Periodic integral and pseudodifferential equations with numerical approximation:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2002
|
| Schriftenreihe: | Springer monographs in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XI, 452 S. |
| ISBN: | 3540418784 9783642075384 |
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| 100 | 1 | |a Saranen, Jukka |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Periodic integral and pseudodifferential equations with numerical approximation |c Jukka Saranen ; Gennadi Vainikko |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2002 | |
| 300 | |a XI, 452 S. | ||
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| 490 | 0 | |a Springer monographs in mathematics | |
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| 650 | 4 | |a Boundary value problems | |
| 650 | 4 | |a Integral equations | |
| 650 | 4 | |a Pseudodifferential operators | |
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Datensatz im Suchindex
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IMAGE 1
JUKKA SARANEN * GENNADI VAINIKKO
PERIODIC INTEGRAL
AND PSEUDODIFFERENTIAL EQUATIONS WITH NUMERICAL APPROXIMATION
JH3 SPRINGER
IMAGE 2
CONTENTS
1. PRELIMINARIES 1
1.1 PRELIMINARIES FROM OPERATOR THEORY 1
1.1.1 LINEAR CONTINUOUS OPERATORS 2
1.1.2 LINEAR COMPACT OPERATORS 3
1.1.3 EXAMPLES OF COMPACT INTEGRAL OPERATORS 4
1.1.4 CHARACTERIZATION OF CONTINUOUS PROJECTORS 5
1.1.5 COMPLEMENTABLE SUBSPACES 5
1.2 FREDHOLM THEOREMS 7
1.2.1 RESULTS INVOLVING ONLY THE BANACH SPACE X 7
1.2.2 RESULTS INVOLVING X AND ITS DUAL X' = C{X, C) 8
1.2.3 DUAL SYSTEMS AND DUAL OPERATORS 9
1.2.4 RESULTS INVOLVING DUAL SYSTEMS 11
1.3 THE FREDHOLM OPERATORS 13
1.3.1 FREDHOLM OPERATORS OF INDEX 0 14
1.3.2 THE PRODUCT OF FREDHOLM OPERATORS 15
1.3.3 FREDHOLM OPERATORS OF NONZERO INDEX 17
1.3.4 PERTURBATION OF FREDHOLM OPERATORS 18
1.4 THE REGULARIZERS 20
1.4.1 CONNECTIONS TO FREDHOLMNESS 20
1.4.2 RESULTS OF FREDHOLM TYPE 22
1.5 KRYLOV SUBSPACE METHODS 25
1.5.1 GMRES 25
1.5.2 ANOTHER ALGORITHM OF GMRES 29
1.5.3 CONJUGATE GRADIENTS (CGMR AND CGME) 30
2. SINGLE LAYER AND DOUBLE LAYER POTENTIALS 35
2.1 CLASSICAL BOUNDARY VALUE PROBLEMS 35
2.2 FUNDAMENTAL SOLUTION 39
2.3 AN INTEGRAL REPRESENTATION OF FUNCTIONS 44
2.4 JORDAN ARES AND CURVES 48
2.5 BOUNDARY POTENTIALS 54
2.5.1 SINGLE LAYER POTENTIAL 54
2.5.2 DOUBLE LAYER POTENTIAL 56
2.5.3 NORMAL DERIVATIVE OF THE SINGLE LAYER POTENTIAL 61
IMAGE 3
CONTENTS
2.5.4 NORMAL DERIVATIVE OF THE DOUBLE LAYER POTENTIAL 66
2.5.5 CALDERON PROJECTOR 69
SOLUTION OF BOUNDARY VALUE PROBLEMS BY INTEGRAL EQUATIONS 71 3.1
INTEGRAL EQUATIONS FOR BOUNDARY VALUE PROBLEMS 71
3.2 SOLUTION OF THE LAPLACE EQUATION BY INTEGRAL EQUATIONS OF THE SECOND
KIND 74
3.2.1 SOLVABILITY OF THE BOUNDARY INTEGRAL EQUATIONS 76
