The Numerical Solution of Integral Equations of the Second Kind

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9780521583916 | 0521583918
Cover: Hardcover
Copyright: 6/28/1997

Rent

(Recommended)

$106.42

Term

Due

Price

Semester

Aug 2

$118.24

Quarter

Jul 22

$112.33

Short Term

Jun 7

$106.42

*This item is part of an exclusive publisher rental program and requires an additional convenience fee. This fee will be reflected in the shopping cart.
Buy New

Special Order: 1-2 Weeks

$161.55
eBook

Available Instantly

Online: 180 Days

Downloadable: 180 Days

$70.20

This book provides an extensive introduction to the numerical solution of a large class of integral equations. The initial chapters provide a general framework for the numerical analysis of Fredholm integral equations of the second kind, covering degenerate kernel, projection and Nystrom methods. Additional discussions of multivariable integral equations and iteration methods update the reader on the present state of the art in this area. The final chapters focus on the numerical solution of boundary integral equation (BIE) reformulations of Laplace's equation, in both two and three dimensions. Two chapters are devoted to planar BIE problems, which include both existing methods and remaining questions. Practical problems for BIE such as the set up and solution of the discretised BIE are also discussed. Each chapter concludes with a discussion of the literature and a large bibliography serves as an extended resource for students and researchers needing more information on solving particular integral equations.

Preface

1 A brief discussion of integral equations

(22)

1.1 Types of integral equations

(5)

1.1.1 Volterra integral equations of the second kind

(1)

1.1.2 Volterra integral equations of the first kind

(1)

1.1.3 Abel integral equations of the first kind

(1)

1.1.4 Fredholm integral equations of the second kind

(1)

1.1.5 Fredholm integral equations of the first kind

(1)

1.1.6 Boundary integral equations

(1)

1.1.7 Wiener-Hopf integral equations

(1)

1.1.8 Cauchy singular integral equations

(1)

1.2 Compact integral operators

(7)

1.2.1 Compact integral operators on C(D)

(1)

1.2.2 Properties of compact operators

(3)

1.2.3 Integral operators on L(2)(a, b)

(2)

1.3 The Fredholm alternative theorem

(4)

1.4 Additional results on Fredholm integral equations

(3)

1.5 Noncompact integral operators

(1)

1.5.1 An Abel integral equation

(1)

1.5.2 Cauchy singular integral operators

(1)

1.5.3 Wiener-Hopf integral operators

(1)

Discussion of the literature

(2)

2 Degenerate kernel methods

(26)

2.1 General theory

(6)

2.1.1 Solution of degenerate kernel integral equation

(3)

2.2 Taylor series approximations

(7)

2.2.1 Conditioning of the linear system

(2)

2.3 Interpolatory degenerate kernel approximations

(9)

2.3.1 Interpolation with respect to the variable t

(1)

2.3.2 Interpolation with respect to the variable s

(1)

2.3.3 Piecewise linear interpolation

(4)

2.3.4 Approximate calculation of the linear system

(3)

2.4 Orthonormal expansions

(2)

Discussion of the literature

(2)

3 Projection methods

(51)

3.1 General theory

(9)

3.1.1 Collocation methods

(2)

3.1.2 Galerkin's method

(2)

3.1.3 The general framework

(4)

3.2 Examples of the collocation method

(8)

3.2.1 Piecewise linear interpolation

(3)

3.2.2 Collocation with trigonometric polynomials

(4)

3.3 Examples of Galerkin's method

(5)

3.3.1 Piecewise linear approximations

(2)

3.3.2 Galerkin's method with trigonometric polynomials

(2)

3.3.3 Uniform convergence

(1)

3.4 Iterated projection methods

(15)

3.4.1 The iterated Galerkin solution

(1)

3.4.2 Uniform convergence of iterated Galerkin approximations

(2)

3.4.3 The iterated collocation solution

(4)

3.4.4 Piecewise polynomial collocation at Gauss-Legendre nodes

(4)

3.4.5 The linear system for the iterated collocation solution

(1)

3.5 Regularization of the solution

(2)

3.6 Condition numbers

(10)

3.6.1 Condition numbers for the collocation method

(4)

3.6.2 Condition numbers based on the iterated collocation solution

(1)

3.6.3 Condition numbers for the Galerkin method

(4)

Discussion of the literature

(2)

4 The Nystrom method

100

(57)

4.1 The Nystrom method for continuous kernel functions

100

(16)

