The numerical solution of integral equations of the second kind:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1997
|
| Edition: | 1. publ. |
| Series: | Cambridge monographs on applied and computational mathematics
4 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 552 S. graph. Darst. |
| ISBN: | 9780521583916 0521583918 |
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MARC
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| 100 | 1 | |a Atkinson, Kendall E. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The numerical solution of integral equations of the second kind |c Kendall E. Atkinson |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 1997 | |
| 300 | |a XVI, 552 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Cambridge monographs on applied and computational mathematics |v 4 | |
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| 650 | 7 | |a Numerieke methoden |2 gtt | |
| 650 | 7 | |a Équations intégrales - Solutions numériques |2 ram | |
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Record in the Search Index
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| adam_text | Contents
Preface page xv
1 A brief discussion of integral equations 1
1.1 Types of integral equations 1
1.1.1 Volterra integral equations of the second kind 1
1.1.2 Volterra integral equations of the first kind 2
1.1.3 Abel integral equations of the first kind 3
1.1.4 Fredholm integral equations of the second kind 3
1.1.5 Fredholm integral equations of the first kind 3
1.1.6 Boundary integral equations 4
1.1.7 Wiener Hopf integral equations 5
1.1.8 Cauchy singular integral equations 5
1.2 Compact integral operators 6
1.2.1 Compact integral operators on C(D) 1
1.2.2 Properties of compact operators 8
1.2.3 Integral operators on L2(a, b) 11
1.3 The Fredholm alternative theorem 13
1.4 Additional results on Fredholm integral equations 17
1.5 Noncompact integral operators 20
1.5.1 An Abel integral equation 20
1.5.2 Cauchy singular integral operators 20
1.5.3 Wiener Hopf integral operators 21
Discussion of the literature 21
2 Degenerate kernel methods 23
2.1 General theory 23
2.1.1 Solution of degenerate kernel integral equation 26
vii
viii Contents
2.2 Taylor series approximations 29
2.2.1 Conditioning of the linear system 34
2.3 Interpolatory degenerate kernel approximations 36
2.3.1 Interpolation with respect to the variable t 37
2.3.2 Interpolation with respect to the variable s 38
2.3.3 Piecewise linear interpolation 38
2.3.4 Approximate calculation of the linear system 42
2.4 Orthonormal expansions 45
Discussion of the literature 47
3 Projection methods 49
3.1 General theory 49
3.1.1 Collocation methods 50
3.1.2 Galerkin s method 52
3.1.3 The general framework 54
3.2 Examples of the collocation method 58
3.2.1 Piecewise linear interpolation 59
3.2.2 Collocation with trigonometric polynomials 62
3.3 Examples of Galerkin s method 66
3.3.1 Piecewise linear approximations 66
3.3.2 Galerkin s method with trigonometric polynomials 68
3.3.3 Uniform convergence 70
3.4 Iterated projection methods 71
3.4.1 The iterated Galerkin solution 74
3.4.2 Uniform convergence of iterated Galerkin
approximations 75
3.4.3 The iterated collocation solution 77
3.4.4 Piecewise polynomial collocation at Gauss Legendre
nodes 81
3.4.5 The linear system for the iterated collocation solution 85
3.5 Regularization of the solution 86
3.6 Condition numbers 88
3.6.1 Condition numbers for the collocation method 90
3.6.2 Condition numbers based on the iterated collocation
solution 94
3.6.3 Condition numbers for the Galerkin method 94
Discussion of the literature 98
4 The Nystrom method 100
4.1 The Nystrom method for continuous kernel functions 100
Contents ix
4.1.1 Properties and error analysis of the Nystrom method 103
An asymptotic error estimate 111
Conditioning of the linear system 112
4.1.2 Collectively compact operator approximations 114
4.2 Product integration methods 116
4.2.1 Computation of the quadrature weights 118
4.2.2 Error analysis 120
4.2.3 Generalizations to other kernel functions 122
4.2.4 Improved error results for special kernels 124
4.2.5 Product integration with graded meshes 125
Application to integral equations 132
The relationship of product integration and collocation
methods 134
4.3 Discrete collocation methods 135
4.3.1 Convergence analysis for {t*} (_ {t{ 139
4.4 Discrete Galerkin methods 142
4.4.1 The discrete orthogonal projection operator 144
4.4.2 An abstract formulation 147
