Verfügbarkeit: The numerical solution of integral equations of the second kind

The numerical solution of integral equations of the second kind:

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Bibliographische Detailangaben
Beteilige Person:	Atkinson, Kendall E. 1940- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Cambridge [u.a.] Cambridge Univ. Press 2009
Ausgabe:	Digitally print. version
Schriftenreihe:	Cambridge monographs on applied and computational mathematics 4
Schlagwörter:	Integral equations > Numerical solutions Integralgleichung Numerisches Verfahren
Links:	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
Umfang:	XVI, 552 S. graph. Darst.
ISBN:	0521102839 9780521102834

Internformat

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Datensatz im Suchindex

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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) 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
any_adam_object	1
author	Atkinson, Kendall E. 1940-
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id	DE-604.BV035381409
illustrated	Illustrated
indexdate	2024-12-20T13:29:45Z
institution	BVB
isbn	0521102839 9780521102834
language	English
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physical	XVI, 552 S. graph. Darst.
publishDate	2009
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publisher	Cambridge Univ. Press
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series	Cambridge monographs on applied and computational mathematics
series2	Cambridge monographs on applied and computational mathematics
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

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