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