3.2.2 SOLUTION OF THE INTERIOR BOUNDARY VALUE PROBLEMS . . . 78 3.2.3
SOLUTION OF THE EXTERIOR BOUNDARY VALUE PROBLEMS . . 81 3.3 SOLUTION OF
THE HELMHOLTZ EQUATION BY INTEGRAL EQUATIONS OF THE SECOND KIND 89
3.3.1 SOLVABILITY OF THE BOUNDARY INTEGRAL EQUATIONS 89
3.3.2 SOLUTION OF THE BOUNDARY VALUE PROBLEMS 95
3.3.3 RADIATION CONDITION 99
SINGULAR INTEGRAL EQUATIONS 105
4.1 SINGULAR INTEGRAL EQUATIONS IN HOLDER SPACES 105
4.1.1 HOLDER SPACES 105
4.1.2 CAUCHY INTEGRAL 106
4.1.3 CAUCHY SINGULAR INTEGRAL OPERATOR 110
4.1.4 OPERATOR S K 112
4.1.5 THE COMMUTATOR OF S AND M A 115
4.1.6 PROJECTION OPERATORS P_ AND P + 116
4.1.7 WINDING NUMBER 117
4.1.8 THE INDEX OF THE SINGULAR OPERATOR 119
4.1.9 THE INDEX OF MORE GENERAL SINGULAR INTEGRAL OPERATOR 122 4.2 L 2
-THEORY OF SINGULAR INTEGRAL EQUATIONS 124
4.2.1 LAX THEOREM 124
4.2.2 SINGULAR INTEGRAL OPERATORS ON L 2 ( R) 127
BOUNDARY INTEGRAL OPERATORS IN PERIODIC SOBOLEV SPACES . 133 5.1
DISTRIBUTIONS ON THE REAL LINE 133
5.2 PERIODIC DISTRIBUTIONS 136
5.3 PERIODIC SOBOLEV SPACES 141
5.4 FINITE PART OF HYPERSINGULAR INTEGRALS 143
5.5 SPECTRAL REPRESENTATION OF A CONVOLUTION INTEGRAL OPERATOR . 148
5.6 SYMM'S INTEGRAL OPERATOR 150
5.7 HUBERT INTEGRAL OPERATOR 152
5.8 CAUCHY INTEGRAL OPERATOR ON THE UNIT CIRCLE 154
5.9 CAUCHY INTEGRAL OPERATOR ON A JORDAN CURVE 155
5.10 HYPERSINGULAR INTEGRAL OPERATOR 156
5.11 BIHARMONIC PROBLEM 157
5.12 OPERATOR INTERPOLATION 161
5.13 MULTIPLICATION OF FUNCTIONS IN H X 163
IMAGE 4
CONTENTS IX
PERIODIC INTEGRAL EQUATIONS 167
6.1 BOUNDEDNESS OF INTEGRAL OPERATORS BETWEEN SOBOLEV SPACES . 167
6.1.1 PRODUCT OF BIPERIODIC FUNCTIONS 168
6.1.2 BOUNDEDNESS OF INTEGRAL OPERATORS 171
6.2 PREDHOLMNESS OF INTEGRAL OPERATORS BETWEEN SOBOLEV SPACES . 175 6.3
A CLASS OF PERIODIC INTEGRAL EQUATIONS 176
6.4 EXAMPLES OF PERIODIC INTEGRAL EQUATIONS 177
6.5 ANALYSIS OF THE MODIFIED SYMM'S EQUATIONS 180
6.6 A GENERAL CLASS OF PERIODIC INTEGRAL EQUATIONS 182
6.6.1 WINDING NUMBER OF A PERIODIC FUNCTION 182
6.6.2 INVERSION OF CAUCHY SINGULAR OPERATORS 185
6.6.3 THE CLASS OF INTEGRAL EQUATIONS 187
6.6.4 EXAMPLE: THE CAUCHY INTEGRAL EQUATION 192
6.7 EQUATIONS WITH ANALYTIC COEMCIENT FUNCTIONS 193
6.7.1 SPACES OF PERIODIC ANALYTIC FUNCTIONS 193
6.7.2 MAPPING PROPERTIES OF PERIODIC INTEGRAL O P E R A T O R S . . .
195 6.7.3 ANALYTIC SOLUTIONS OF INTEGRAL EQUATIONS 197
PERIODIC PSEUDODIFFERENTIAL OPERATORS 199
7.1 PROLONGATION OF A FUNCTION DEFINED ON Z 199
7.2 TWO DEFINITIONS OF PPDO AND THEIR EQUIVALENCE 202
7.3 BOUNDEDNESS OF A PPDO 205
7.4 ASYMPTOTIC EXPANSION OF THE SYMBOL 206
7.5 AMPLITUDES 207
7.6 ASYMPTOTIC EXPANSION OF INTEGRAL OPERATORS 214
7.6.1 OPERATOR (AU)(T) = /** A(*,S) LOG|SINIR(I - S)\U(S)DS . . 215
7.6.2 OPERATOR (AU){T) = IF* A(T, S) COTIR(T - S)U(S) DS 216
