Linear integral equations:
This book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in di...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2014
|
| Ausgabe: | 3. ed. |
| Schriftenreihe: | Applied mathematical sciences
82 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | This book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in differential and integral calculus. The functional analysis which is necessary for an adequate treatment of the theory and the numerical solution of integral equations is developed within the book itself. Problems are included at the end of each chapter. |
| Beschreibung: | XVI, 412 S. graph. Darst. |
| ISBN: | 9781461495925 9781493950164 |
Internformat
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| 245 | 1 | 0 | |a Linear integral equations |c Rainer Kress |
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| 300 | |a XVI, 412 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Applied mathematical sciences |v 82 | |
| 520 | 3 | |a This book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in differential and integral calculus. The functional analysis which is necessary for an adequate treatment of the theory and the numerical solution of integral equations is developed within the book itself. Problems are included at the end of each chapter. | |
| 650 | 4 | |a Lineare Integralgleichung - Lehrbuch | |
| 650 | 4 | |a Integral equations | |
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Datensatz im Suchindex
| _version_ | 1804151772808216576 |
|---|---|
| adam_text | Titel: Linear integral equations
Autor: Kress, Rainer
Jahr: 2014
Contents
1 Introduction and Basic Functional Analysis..............................................1
1.1 Abel s Integral Equation..........................................................................2
1.2 Convergence and Continuity..................................................................5
1.3 Completeness............................................................................................8
1.4 Compactness............................................................................................8
1.5 Scalar Products........................................................................................11
1.6 Best Approximation................................................................................13
Problems............................................................................................................16
2 Bounded and Compact Operators................................................................17
2.1 Bounded Operators..................................................................................17
2.2 The Dual Space........................................................................................20
2.3 Integral Operators....................................................................................22
2.4 Neumann Series........................................................................................23
2.5 Compact Operators..................................................................................25
Problems............................................................................................................32
3 Riesz Theory....................................................................................................33
3.1 Riesz Theory for Compact Operators....................................................33
3.2 Spectral Theory for Compact Operators................................................40
3.3 Volterra Integral Equations......................................................................41
Problems............................................................................................................44
4 Dual Systems and Fredholm Alternative....................................................45
4.1 Dual Systems via Bilinear Forms..........................................................45
4.2 Dual Systems via Sesquilinear Forms....................................................48
4.3 The Fredholm Alternative ......................................................................52
4.4 Boundary Value Problems......................................................................58
Problems............................................................................................................62
xiii
xiv Contents
5 Regularization in Dual Systems....................................................................63
5.1 Regularizers..............................................................................................63
5.2 Normal Solvability..................................................................................65
5.3 Index and Fredholm Operators ..............................................................69
Problems............................................................................................................74
6 Potential Theory..............................................................................................75
6.1 Harmonie Functions................................................................................75
6.2 Boundary Value Problems: Uniqueness................................................83
6.3 Surface Potentials....................................................................................87
6.4 Boundary Value Problems: Existence....................................................91
6.5 Nonsmooth Boundaries ..........................................................................96
Problems......................................................101
7 Singular Boundary Integral Equations...........................103
7.1 Holder Continuity..........................................103
7.2 The Cauchy Integral Operator................................108
7.3 The Riemann Problem......................................116
7.4 Integral Equations with Cauchy Kernel ........................118
7.5 Cauchy Integral and Logarithmic Potential.....................125
7.6 Boundary Integral Equations in Holder Spaces..................129
7.7 Boundary Integral Equations of the First Kind ..................132
7.8 Logarithmic Single-Layer Potential on an Arc...................135
Problems......................................................139
8 Sobolev Spaces.................................................141
8.1 The Sobolev Space HP[0, 2tt].................................141
8.2 The Sobolev Space Hp(r)...................................151
8.3 Weak Solutions to Boundary Value Problems...................159
Problems......................................................170
9 The Heat Equation.............................................171
9.1 Initial Boundary Value Problem: Uniqueness....................171
9.2 Heat Potentials.............................................174
9.3 Initial Boundary Value Problem: Existence.....................179
Problems......................................................181
