Integral Equations: Theory and Numerical Treatment

Front Cover
Springer Science & Business Media, Jun 1, 1995 - Mathematics - 362 pages
0 Reviews
The theory of integral equations has been an active research field for many years and is based on analysis, function theory, and functional analysis. On the other hand, integral equations are of practical interest because of the «boundary integral equation method», which transforms partial differential equations on a domain into integral equations over its boundary. This book grew out of a series of lectures given by the author at the Ruhr-Universitat Bochum and the Christian-Albrecht-Universitat zu Kiel to students of mathematics. The contents of the first six chapters correspond to an intensive lecture course of four hours per week for a semester. Readers of the book require background from analysis and the foundations of numeri cal mathematics. Knowledge of functional analysis is helpful, but to begin with some basic facts about Banach and Hilbert spaces are sufficient. The theoretical part of this book is reduced to a minimum; in Chapters 2, 4, and 5 more importance is attached to the numerical treatment of the integral equations than to their theory. Important parts of functional analysis (e. g. , the Riesz-Schauder theory) are presented without proof. We expect the reader either to be already familiar with functional analysis or to become motivated by the practical examples given here to read a book about this topic. We recall that also from a historical point of view, functional analysis was initially stimulated by the investigation of integral equations.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Introduction
1
12 Basics from Analysis
3
123 Holder Continuous Functions
4
13 Basics from Functional Analysis
5
132 Banach Spaces CD CD CD
6
133 Banach Spaces L1D L2D LiD
7
134 Dense Subspaces
8
136 Linear Operators
9
526 Convergence
167
53 The Hierarchy of Discrete Problems
168
533 Relative Consistency
172
534 Convergence
173
54 TwoGrid Iteration
174
542 Convergence Analysis
175
543 Amount of Computational Work
176
544 Variant for ApI
178