Discussion of the literature 154
5 Solving multivariable integral equations 157
5.1 Multivariable interpolation and numerical integration 157
5.1.1 Interpolation over triangles 160
Piecewise polynomial interpolation 163
Interpolation error formulas over triangles 165
5.1.2 Numerical integration over triangles 167
Some quadrature formulas based on interpolation 169
Other quadrature formulas 170
Error formulas for composite numerical integration
formulas 171
How to refine a triangulation 173
5.2 Solving integral equations on polygonal regions 175
5.2.1 Collocation methods 176
The iterated collocation method and superconvergence 178
5.2.2 Galerkin methods 181
Uniform convergence 183
5.2.3 The Nystrom method 184
Discrete Galerkin methods 186
5.3 Interpolation and numerical integration on surfaces 188
5.3.1 Interpolation over a surface 189
x Contents
5.3.2 Numerical integration over a surface 191
5.3.3 Approximating the surface 192
5.3.4 Nonconforming triangulations 204
5.4 Boundary element methods for solving integral equations 205
5.4.1 The Nystrom method 205
Using the approximate surface 207
5.4.2 Collocation methods 213
Using the approximate surface 215
Discrete collocation methods 217
5.4.3 Galerkin methods 218
Discrete Galerkin methods 221
5.5 Global approximation methods on smooth surfaces 222
5.5.1 Spherical polynomials and spherical harmonics 224
Best approximations 228
5.5.2 Numerical integration on the sphere 229
A discrete orthogonal projection operator 232
5.5.3 Solution of integral equations on the unit sphere 235
A Galerkin method 236
A discrete Galerkin method 237
Discussion of the literature 239
6 Iteration methods 241
6.1 Solving degenerate kernel integral equations by iteration 242
6.1.1 Implementation 244
6.2 Two grid iteration for the Nystrom method 248
6.2.1 Iteration method 1 for Nystrom s method 249
Implementation for solving the linear system 254
Operations count 256
6.2.2 Iteration method 2 for Nystrom s method 258
Implementation for solving the linear system 261
Operations count 265
An algorithm with automatic error control 266
6.3 Two grid iteration for collocation methods 267
6.3.1 Prolongation and restriction operators 269
6.3.2 The two grid iteration method 272
An alternative formulation 280
Operations count 280
6.4 Multigrid iteration for collocation methods 281
6.4.1 Operations count 288
6.5 The conjugate gradient method 291
Contents xi
6.5.1 The conjugate gradient method for the undiscretized
integral equation 291
Bounds on q 296
6.5.2 The conjugate gradient iteration for Nystrom s method 298
The conjugate gradient method and its convergence 299
6.5.3 Nonsymmetric integral equations 301
Discussion of the literature 303
7 Boundary integral equations on a smooth planar boundary 306
7.1 Boundary integral equations 307
7.1.1 Green s identities and representation formula 308
7.1.2 The Kelvin transformation and exterior problems 310
7.1.3 Boundary integral equations of direct type 314
The interior Dirichlet problem 315
The interior Neumann problem 315
The exterior Neumann problem 316
The exterior Dirichlet problem 317
7.1.4 Boundary integral equations of indirect type 317
Double layer potentials 318
Single layer potentials 319
7.2 Boundary integral equations of the second kind 320
7.2.1 Evaluation of the double layer potential 324
7.2.2 The exterior Neumann problem 328
7.2.3 Other boundary value problems 333
7.3 Boundary integral equations of the first kind 338
7.3.1 Sobolev spaces 338
The trapezoidal rule and trigonometric interpolation 341
7.3.2 Some pseudodifferential equations 342
The Cauchy singular integral operator 344
A hypersingular integral operator 346
Pseudodifferential operators 349
7.3.3 Two numerical methods 349
A discrete Galerkin method 351
7.4 Finite element methods 359
7.4.1 Sobolev spaces — A further discussion 360
Extensions of boundary integral operators 363
7.4.2 An abstract framework 364
A general existence theorem 367
An abstract finite element theory 372
The finite element solution as a projection 375
xii Contents
7.4.3 Boundary element methods for boundary integral equations 376
Additional remarks 380
Discussion of the literature 3 81
8 Boundary integral equations on a piecewise smooth planar
boundary 384
8.1 Theoretical behavior 385
8.1.1 Boundary integral equations for the interior Dirichlet
problem 387
8.1.2 An indirect method for the Dirichlet problem 389
8.1.3 A BIE on an open wedge 390
8.1.4 A decomposition of the boundary integral equation 394