7.7 THE SYMBOL OF DUAL AND ADJOINT OPERATORS 217
7.8 THE SYMBOL OF THE COMPOSITION OF PPDOS 218
7.9 PSEUDOLOCALITY 221
7.10 ELLIPTIC PPDOS 224
7.11 GAERDING'S INEQUALITY 227
7.12 ESTIMATION OF THE OPERATOR NORM 230
7.13 CLASSICAL PPDOS 231
7.14 INTEGRAL OPERATOR REPRESENTATION OF CLASSICAL PPDOS 234
7.15 FUNCTIONS (*) 237
T R I G O N O M E T R IE I N T E R P O L A T I ON 239
8.1 SUBSPACE T N 239
8.2 ORTHOGONAL PROTECTION 241
8.3 INTERPOLATION PROJECTION 242
8.3.1 INTERPOLATION OF FUNCTIONS U 6 H SS 242
8.3.2 INTERPOLATION OF EVEN AND ODD FUNCTIONS 244
8.3.3 DISCRETE FOURIER TRANSFORM 245
8.3.4 INTERPOLATION OF FUNCTIONS AU N , U* &T* 246
IMAGE 5
X CONTENTS
8.4 EXPONENTIAL APPROXIMATION ORDER 248
8.5 TWO DIMENSIONAL INTERPOLATION 250
8.5.1 ESTIMATES FOR U 6 H» 1 * 3 250
8.5.2 ESTIMATES FOR U E HF(R 2 ) 252
9. GALERKIN METHOD AND FAST SOLVERS 255
9.1 PRECONDITION OF THE PROBLEM 255
9.2 GALERKIN METHOD FOR THE PRECONDITIONED PROBLEM 257
9.3 MATRIX REPRESENTATION OF A PIO 262
9.4 A FUELL DISCRETIZATION 264
9.4.1 APPROXIMATION OF THE GALERKIN EQUATION 264
9.4.2 COMPUTATIONAL COSTS 268
9.4.3 FAST SOLVERS 270
9.5 USING ASYMPTOTIC EXPANSIONS 273
9.5.1 APPROXIMATION OF THE GALERKIN EQUATION 273
9.5.2 TWO GRID ITERATION METHOD 276
9.5.3 FAST SOLVERS 277
9.6 STABILITY ESTIMATES 279
9.7 REGULARIZATION VIA DISCRETIZATION 286
9.8 STANDARD GALERKIN METHOD 286
9.8.1 STABILITY INEQUALITY 287
9.8.2 CONVERGENCE OF THE GALERKIN METHOD 290
10. TRIGONOMETRIE COLLOCATION 293
10.1 COLLOCATION PROBLEM 293
10.2 FUELL DISCRETIZATION 298
10.3 MODIFICATIONS 302
10.4 FURTHER DISCRETE VERSIONS 303
10.4.1 PROBLEM, METHODS, CONVERGENCE 304
10.4.2 MATRIX FORM OF THE METHOD 308
10.4.3 PRECONDITIONING AND ITERATION SOLUTION 311
10.5 FAST SOLVERS FOR LIPPMANN-SCHWINGER EQUATION 314
10.5.1 LIPPMANN-SCHWINGER EQUATION 314
10.5.2 COLLOCATION SOLUTION 317
10.5.3 TWO-GRID ITERATIONS 319
10.5.4 MATRIX FORMS OF THE METHODS 321
10.5.5 APPENDIX: FOURIER COEFFICIENTS OF K.(X) 323
11. INTEGRAL EQUATIONS ON AN OPEN ARE 325
11.1 EQUATIONS ON AN INTERVAL 325
11.2 PERIODIZATION 329
11.2.1 EQUATIONS WITH A LOGARITHMIC SINGULAR KERNEL 329
11.2.2 CAUCHY-SINGULAR EQUATIONS 330
11.2.3 HYPERSINGULAR EQUATIONS 332
11.3 EVEN AND ODD OPERATORS 333
IMAGE 6
CONTENTS XI
11.4 ANALYSIS OF THE PERIODIC PROBLEM 337
11.4.1 CASE OF EVEN OPERATOR 338
11.4.2 CASE OF ODD OPERATOR 340
11.5 MORE ABOUT CONVOLUTION OPERATORS ON (-1,1) 343
11.6 COLLOCATION SOLUTION 348
11.7 FULLY DISCRETE COLLOCATION METHODS 353
12. QUADRATURE M E T H O DS 355
12.1 THE IDEA OF A QUADRATURE METHOD 355
12.2 A SIMPLE QUADRATURE METHOD 357
12.2.1 CONSISTENCY 357
12.2.2 STABILITY 361
12.2.3 CONVERGENCE 365
12.3 THE E-QUADRATURE METHOD 366
12.4 A MODIFIED QUADRATURE METHOD 380
12.5 INTEGRAL EQUATIONS OF THE SECOND KIND 390