137 Theorem of Uniform Boundedness
10
138 Compact Sets and Compact Mappings
11
139 RieszSchauder Theory
14
14 Basics from Numerical Mathematics
15
142 Quadrature
19
143 Condition Number of a System of Equations
23
Volterra Integral Equations
25
212 Regularity of the Solution
27
22 Numerical Solution by Quadrature Methods
29
222 Error Estimate
30
23 Further Numerical Methods
37
24 Linear Volterra Integral Equations of Convolution Type
39
25 Volterra Integral Equations of the First Kind
41
Theory of Fredholm Integral Equations of the Second Kind
42
32 Compactness of the Integral Operator X
43
322 The Case XCD
45
323 The Case XLD
47
33 Finite Approximability of the Integral Operator 1C
48
332 Degenerate Kernels
49
34 The Image Space of 1C
50
342 The Image Kf for
52
343 Kernels with Integrable Singularity
54
344 Compactness
57
35 Solution of the Fredholm Integral Equation of the Second Kind
58
Numerical Treatment of Fredholm Integral Equations of the Second Kind
59
412 Consistency and Stability
60
413 Convergence
61
414 Stability and Convergence Theorem
62
415 Error Estimates
64
42 Discretisation by Kernel Approximation
65
422 Setting Up the System of Equations
66
423 Kernel Approximation by Interpolation
67
424 Tensor Approximation of k
68
425 Examples of Kernel Approximations
69
426 A Variant of the Kernel Approximation
70
428 Numerical Examples
73
43 Projection Methods in General
75
433 Lemmata
76
434 Discretisation by means of a Projection
77
435 Convergence Analysis
79
44 Collocation Method
81
442 Setting up the System of Linear Equations
82
443 Examples for Interpolations
84
444 Condition Number of the System of Equations
86
445 Numerical Examples
88
45 Galerkin Method
91
452 Derivation of the System of Equations
92
453 Convergence in LD and LeoD
93
454 Error Estimates
96
4SS Condition Number of the System of Equations
97
Piecewise Constant Functions
100
Piecewise Linear Functions
104
458 General Analysis of Projection Errors
105
Piecewise Linear Functions
107
4510 Numerical Examples
109
46 Additional Comments Concerning Projection Methods
111
462 Estimates with Respect to Weaker Norms
112
463 The Iterated Approximation
116
464 Superconvergence
118
465 More General Formulations of the Projection Method
121
466 Numerical Quadrature
123
467 Product Integration
126
The Nystrom Method
128
472 Convergence Analysis
130
473 Stability
132
474 Consistency Order
136
475 Condition Number of the System of Equations
137
476 Regularisation
138
477 Numerical Examples
139
48 Supplements
141
4813 From the Collocation to the Nystrom Method
142
4814 From the Collocation to the Galerkin Method
143
483 Extrapolation Method
144
484 Eigenvalue Problems
147
485 Complementary Integral Equations
152
Perturbation Theorem for Stability
154
MultiGrid Methods for Solving Systems Arising from Integral Equations of the Second Kind
155
512 Direct Solution of the System of Equations
156
514 Conjugate Gradient Method
158
52 Stability and Convergence Discrete Formulation
160
522 The Banach Space Y and the Discrete Spaces Yn
163
523 The Interpolation Error or Projection Error
165
525 Stability
166
545 Numerical Examples
179
55 MultiGrid Iteration
181
552 Amount of Computational Work
183
553 Convergence
184
554 Numerical Examples
188
555 Variants of the MultiGrid Methods
190
56 Nested Iteration
195
562 Amount of Computational Work
196
563 Convergence
197
564 Numerical Examples
199
6 Abels Integral Equation
201
613 Improper Integrals
203
62 A Necessary Condition for a Bounded Solution
206
63 Eulers Integrals
207
64 Inversion of Abels Integral Equation
208
65 Reformulation for Kernels kxyxyfi
213
66 Numerical Methods for Abels Integral Equation
214
Singular Integral Equations
216
712 Curvilinear Integrals
220
713 Cauchys Principal Value for Curvilinear Integrals
222
714 The Example ffCJ2fCzJ
224
72 The Cauchy Kernel
230
722 Regularity Properties
234
723 Properties of the Generated Holomorphic Function
236
724 Representation of K
244
725 The Cauchy Integral on the Unit Circle
245
73 The Singular Integral Equation
247
733 General Singular Integral Equations
248
734 Approximation of the Cauchy Integral on the Unit Circle
249
73 S Approximation of the Cauchy Integral on an Arbitrary Curve F
250
736 MultiGrid Methods for Equations of a Special Form
251
74 Application to the Dirichlet Problem for Laplaces Equation
253
743 Uniqueness and Representation Theorem
257
744 The Case of a Smooth Boundary T
259
745 The DoubleLayer Potential for Solving the Exterior Problem
260
746 The Tangential Derivative of the SingleLayer Potential
262
75 Hypersingular Integrals
264
The Integral Equation Method
266
812 Continuity of the SingleLayer Potential
268
8123 Improper Integrals on Surfaces
269
8124 Properties of the SingleLayer Potential
271
8132 The Cauchy Principal Value for Surface Integrals
275
8133 Other Directional Derivatives
279
814 Formulation of the Di rich let Boundary Value Problem as Integral Equation of the First Kind for the SingleLayer Potential
281
8142 The Integral Equation of the First Kind
283
81S Formulation of the Neumann Boundary Value Problem as Integral Equation of the Second Kind for the SingleLayer Potential
284
82 The DoubleLayer Potential
287
822 Regularity Properties of the DoubleLayer Integral Operator
289
823 Jump Properties of the DoubleLayer Potential
292
824 Further Properties of the DoubleLayer Potential
294
8242 The Potential close to the Jump Discontinuity of the Density
295
8243 The DoubleLayer Potential of the Density f1
296
825 Derivatives of the DoubleLayer Potential
299
826 Integral Equations with the DoubleLayer Operator
303
8262 Formulation of the Neumann Boundary Value Problem as Integral Equation of the Second Kind with the Double Layer Operator
305
827 NonSmooth Curves or Surfaces
306
83 The Hypersingular Integral Equation
309
Integral Equations for the Laplace Equation
313
85 The Integral Equation Method for Other Differential Equations
314
852 Equations of Higher Order
315
853 Systems of Differential Equations
316
The Boundary Element Method
318
913 Collocation Method
319
914 Convergence in the Compact Case
320
92 The Boundary Elements
323
922 Geometric Discretisation
324
923 Elements in the ThreeDimensional Case
325
924 Error Considerations
326
93 MultiGrid Methods
327
932 Equations of the First Kind
328
94 Integration and Numerical Quadrature
330
943 Nearly Singular Integrals
331
944 Strongly Singular Integrals
333
95 Solution of Inhomogeneous Equations
335
96 Computation of the Potential
336
964 Extrapolation
337
972 Panels
338
9733 Admissible Clusters and Admissible Coverings
339
9734 The Algorithm for MatrixVector Multiplication
340
974 The Additional Quadrature Error
342
Bibliography
344
Index
353
Copyright

Other editions - View all

Common terms and phrases

References to this book

All Book Search results »

Bibliographic information