with a Logarithmic Kernel580

18.3.3 Another First Kind Fredholm Integral Equation with a Logarithmic Kernel583

18.4 The Fresnel Integrals584

18.5 The Thomas-Fermi Equation587

18.6 Heat Transfer and Heat Radiation590

18.6.1 Heat Transfer:Lighthill Singular Integral Equation590

18.6.2 Heat Radiation in a Semi-Infinite Solid592

References594

Appendix A Table of Indefinite Integrals597

A.1 Basic Forms597

A.2 Trigonometric Forms597

A.3 Inverse Trigonometric Forms598

A.4 Exponential and Logarithmic Forms598

A.5 Hyperbolic Forms599

A.6 Other Forms599

Appendix B Integrals Involving Irrational Algebraic Functions600

B.1 Integrals Involving？,n is an integer,n≥0600

B.2 Integrals Involving？,n is an odd integer,n≥1600

Appendix C Series Representations601

C.1 Exponential Functions Series601

C.2 Trigonometric Functions601

C.3 Inverse Trigonometric Functions602

C.4 Hyperbolic Functions602

C.5 Inverse Hyperbolic Functions602

C.6 Logarithmic Functions602

Appendix D The Error and the Complementary Error Functions603

D.1 The Error Function603

D.2 The Complementary Error Function603

Appendix E Gamma Function604

Appendix F Infinite Series605

F.1 Numerical Series605

F.2 Trigonometric Series605

Appendix G The Fresnel Integrals607

G.1 The Fresnel Cosine Integral607

G.2 The Fresnel Sine Integral607

Answers609

Index637