10 Operator Approximations.......................................183
10.1 Approximations via Norm Convergence........................183
10.2 Uniform Boundedness Principle..............................185
10.3 Collectively Compact Operators..............................190
10.4 Approximations via Pointwise Convergence....................191
10.5 Successive Approximations..................................193
Problems......................................................198
Contents xv
11 Degenerate Kernel Approximation...............................199
11.1 Degenerate Operators and Kernels............................199
11.2 Interpolation...............................................201
11.3 Trigonometrie Interpolation..................................204
11.4 Degenerate Kernels via Interpolation..........................210
11.5 Degenerate Kernels via Expansions...........................215
Problems......................................................218
12 Quadrature Methods...........................................219
12.1 Numerical Integration.......................................219
12.2 Nyström s Method..........................................224
12.3 Weakly Singular Kernels....................................229
12.4 Nyström s Method in Sobolev Spaces..........................237
Problems......................................................239
13 Projection Methods ............................................241
13.1 The Projection Method......................................241
13.2 Projection Methods for Equations of the Second Kind............246
13.3 The Collocation Method.....................................249
13.4 Collocation Methods for Equations of the First Kind.............255
13.5 A Collocation Method for Hypersingular Equations..............263
13.6 The Galerkin Method.......................................267
13.7 The Lippmann-Schwinger Equation...........................272
Problems......................................................278
14 Iterative Solution and Stability..................................279
14.1 Stability of Linear Systems..................................279
14.2 Two-Grid Methods.........................................283
14.3 Multigrid Methods .........................................287
14.4 Fast Matrix-Vector Multiplication.............................292
Problems......................................................296
15 Equations of the First Kind .....................................297
15.1 Ill-Posed Problems.........................................297
15.2 Regularizaron of Ill-Posed Problems..........................301
15.3 Compact Self-Adjoint Operators..............................303
15.4 Singular Value Decomposition ...............................309
15.5 Regularization Schemes.....................................313
Problems......................................................321
16 Tikhonov Regularization........................................323
16.1 Weak Convergence.........................................323
16.2 The Tikhonov Functional....................................324
16.3 Quasi-Solutions............................................329
xvi Contents
16.4 Minimum Norm Solutions...................................334
16.5 Classical Tikhonov Regularization............................338
16.6 Ill-Posed Integral Equations in Potential Theory.................343
Problems......................................................349
17 Regularization by Discretization.................................351
17.1 Projection Methods for Ill-Posed Equations.....................351
17.2 The Moment Method.......................................358
17.3 Hilbert Spaces with Reproducing Kernel.......................360
17.4 Moment Collocation........................................361
Problems......................................................363
18 Inverse Boundary Value Problems...............................365
18.1 An Inverse Problem for the Laplace Equation...................365
18.2 Decomposition Methods.....................................368
18.3 Differentiability with Respect to the Boundary..................378
18.4 Iterative Methods...........................................383
18.5 Sampling Methods.........................................390
Problems......................................................398
References.........................................................399
Index.............................................................409
|
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
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| dewey-search | 510 s 515/.45 21 510 |
| dewey-sort | 3510 S 3515 245 221 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-10T01:00:08Z |
| institution | BVB |
| isbn | 9781461495925 9781493950164 |
| language | English |
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| physical | XVI, 412 S. graph. Darst. |
| publishDate | 2014 |
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| publisher | Springer |
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| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Kress, Rainer 1941- Verfasser (DE-588)115774416 aut Linear integral equations Rainer Kress 3. ed. New York [u.a.] Springer 2014 XVI, 412 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 82 This book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in differential and integral calculus. The functional analysis which is necessary for an adequate treatment of the theory and the numerical solution of integral equations is developed within the book itself. Problems are included at the end of each chapter. Lineare Integralgleichung - Lehrbuch Integral equations Lineare Integralgleichung (DE-588)4114426-0 gnd rswk-swf Lineare Integralgleichung (DE-588)4114426-0 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4614-9593-2 Applied mathematical sciences 82 (DE-604)BV000005274 82 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027028331&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kress, Rainer 1941- Linear integral equations Applied mathematical sciences Lineare Integralgleichung - Lehrbuch Integral equations Lineare Integralgleichung (DE-588)4114426-0 gnd |
| subject_GND | (DE-588)4114426-0 |
| title | Linear integral equations |
| title_auth | Linear integral equations |
| title_exact_search | Linear integral equations |
| title_full | Linear integral equations Rainer Kress |
| title_fullStr | Linear integral equations Rainer Kress |
| title_full_unstemmed | Linear integral equations Rainer Kress |
| title_short | Linear integral equations |
| title_sort | linear integral equations |
| topic | Lineare Integralgleichung - Lehrbuch Integral equations Lineare Integralgleichung (DE-588)4114426-0 gnd |
| topic_facet | Lineare Integralgleichung - Lehrbuch Integral equations Lineare Integralgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027028331&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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