Introduction to integral equations with applications / Abdul J. Jerri.

Format:

Book

Author/Creator:

Jerri, Abdul J., 1932-

Contributor:

Class of 1932 Fund.

Language:

English

Subjects (All):

Integral equations--Numerical solutions.

Integral equations.

Integral equations--Problems, exercises, etc.

Local Subjects:

Integral equations.

Integral equations--Problems, exercises, etc.

Physical Description:

xviii, 433 pages : illustrations ; 24 cm

Edition:

Second edition.

Place of Publication:

New York : Wiley, 1999.

Contents:

1 Integral Equations, Origin, and Basic Tools 1

1.1 Various Problems as Integral Equations 3

1.2 Classification of Integral Equations 24

1.3 Some Important Identities and Basic Definitions 31

1.3.1 Multiple Integrals Reduced to Single Integrals 31

1.3.2 Generalized Leibnitz Formula 33

1.3.3 Convergence of Integrals and Basic Definitions 36

1.4 Laplace, Fourier, and Other Transforms 42

1.4.1 The Laplace Transform 43

1.4.2 Fourier Transforms 53

1.4.3 Other Transforms 70

1.5 Basic Numerical Integration Formulas 79

1.5.1 Basic (Elementary) Integration Formulas 80

1.5.2 The Smoothing Effect of Integration 86

1.5.3 Interpolation of the Numerical Solutions of Integral Equations 87

1.5.4 Review of Cramer's Rule 93

2 Modeling of Problems as Integral Equations 97

2.1 Population Dynamics 98

2.1.1 Human Population 98

2.1.2 Biological Species Living Together 100

2.2 Control and Other Problems 104

2.2.1 Mortality of Equipment and Rate of Replacement 104

2.3 Mechanics Problems 107

2.3.1 Hanging Chain 107

2.3.2 Sliding a Bead Along a Wire: Abel's Problem 110

2.4 Initial Value Problems Reduced to Volterra Integral Equations 113

2.5 Boundary Value Problems Reduced to Fredholm Integral Equations 118

2.6 Mixed Boundary Conditions: Dual Integral Equations 124

2.6.1 Electrified Infinite Plane 124

2.6.2 Electrified Disc 126

2.7 Integral Equations in Higher Dimensions 128

2.7.1 Schrodinger Equation as an Integral Equation in the Three-Dimensional Momentum Space 129

3 Volterra Integral Equations 133

3.1 Volterra Equations of the Second Kind 134

3.1.1 Resolvent Kernel Method: Neumann Series 134

3.1.2 Method of Successive Approximations (Iterations) 141

3.1.3 Laplace Transform Method: Difference Kernel 143

3.2 Volterra Integral Equations of the First Kind 148

3.2.1 Volterra Integral Equation of the First Kind with a Difference Kernel

Laplace Transform Method 150

3.3 Numerical Solution of Volterra Integral Equations 156

3.3.1 Numerical Approximation Setting of Volterra Equations 156

4 The Green's Function 165

4.1 Construction of the Green's Function 165

4.1.1 Nonhomogeneous Differential Equations 166

4.1.2 Construction of the Green's Function

Variation of Parameters Method 169

4.1.3 Orthogonal Series Representation of Green's Function 187

4.1.4 Green's Function in Two Dimensions 192

4.2 Fredholm Integral Equations and the Green's Function 202

5 Fredholm Integral Equations 209

5.1 Fredholm Integral Equations with Degenerate Kernel 211

5.1.1 Nonhomogeneous Fredholm Equations with Degenerate Kernel 211

5.1.2 Fredholm Alternative 215

5.1.3 Approximating a Kernel by a Degenerate One 231

5.2 Fredholm Integral Equations with Symmetric Kernel 237

5.2.1 Homogeneous Fredholm Equations with Symmetric Kernel 237

5.2.2 Solution of Fredholm Equations of the Second Kind with Symmetric Kernel 245

5.3 Fredholm Integral Equations of the Second Kind 253

5.3.1 Method of Fredholm Resolvent Kernel 253

5.3.2 Method of Iterated Kernels 256

5.3.3 Some Basic Approximate Methods 262

5.4 Fredholm Integral Equations of the First Kind 271

5.4.1 Fredholm Equations of the First Kind with Symmetric Kernels 272

5.4.2 Ill-Posed Problems and the Fredholm Equation of the First Kind 277

5.5 Numerical Solution of Fredholm Integral Equations 285

5.5.1 Numerical Approximation Setting of Fredholm Integral Equations 286

5.5.2 Homogeneous Fredholm Equations 291

6 Existence of the Solutions: Basic Fixed Point Theorems 299

6.1 Preliminaries: Toward a Contractive Mapping 299

6.1.1 Basic Definitions: Complete Metric Spaces 305

6.1.2 Contractive Mapping for Linear Fredholm Equations 308

6.1.3 Contractive Mapping for Linear Volterra Equations 310

6.2 Fixed Point Theorem of Banach 313

6.2.1 Existence of the Solution for Linear Integral Equations 316

6.2.2 Existence of the Solution for Nonlinear Integral Equations 319

6.2.3 Existence of the Solution for Nonlinear Differential Equations 324

7 Higher Quadrature Rules for the Numerical Solutions 327

7.1 Higher Quadrature Rules of Integration with Tables 327

7.2 Higher Quadrature Rules for Volterra Equations 350

7.3 Higher Quadrature Rules for Fredholm Equations 358

7.3.1 Comments on Higher Quadrature Rules for Some Singular Fredholm Equations 365

Appendix A The Hankel Transforms 373

A.1 The Hankel Transform for the Electrified Disc 373

A.2 The Finite Hankel Transform 375

Appendix B Green's Function for Various Boundary Value Problems 379

B.1 Green's Functions in Terms of Simple Functions 379

B.2 Green's Function in Terms of Special Functions 382.

Notes:

"A Wiley-Interscience Publication."

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1932 Fund.

ISBN:

0471317349

OCLC:

40979823