12.6 SINGULAR INTEGRAL EQUATIONS 392
12.7 HYPERSINGULAR EQUATIONS 396
12.8 EXTENSIONS BY LOCALIZATION 399
13. SPLINE APPROXIMATION METHODS 401
13.1 SPLINE SPACES 401
13.2 SPLINES ON UNIFORM MESHES 403
13.3 APPROXIMATION AND INVERSE PROPERTIES OF SPLINES 405
13.4 DISCRETE FOURIER TRANSFORM OF PERIODIC FUNCTIONS 411
13.5 SPLINE COLLOCATION ON UNIFORM MESH 413
13.6 AN ABSTRACT GALERKIN METHOD 428
13.7 THE SPLINE GALERKIN METHOD 433
13.8 SOME EXTENSIONS OF THE BASIC METHODS 435
13.8.1 SPLINE BASED PETROV-GALERKIN METHODS 435
13.8.2 QUALOCATION AND RELATED METHODS 436
13.8.3 DIRAC'S DISTRIBUTIONS AS TRIAL FUNCTIONS 438
BIBLIOGRAPH/ 441
INDEX 449 |
| any_adam_object | 1 |
| author | Saranen, Jukka Vainikko, Gennadi 1938- |
| author_GND | (DE-588)103337075 |
| author_facet | Saranen, Jukka Vainikko, Gennadi 1938- |
| author_role | aut aut |
| author_sort | Saranen, Jukka |
| author_variant | j s js g v gv |
| building | Verbundindex |
| bvnumber | BV013953005 |
| callnumber-first | Q - Science |
| callnumber-label | QA329 |
| callnumber-raw | QA329.7 |
| callnumber-search | QA329.7 |
| callnumber-sort | QA 3329.7 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 640 |
| classification_tum | MAT 410f MAT 677f |
| ctrlnum | (OCoLC)48221476 (DE-599)BVBBV013953005 |
| dewey-full | 515/.7242 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.7242 |
| dewey-search | 515/.7242 |
| dewey-sort | 3515 47242 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV013953005 |
| illustrated | Not Illustrated |
| indexdate | 2025-01-30T09:01:05Z |
| institution | BVB |
| isbn | 3540418784 9783642075384 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009548889 |
| oclc_num | 48221476 |
| open_access_boolean | |
| owner | DE-703 DE-92 DE-355 DE-BY-UBR DE-824 DE-91G DE-BY-TUM DE-706 DE-634 DE-83 DE-11 DE-20 DE-19 DE-BY-UBM |
| owner_facet | DE-703 DE-92 DE-355 DE-BY-UBR DE-824 DE-91G DE-BY-TUM DE-706 DE-634 DE-83 DE-11 DE-20 DE-19 DE-BY-UBM |
| physical | XI, 452 S. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Saranen, Jukka Verfasser aut Periodic integral and pseudodifferential equations with numerical approximation Jukka Saranen ; Gennadi Vainikko Berlin [u.a.] Springer 2002 XI, 452 S. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Hier auch später erschienene, unveränderte Nachdrucke Boundary value problems Integral equations Pseudodifferential operators Integralgleichung (DE-588)4027229-1 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Randelemente-Methode (DE-588)4076508-8 gnd rswk-swf Pseudodifferentialgleichung (DE-588)4121533-3 gnd rswk-swf Periodische Integralgleichung (DE-588)4666016-1 gnd rswk-swf Pseudodifferentialoperator (DE-588)4047640-6 gnd rswk-swf Periodische Integralgleichung (DE-588)4666016-1 s Pseudodifferentialoperator (DE-588)4047640-6 s DE-604 Randelemente-Methode (DE-588)4076508-8 s Integralgleichung (DE-588)4027229-1 s Pseudodifferentialgleichung (DE-588)4121533-3 s Randwertproblem (DE-588)4048395-2 s Vainikko, Gennadi 1938- Verfasser (DE-588)103337075 aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009548889&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Saranen, Jukka Vainikko, Gennadi 1938- Periodic integral and pseudodifferential equations with numerical approximation Boundary value problems Integral equations Pseudodifferential operators Integralgleichung (DE-588)4027229-1 gnd Randwertproblem (DE-588)4048395-2 gnd Randelemente-Methode (DE-588)4076508-8 gnd Pseudodifferentialgleichung (DE-588)4121533-3 gnd Periodische Integralgleichung (DE-588)4666016-1 gnd Pseudodifferentialoperator (DE-588)4047640-6 gnd |
| subject_GND | (DE-588)4027229-1 (DE-588)4048395-2 (DE-588)4076508-8 (DE-588)4121533-3 (DE-588)4666016-1 (DE-588)4047640-6 |
| title | Periodic integral and pseudodifferential equations with numerical approximation |
| title_auth | Periodic integral and pseudodifferential equations with numerical approximation |
| title_exact_search | Periodic integral and pseudodifferential equations with numerical approximation |
| title_full | Periodic integral and pseudodifferential equations with numerical approximation Jukka Saranen ; Gennadi Vainikko |
| title_fullStr | Periodic integral and pseudodifferential equations with numerical approximation Jukka Saranen ; Gennadi Vainikko |
| title_full_unstemmed | Periodic integral and pseudodifferential equations with numerical approximation Jukka Saranen ; Gennadi Vainikko |
| title_short | Periodic integral and pseudodifferential equations with numerical approximation |
| title_sort | periodic integral and pseudodifferential equations with numerical approximation |
| topic | Boundary value problems Integral equations Pseudodifferential operators Integralgleichung (DE-588)4027229-1 gnd Randwertproblem (DE-588)4048395-2 gnd Randelemente-Methode (DE-588)4076508-8 gnd Pseudodifferentialgleichung (DE-588)4121533-3 gnd Periodische Integralgleichung (DE-588)4666016-1 gnd Pseudodifferentialoperator (DE-588)4047640-6 gnd |
| topic_facet | Boundary value problems Integral equations Pseudodifferential operators Integralgleichung Randwertproblem Randelemente-Methode Pseudodifferentialgleichung Periodische Integralgleichung Pseudodifferentialoperator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009548889&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT saranenjukka periodicintegralandpseudodifferentialequationswithnumericalapproximation AT vainikkogennadi periodicintegralandpseudodifferentialequationswithnumericalapproximation |