Computational Methods for Linear Integral Equations

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Springer Science & Business Media, Apr 26, 2002 - Mathematics - 508 pages
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This book presents numerical methods and computational aspects for linear integral equations. Such equations occur in various areas of applied mathematics, physics, and engineering. The material covered in this book, though not exhaustive, offers useful techniques for solving a variety of problems. Historical information cover ing the nineteenth and twentieth centuries is available in fragments in Kantorovich and Krylov (1958), Anselone (1964), Mikhlin (1967), Lonseth (1977), Atkinson (1976), Baker (1978), Kondo (1991), and Brunner (1997). Integral equations are encountered in a variety of applications in many fields including continuum mechanics, potential theory, geophysics, electricity and mag netism, kinetic theory of gases, hereditary phenomena in physics and biology, renewal theory, quantum mechanics, radiation, optimization, optimal control sys tems, communication theory, mathematical economics, population genetics, queue ing theory, and medicine. Most of the boundary value problems involving differ ential equations can be converted into problems in integral equations, but there are certain problems which can be formulated only in terms of integral equations. A computational approach to the solution of integral equations is, therefore, an essential branch of scientific inquiry.
 

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Contents

Introduction
1
12 Classification
7
122 Convolution of Kernels
13
13 Function Spaces
16
14 Convergence
20
15 Inverse Operator
21
16 Nystrom System
23
17 Other Types of Kernels
28
854 A Class of Singular Integrals
242
86 Weakly Singular Volterra Equations
244
Cauchy Singular Equations
252
92 Approximations by Trigonometric Polynomials
254
93 Cauchy Singular Equations of the Second Kind
258
94 From CSK2 to FK2
262
95 Gauss Jacobi Quadrature
264
952 Collocation Method for CSK2
267

172 Degenerate Kernels
31
18 Neumann Series
33
19 Resolvent Operator
35
110 Fredholm Alternative
40
Eigenvalue Problems
44
21 Linear Symmetric Equations
45
211 Ritz Method
46
212 Method of Moments
48
213 Kelloggs Method
51
214 Trace Method
52
22 Residual Methods
55
222 LeastSquares Method
57
223 Galerkin Method
60
23 Degenerate Kernels
63
24 Replacement by a Degenerate Kernel
67
25 Batemanls Method
72
26 Generalized Eigenvalue Problem
77
27 Applications
82
Equations of the Second Kind
90
311 Systems of Integral Equations A system of integral equations
91
312 Taylors Series Method
93
32 Volterra Equations
96
321 Quadrature Methods
98
322 BlockbyBlock Method
102
Classical Methods for FK2
106
42 ProductIntegration Method
108
43 Quadrature Method
111
44 Deferred Correction Methods
115
45 A Modified Quadrature Method
121
46 Collocation Methods
123
47 Elliotts Modification
127
Variational Methods
130
52 RitzGalerkin Methods
136
53 Special Cases
139
54 FredholmNystrom System
144
Iteration Methods
146
62 Quadrature Formulas
150
63 Error Analysis
153
64 Iterative Scheme
162
65 KrylovBogoliubov Method
167
Singular Equations
175
72 Fredholm Theorems
178
73 Modified Quadrature Rule
179
74 ConvolutionType Kernels
181
75 VolterraType Singular Equations
186
76 Convolution Methods
188
762 Convolution of an SVK2
190
764 Sonines Equation
192
765 Logarithmic Kernel
193
77 Asymptotic Methods for LogSingular Equations
194
78 Iteration Methods
197
781 Atkinsons Scheme
198
782 Brakhages Scheme
199
783 Atkinsons Direct Scheme
201
784 Kelleys Algorithm
204
79 Singular Equations with the Hilbert Kernel
207
792 SecondKind Equations
209
710 FinitePart Singular Equations
210
Weakly Singular Equations
213
82 Taylors Series Method
215
83 LpApproximation Method
216
84 ProductIntegration Method
218
841 Atkinsons Method
220
842 Generalization
226
843 Atkinsons Modification
228
834 Asymptotic Expansions
231
85 Splines Method
233
851 From SK2 to IDE
234
852 BSpline Method
235
853 From LogSingular to CauchySingular
241
953 A Special Case
271
954 Generalized Cauchy Kernel
273
96 Collocation Method for CSKl
276
961 Canonical Equation
280
SincGalerkin Methods
286
102 Conformal Maps and Interpolation
289
103 Approximation Theory
296
104 Convergence
297
105 SincGalerkin Scheme
300
106 Computation Guidelines
303
107 SincCollocation Method
304
108 SingleLayer Potential
310
109 DoubleLayer Problem
320
Equations of the First Kind
321
112 Separable Kernels
323
113 Some Theorems
325
114 Numerical Methods
327
1142 Method of Moments
332
1143 Regularization Methods
333
1144 Other Regularizations
337
1145 Comparison
342
1146 Prom FKl to FK2
344
1147 Equations over a Contour
346
115 Volterra Equations of the First Kind
347
1151 Quadrature Method
348
1152 Method of Linz
349
1153 ProductIntegration Method
354
116 Abells Equation
357
117 Iterative Schemes
361
1172 van den Bergs Schemence
367
Inversion of Laplace Transforms
375
122 General Interpolating Scheme
377
1221 Inverse in Terms of Legendre Polynomials
378
1222 Inverse in Terms of Shifted Legendre Polynomials
380
1223 Method of Bellman Kalaba and Locket
381
1224 Solution of the System
382
1225 Inverse in Terms of Laguerre Polynomials
385
1226 Inverse in Terms of Chebyshev Polynomials
393
1227 Inverse in Terms of Jacobi Polynomials
396
123 Inversion by Fourier Series
399
1232 Error Analysis
401
1233 Examples
402
1234 Durbins Improvement
403
1235 Error Analysis
404
1236 Examples
405
124 Inversion by the Riemann Sum
410
125 Approximate Formulas
412
Quadrature Rules
416
A2 Gaussian Quadrature
420
A3 Integration of Products
423
A4 Singular Integrals
430
A5 InfiniteRange Integrals
432
A6 Linear Transformation of Quadratures
433
A7 Trigonometric Polynomials
434
A8 Condition Number
435
A9 Quadrature Tables
438
Orthogonal Polynomials
444
B1 Zeros of Some Orthogonal Polynomials
446
Whittakerls Cardinal Function
452
C2 Approximation of an Integral
457
Singular Integrals
459
D2 PV of a Singular Integral on a Contour
461
D3 Hadamards FinitePart Integrals
464
D4 TwoSided FinitePart Integrals
465
D5 OneSided FinitePart Integrals
467
D6 Examples of Cauchy PV Integrals
468
D7 Examples of Hadamards FinitePart Integrals
471
Bibliography
473
Subject Index
501
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Page 479 - Technical Details", NIREX Safety Studies Report NSS/336, Part B, Version 3C. 1994. [8] A. Talbot, "The Accurate Numerical Inversion of Laplace Transforms", J. Inst. Math. Applications. Vol. 23, pp 97-120, 1979. [9] FR Dehoog, JH Knight and AN Stokes "An Improved Method for Numerical Inversion of Laplace Transforms", SIAM J. Stat. Comput.. Vol.3, no. 3, 1982. EXPERIMENTAL STUDY ON GROUNDWATER FLOW AND MASS TRANSPORT IN A HETEROGENEOUS POROUS MEDIUM K.OICHIRO HATANAKA*. SHINGO WATARI*, MASAHIRO UCHIDA',...
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