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
2009
|
| Edition: | Digitally print. version |
| Series: | Cambridge monographs on applied and computational mathematics
4 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 552 S. graph. Darst. |
| ISBN: | 0521102839 9780521102834 |
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MARC
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| 100 | 1 | |a Atkinson, Kendall E. |d 1940- |e Verfasser |0 (DE-588)12286977X |4 aut | |
| 245 | 1 | 0 | |a The numerical solution of integral equations of the second kind |c Kendall E. Atkinson |
| 250 | |a Digitally print. version | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2009 | |
| 300 | |a XVI, 552 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Cambridge monographs on applied and computational mathematics |v 4 | |
| 650 | 4 | |a Integral equations |x Numerical solutions | |
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Record in the Search Index
| _version_ | 1804138711135289344 |
|---|---|
| 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)
7
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 mtegral 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
Nyström
method
100
4.1
The
Nyström
method for continuous kernel functions
100
Contents ix
4.1.1
Properties and error analysis of the
Nyström
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
{τ*}
<£
{í,}
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
Nyström
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
Nyström
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
б
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
Nyström
method
248
6.2.1
Iteration method
1
for
Nyström s
method
249
Implementation for solving the linear system
254
Operations count
256
6.2.2
Iteration method
2
for
Nyström 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 Ck
296
6.5.2
The conjugate gradient iteration for
Nyström
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 38°
Discussion of the literature 381
8
Boundary integral equations on a piecewise smooth planar
boundary
^84
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 39°
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
Nyström
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
5
when
S
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
(2π
+
Vnt) pn
=
Pnf
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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| id | DE-604.BV035381409 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:32:31Z |
| institution | BVB |
| isbn | 0521102839 9780521102834 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017185665 |
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| owner | DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-11 |
| physical | XVI, 552 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge monographs on applied and computational mathematics |
| series2 | Cambridge monographs on applied and computational mathematics |
| spelling | Atkinson, Kendall E. 1940- Verfasser (DE-588)12286977X aut The numerical solution of integral equations of the second kind Kendall E. Atkinson Digitally print. version Cambridge [u.a.] Cambridge Univ. Press 2009 XVI, 552 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on applied and computational mathematics 4 Integral equations Numerical solutions Integralgleichung (DE-588)4027229-1 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 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 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017185665&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Atkinson, Kendall E. 1940- The numerical solution of integral equations of the second kind Cambridge monographs on applied and computational mathematics Integral equations Numerical solutions Integralgleichung (DE-588)4027229-1 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4027229-1 (DE-588)4128130-5 |
| 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 | Integral equations Numerical solutions Integralgleichung (DE-588)4027229-1 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Integral equations Numerical solutions Integralgleichung Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017185665&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011073737 |
| work_keys_str_mv | AT atkinsonkendalle thenumericalsolutionofintegralequationsofthesecondkind |