4.1.1 Properties and error analysis of the Nystrom method

103

(8)

An asymptotic error estimate

111

(1)

Conditioning of the linear system

112

(2)

4.1.2 Collectively compact operator approximations

114

(2)

4.2 Product integration methods

116

(19)

4.2.1 Computation of the quadrature weights

118

(2)

4.2.2 Error analysis

120

(2)

4.2.3 Generalizations to other kernel functions

122

(2)

4.2.4 Improved error results for special kernels

124

(1)

4.2.5 Product integration with graded meshes

125

(7)

Application to integral equations

132

(2)

The relationship of product integration and collocation methods

134

(1)

4.3 Discrete collocation methods

135

(7)

4.3.1 Convergence analysis for {XXX(k)} XXX {t(i)}

139

(3)

4.4 Discrete Galerkin methods

142

(12)

4.4.1 The discrete orthogonal projection operator

144

(3)

4.4.2 An abstract formulation

147

(7)

Discussion of the literature

154

(3)

5 Solving multivariable integral equations

157

(84)

5.1 Multivariable interpolation and numerical integration

157

(18)

5.1.1 Interpolation over triangles

160

(3)

Piecewise polynomial interpolation

163

(2)

Interpolation error formulas over triangles

165

(2)

5.1.2 Numerical integration over triangles

167

(2)

Some quadrature formulas based on interpolation

169

(1)

Other quadrature formulas

170

(1)

Error formulas for composite numerical integration formulas

171

(2)

How to refine a triangulation

173

(2)

5.2 Solving integral equations on polygonal regions

175

(13)

5.2.1 Collocation methods

176

(2)

The iterated collocation method and superconvergence

178

(3)

5.2.2 Galerkin methods

181

(2)

Uniform convergence

183

(1)

5.2.3 The Nystrom method

184

(2)

Discrete Galerkin methods

186

(2)

5.3 Interpolation and numerical integration on surfaces

188

(17)

5.3.1 Interpolation over a surface

189

(2)

5.3.2 Numerical integration over a surface

191

(1)

5.3.3 Approximating the surface

192

(2)

5.3.4 Nonconforming triangulations

204

(1)

5.4 Boundary element methods for solving integral equations

205

(17)

5.4.1 The Nystrom method

205

(2)

Using the approximate surface

207

(6)

5.4.2 Collocation methods

213

(2)

Using the approximate surface

215

(2)

Discrete collocation methods

217

(1)

5.4.3 Galerkin methods

218

(3)

Discrete Galerkin methods

221

(1)

5.5 Global approximation methods on smooth surfaces

222

(17)

5.5.1 Spherical polynomials and spherical harmonics

224

(4)

Best approximations

228

(1)

5.5.2 Numerical integration on the sphere

229

(3)

A discrete orthogonal projection operator

232

(3)

5.5.3 Solution of integral equations on the unit sphere

235

(1)

A Galerkin method

236

(1)

A discrete Galerkin method

237

(2)

Discussion of the literature

239

(2)

6 Iteration methods

241

(65)

6.1 Solving degenerate kernel integral equations by iteration

242

(6)

6.1.1 Implementation

244

(4)

6.2 Two-grid iteration for the Nystrom method

248

(19)

6.2.1 Iteration method 1 for Nystrom's method

249

(5)

Implementation for solving the linear system

254

(2)

Operations count

256

(2)

6.2.2 Iteration method 2 for Nystrom's method

258

(3)

Implementation for solving the linear system

261

(4)

Operations count

265

(1)

An algorithm with automatic error control

266

(1)

6.3 Two-grid iteration for collocation methods

267

(14)

6.3.1 Prolongation and restriction operators

269

(3)

6.3.2 The two-grid iteration method

272

(8)

An alternative formulation

280

(1)

Operations count

280

(1)

6.4 Multigrid iteration for collocation methods

281

(10)

6.4.1 Operations count

288

(3)

6.5 The conjugate gradient method

291

(12)

6.5.1 The conjugate gradient method for the undiscretized integral equation

291

(5)

Bounds on C(k)

296

(2)

6.5.2 The conjugate gradient iteration for Nystrom's method

298

(1)

The conjugate gradient method and its convergence

299

(2)

6.5.3 Nonsymmetric integral equations

301

(2)

Discussion of the literature

303

(3)

7 Boundary integral equations on a smooth planar boundary

306

(78)

7.1 Boundary integral equations

307

(13)

7.1.1 Green's identities and representation formula

308

(2)