8.2 The Galerkin method 397
8.2.1 Superconvergence results 403
8.3 The collocation method 404
8.3.1 Preliminary definitions and assumptions 406
Graded meshes 408
8.3.2 The collocation method 410
A modified collocation method 412
8.4 The Nystrom method 418
8.4.1 Error analysis 421
Discussion of the literature 425
9 Boundary integral equations in three dimensions 427
9.1 Boundary integral representations 428
9.1.1 Green s representation formula 430
The existence of the single and double layer potentials 431
Exterior problems and the Kelvin transform 432
Green s representation formula for exterior regions 434
9.1.2 Direct boundary integral equations 435
9.1.3 Indirect boundary integral equations 437
9.1.4 Properties of the integral operators 439
9.1.5 Properties of K. and S when 5 is only piecewise
smooth 442
9.2 Boundary element collocation methods on smooth surfaces 446
9.2.1 The linear system 455
Numerical integration of singular integrals 457
Numerical integration of nonsingular integrals 460
9.2.2 Solving the linear system 462
9.2.3 Experiments for a first kind equation 467
Contents xiii
9.3.1 The collocation method 472
Applications to various interpolatory projections 474
Numerical integration and surface approximation 474
9.3.2 Iterative solution of the linear system 479
9.3.3 Collocation methods for polyhedral regions 486
9.4 Boundary element Galerkin methods 489
9.4.1 A finite element method for an equation of the first kind 492
Generalizations to other boundary integral equations 496
9.5 Numerical methods using spherical polynomial approximations 496
9.5.1 The linear system for (2jr + VnK.)pn = VJ 501
9.5.2 Solution of an integral equation of the first kind 504
Implementation of the Galerkin method 509
Other boundary integral equations and general
comments 511
Discussion of the literature 512
Appendix: Results from functional analysis 516
Bibliography 519
Index 547
|
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| spelling | Atkinson, Kendall E. Verfasser aut The numerical solution of integral equations of the second kind Kendall E. Atkinson 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1997 XVI, 552 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on applied and computational mathematics 4 Integraalvergelijkingen gtt Numerieke methoden gtt Équations intégrales - Solutions numériques ram Integral equations Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Integralgleichung (DE-588)4027229-1 gnd rswk-swf Integralgleichung (DE-588)4027229-1 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Cambridge monographs on applied and computational mathematics 4 (DE-604)BV011073737 4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007756445&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Atkinson, Kendall E. The numerical solution of integral equations of the second kind Cambridge monographs on applied and computational mathematics Integraalvergelijkingen gtt Numerieke methoden gtt Équations intégrales - Solutions numériques ram Integral equations Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd Integralgleichung (DE-588)4027229-1 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4027229-1 |
| title | The numerical solution of integral equations of the second kind |
| title_auth | The numerical solution of integral equations of the second kind |
| title_exact_search | The numerical solution of integral equations of the second kind |
| title_full | The numerical solution of integral equations of the second kind Kendall E. Atkinson |
| title_fullStr | The numerical solution of integral equations of the second kind Kendall E. Atkinson |
| title_full_unstemmed | The numerical solution of integral equations of the second kind Kendall E. Atkinson |
| title_short | The numerical solution of integral equations of the second kind |
| title_sort | the numerical solution of integral equations of the second kind |
| topic | Integraalvergelijkingen gtt Numerieke methoden gtt Équations intégrales - Solutions numériques ram Integral equations Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd Integralgleichung (DE-588)4027229-1 gnd |
| topic_facet | Integraalvergelijkingen Numerieke methoden Équations intégrales - Solutions numériques Integral equations Numerical solutions Numerisches Verfahren Integralgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007756445&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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