First Course In Integral Equations, A (Second Edition)

by Abdul-majid Wazwaz

Overview

This second edition integrates the newly developed methods with classical techniques to give both modern and powerful approaches for solving integral equations. It provides a comprehensive treatment of linear and nonlinear Fredholm and Volterra integral equations of the first and second kinds. The materials are presented in an accessible and straightforward manner to readers, particularly those from non-mathematics backgrounds. Numerous well-explained applications and examples as well as practical exercises are presented to guide readers through the text. Selected applications from mathematics, science and engineering are investigated by using the newly developed methods.This volume consists of nine chapters, pedagogically organized, with six chapters devoted to linear integral equations, two chapters on nonlinear integral equations, and the last chapter on applications. It is intended for scholars and researchers, and can be used for advanced undergraduate and graduate students in applied mathematics, science and engineering.Click here for solutions manual.

Product Details

ISBN-13:	9789814675123
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/16/2015
Edition description:	Revised ed.
Pages:	328
Product dimensions:	6.00(w) x 8.90(h) x 0.80(d)

Table of Contents

Preface to the Second Edition xi

Preface to the First Edition xiii

1 Introductory Concepts 1

1.1 Definitions 1

1.2 Classification of Linear Integral Equations 2

1.2.1 Fredholm Linear Integral Equations 3

1.2.2 Volterra Linear Integral Equations 4

1.2.3 Integra-Differential Equations 6

1.2.4 Singular Integral Equations 7

1.2.5 Volterra-Fredholm Integral Equations 8

1.2.6 Volterra-Fredholm Integro-Differential Equations 8

1.3 Solution of an Integral Equation 12

1.4 Converting Volterra Equation to an ODE 16

1.4.1 Differentiating Any Integral: Leibniz Rule 17

1.5 Converting IVP to Volterra Equation 20

1.6 Converting BVP to Fredholm Equation 25

1.7 Taylor Series 28

1.8 Infinite Geometric Series 32

2 Fredholm Integral Equations 35

2.1 Introduction 35

2.2 The Adomian Decomposition Method 37

2.2.1 The Modified Decomposition Method 41

2.2.2 ' The Noise Terms Phenomenon 45

2.3 The Variational Iteration Method 49

2.4 The Direct Computation Method 54

2.5 The Successive Approximations Method 60

2.6 The Method of Successive Substitutions 64

2.7 Comparison between Alternative Methods 67

2.8 Homogeneous Fredholm Integral Equations 71

2.9 Fredholm Integral Equations of the First Kind 76

2.9.1 The Method of Regularization 76

3 Volterra Integral Equations 81

3.1 Introduction 81

3.2 The Adomian Decomposition Method 82

3.2.1 The Modified Decomposition Method 87

3.2.2 The Noise Terms Phenomenon 89

3.3 The Variational Iteration Method 92

3.4 The Series Solution Method 96

3.5 Converting Volterra Equation to IVP 100

3.6 Successive Approximations Method 104

3.7 The Method of Successive Substitutions 109

3.8 Comparison between Alternative Methods 112

3.9 Volterra Integral Equations of the First Kind 117

3.9.1 The Series Solution Method 117

3.9.2 Conversion of First Kind to Second Kind 119

4 Fredholm Integro-Differential Equations 123

4.1 Introduction 123

4.2 Fredholm Integro-Differential Equations 124

4.3 The Direct Computation Method 125

4.4 The Adomian Decomposition Method 129

4.4.1 The Modified Decomposition Method 132

4.4.2 The Noise Terms Phenomenon 132

4.5 The Variational Iteration Method 138

4.6 Converting to Fredholm Integral Equations 142

5 Volterra Integro-Differential Equations 145

5.1 Introduction 145

5.2 Volterra Integro-Differential Equations 146

5.3 The Series Solution Method 147

5.4 The Adomian Decomposition Method 151

5.5 The Variational Iteration Method 156

5.6 Converting to Volterra Integral Equation 159

5.7 Converting to Initial Value Problems 162

5.8 Volterra Integro-Differential Equations of the First Kind 165

6 Singular Integral Equations 171

6.1 Introduction 171

6.2 Abel's Problem 173

6.3 The Generalized Abel's Integral Equation 178

6.4 The Weakly-Singular Volterra Integral Equations 181

6.4.1 The Adomian Decomposition Method 182

6.5 The Weakly-Singular Fredholm Integral Equations 187

6.5.1 The Modified Decomposition Method 188

7 Nonlinear Fredholm Integral Equations 191

7.1 Introduction 191

7.2 Nonlinear Fredholm Integral Equations of the Second Kind 192

7.2.1 The Direct Computation Method 192

7.2.2 The Adomian Decomposition Method 197

7.2.3 The Variational Iteration Method 207

7.3 Nonlinear Fredholm Integral Equations of the First Kind 211

7.3.1 The Method of Regularization 212

7.4 Nonlinear Weakly-Singular Fredholm Integral Equations 216

7.4.1 The Modified Decomposition Method 217

8 Nonlinear Volterra Integral Equations 223

8.1 Introduction 223

8.2 Nonlinear Volterra Integral Equations of the Second Kind 224

8.2.1 The Series Solution Method 225

8.2.2 The Adomian Decomposition Method 228

8.2.3 The Variational Iteration Method 233

8.3 Nonlinear Volterra Integral Equations of the First Kind 236

8.3.1 The Series Solution Method 236

8.3.2 Conversion to a Volterra Equation of the Second Kind 239

8.4 Nonlinear Weakly-Singular Volterra Integral Equations 242

8.4.1 The Modified Decomposition Method 243

9 Applications of Integral Equations 249

9.1 Introduction 249

9.2 Volterra Integral Form of the Lane-Emden Equation 250

9.2.1 Lane-Emden Equation of the First Kind 251

9.2.2 Lane-Emden Equation of the Second Kind 254

9.3 The Schlörnilch's Integral Equation 257

9.3.1 The Linear Schlörnilch's Integral Equation 257

9.3.2 The Method of Regularization 258

9.3.3 The Nonlinear Schlörnilch's Integral Equation 260

9.4 Bratu-Type Problems 263

9.4.1 First Bratu-Type Problem 264

9.4.2 Second Bratu-Type Problem 266

9.4.3 Third Bratu-Type Problem 268

9.5 Systems of Integral Equations 269

9.5.1 Systems of Fredholm Integral Equations 270

9.5.2 Systems of Volterra Integral Equations 272

9.6 Numerical Treatment of Fredholm Integral Equations 274

9.7 Numerical Treatment of Volterra Integral Equations 279

A Table of Indefinite Integrals 283

B Integrals Involving Irrational Algebraic Functions 287

C Series 289

D The Error and the Gamma Functions 291

Answers to Exercises 293

Bibliography 305

Index 311