7.1.2 The Kelvin transformation and exterior problems

310

(4)

7.1.3 Boundary integral equations of direct type

314

(1)

The interior Dirichlet problem

315

(1)

The interior Neumann problem

315

(1)

The exterior Neumann problem

316

(1)

The exterior Dirichlet problem

317

(1)

7.1.4 Boundary integral equations of indirect type

317

(1)

Double layer potentials

318

(1)

Single layer potentials

319

(1)

7.2 Boundary integral equations of the second kind

320

(18)

7.2.1 Evaluation of the double layer potential

324

(4)

7.2.2 The exterior Neumann problem

328

(5)

7.2.3 Other boundary value problems

333

(5)

7.3 Boundary integral equations of the first kind

338

(21)

7.3.1 Sobolev spaces

338

(3)

The trapezoidal rule and trigonometric interpolation

341

(1)

7.3.2 Some pseudodifferential equations

342

(2)

The Cauchy singular integral operator

344

(2)

A hypersingular integral operator

346

(3)

Pseudodifferential operators

349

(1)

7.3.3 Two numerical methods

349

(2)

A discrete Galerkin method

351

(8)

7.4 Finite element methods

359

(22)

7.4.1 Sobolev spaces - A further discussion

360

(3)

Extensions of boundary integral operators

363

(1)

7.4.2 An abstract framework

364

(3)

A general existence theorem

367

(5)

An abstract finite element theory

372

(3)

The finite element solution as a projection

375

(1)

7.4.3 Boundary element methods for boundary integral equations

376

(4)

Additional remarks

380

(1)

Discussion of the literature

381

(3)

8 Boundary integral equations on a piecewise smooth planar boundary

384

(43)

8.1 Theoretical behavior

385

(12)

8.1.1 Boundary integral equations for the interior Dirichlet problem

387

(2)

8.1.2 An indirect method for the Dirichlet problem

389

(1)

8.1.3 A BIE on an open wedge

390

(4)

8.1.4 A decomposition of the boundary integral equation

394

(3)

8.2 The Galerkin method

397

(7)

8.2.1 Superconvergence results

403

(1)

8.3 The collocation method

404

(14)

8.3.1 Preliminary definitions and assumptions

406

(2)

Graded meshes

408

(2)

8.3.2 The collocation method

410

(2)

A modified collocation method

412

(6)

8.4 The Nystrom method

418

(7)

8.4.1 Error analysis

421

(4)

Discussion of the literature

425

(2)

9 Boundary integral equations in three dimensions

427

(89)

9.1 Boundary integral representations

428

(18)

9.1.1 Green's representation formula

430

(1)

The existence of the single and double layer potentials

431

(1)

Exterior problems and the Kelvin transform

432

(2)

Green's representation formula for exterior regions

434

(1)

9.1.2 Direct boundary integral equations

435

(2)

9.1.3 Indirect boundary integral equations

437

(2)

9.1.4 Properties of the integral operators

439

(3)

9.1.5 Properties of K and S when S is only piecewise smooth

442

(4)

9.2 Boundary element collocation methods on smooth surfaces

446

(43)

9.2.1 The linear system

455

(2)

Numerical integration of singular integrals

457

(3)

Numerical integration of nonsingular integrals

460

(2)

9.2.2 Solving the linear system

462

(5)

9.2.3 Experiments for a first kind equation

467

(5)

9.3.1 The collocation method

472

(2)

Applications to various interpolatory projections

474

(1)

Numerical integration and surface approximation

474

(5)

9.3.2 Iterative solution of the linear system

479

(7)

9.3.3 Collocation methods for polyhedral regions

486

(3)

9.4 Boundary element Galerkin methods

489

(7)

9.4.1 A finite element method for an equation of the first kind

492

(4)

Generalizations to other boundary integral equations

496

(1)

9.5 Numerical methods using spherical polynomial approximations

496

(16)

9.5.1 The linear system for (2XXX + P(n)K)p(n) = P(n)f

501

(3)

9.5.2 Solution of an integral equation of the first kind

504

(5)

Implementation of the Galerkin method

509

(2)

Other boundary integral equations and general comments

511

(1)

Discussion of the literature

512

(4)

Appendix: Results from functional analysis

516

(3)

Bibliography

519

(28)

Index

547

Please wait while the item is added to your bag...

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

The Numerical Solution of Integral Equations of the Second Kind

Summary

Table of Contents

Supplemental Materials