Spectral methods: fundamentals in single domains ; with ... 19 tables
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2006
|
| Series: | Scientific computation
|
| Subjects: | |
| Online Access: | Beschreibung für Leser Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [529] - 552 |
| Physical Description: | XXII, 563 S. Ill., graph. Darst. |
| ISBN: | 3540307257 9783540307259 |
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MARC
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| 245 | 1 | 0 | |a Spectral methods |b fundamentals in single domains ; with ... 19 tables |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XXII, 563 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 0 | |a Scientific computation | |
| 500 | |a Literaturverz. S. [529] - 552 | ||
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| 650 | 7 | |a Análise numérica |2 larpcal | |
| 650 | 7 | |a Equações diferenciais parciais |2 larpcal | |
| 650 | 7 | |a Spectraaltheorie |2 gtt | |
| 650 | 4 | |a Differential equations, Partial |x Numerical solutions | |
| 650 | 4 | |a Numerical analysis | |
| 650 | 4 | |a Spectral theory (Mathematics) | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1. Introduction 3
1.1 Historical Background 3
1.2 Some Examples of Spectral Methods 7
1.2.1 A Fourier Galerkin Method for the Wave Equation ... 7
1.2.2 A Chebyshev Collocation Method for the Heat Equation 11
1.2.3 A Legendre Galerkin with Numerical Integration
(G NI) Method for the Advection Diffusion Reaction
Equation 16
1.2.4 A Legendre Tau Method for the Poisson Equation .... 21
1.2.5 Basic Aspects of Galerkin, Collocation, G NI
and Tau Methods 24
1.3 Three Dimensional Applications in Fluids: A Look Ahead ... 25
2. Polynomial Approximation 39
2.1 The Fourier System 41
2.1.1 The Continuous Fourier Expansion 41
2.1.2 The Discrete Fourier Expansion 47
2.1.3 Differentiation 52
2.1.4 The Gibbs Phenomenon 56
2.2 Orthogonal Polynomials in ( 1,1) 68
2.2.1 Sturm Liouville Problems 68
2.2.2 Orthogonal Systems of Polynomials 69
2.2.3 Gauss Type Quadratures and Discrete Polynomial
Transforms 70
2.3 Legendre Polynomials 75
2.3.1 Basic Formulas 75
2.3.2 Differentiation 77
2.3.3 Orthogonality, Diagonalization and Localization 81
2.4 Chebyshev Polynomials 84
2.4.1 Basic Formulas 84
2.4.2 Differentiation 87
2.5 Jacobi Polynomials 91
2.6 Approximation in Unbounded Domains 93
2.6.1 Laguerre Polynomials and Laguerre Functions 94
XIV Contents
2.6.2 Hermite Polynomials and Hermite Functions 95
2.7 Mappings for Unbounded Domains 96
2.7.1 Semi Infinite Intervals 96
2.7.2 The Real Line 97
2.8 Tensor Product Expansions 98
2.8.1 Multidimensional Mapping 99
2.9 Expansions on Triangles and Related Domains 103
2.9.1 Collapsed Coordinates and Warped Tensor Product
Expansions 103
2.9.2 Non Tensor Product Expansions 110
2.9.3 Mappings 114
3. Basic Approaches to Constructing Spectral Methods 117
3.1 Burgers Equation 118
3.2 Strong and Weak Formulations of Differential Equations 119
3.3 Spectral Approximation of the Burgers Equation 121
3.3.1 Fourier Galerkin 122
3.3.2 Fourier Collocation 123
3.3.3 Chebyshev Tau 127
3.3.4 Chebyshev Collocation 129
3.3.5 Legendre G NI 130
3.4 Convolution Sums 132
3.4.1 Transform Methods and Pseudospectral Methods 133
3.4.2 Aliasing Removal by Padding or Truncation 134
3.4.3 Aliasing Removal by Phase Shifts 135
3.4.4 Aliasing Removal for Orthogonal Polynomials 136
3.5 Relation Between Collocation, G NI
and Pseudospectral Methods 138
3.6 Conservation Forms 140
3.7 Scalar Hyperbolic Problems 145
3.7.1 Enforcement of Boundary Conditions 145
3.7.2 Numerical Examples 150
3.8 Matrix Construction for Galerkin and G NI Methods 154
3.8.1 Matrix Elements 157
3.8.2 An Example of Algebraic Equivalence between G NI
and Collocation Methods 160
3.9 Polar Coordinates 162
3.10 Aliasing Effects 163
4. Algebraic Systems and Solution Techniques 167
4.1 Ad hoc Direct Methods 169
4.1.1 Fourier Approximations 170
4.1.2 Chebyshev Tau Approximations 173
4.1.3 Galerkin Approximations 177
4.1.4 Schur Decomposition and Matrix Diagonalization 181
Contents XV
4.2 Direct Methods 186
4.2.1 Tensor Products of Matrices 186
4.2.2 Multidimensional Stiffness and Mass Matrices 187
4.2.3 Gaussian Elimination Techniques 192
4.3 Eigen Analysis of Spectral Derivative Matrices 195
4.3.1 Second Derivative Matrices 197
4.3.2 First Derivative Matrices 200
4.3.3 Advection Diffusion Matrices 206
4.4 Preconditioning 208
4.4.1 Fundamentals of Iterative Methods
for Spectral Discretizations 209
4.4.2 Low Order Preconditioning
of Model Spectral Operators in One Dimension 211
4.4.3 Low Order Preconditioning in Several Dimensions .... 227
4.4.4 Spectral Preconditioning 238
4.5 Descent and Krylov Iterative Methods
for Spectral Equations 239
4.5.1 Multidimensional Matrix Vector Multiplication 239
4.5.2 Iterative Methods 241
4.6 Spectral Multigrid Methods 242
4.6.1 One Dimensional Fourier Multigrid Model Problem . .. 243
4.6.2 General Spectral Multigrid Methods 246
4.7 Numerical Examples of Direct and Iterative Methods 251
4.7.1 Fourier Collocation Discretizations 251
4.7.2 Chebyshev Collocation Discretizations 253
4.7.3 Legendre G NI Discretizations 256
4.7.4 Preconditioners for Legendre G NI Matrices 259
4.8 Interlude 265
5. Polynomial Approximation Theory 267
5.1 Fourier Approximation 268
5.1.1 Inverse Inequalities for Trigonometric Polynomials .... 268
5.1.2 Estimates for the Truncation and Best Approximation
Errors 269
5.1.3 Estimates for the Interpolation Error 272
5.2 Sturm Liouville Expansions 275
5.2.1 Regular Sturm Liouville Problems 275
5.2.2 Singular Sturm Liouville Problems 277
5.3 Discrete Norms 279
5.4 Legendre Approximations 281
5.4.1 Inverse Inequalities for Algebraic Polynomials 281
5.4.2 Estimates for the Truncation and Best Approximation
Errors 283
5.4.3 Estimates for the Interpolation Error 289
5.4.4 Scaled Estimates 290
XVI Contents
5.5 Chebyshev Approximations 292
5.5.1 Inverse Inequalities for Polynomials 292
5.5.2 Estimates for the Truncation and Best Approximation
Errors 293
5.5.3 Estimates for the Interpolation Error 296
5.6 Proofs of Some Approximation Results 298
5.7 Other Polynomial Approximations 309
5.7.1 Jacobi Polynomials 309
5.7.2 Laguerre and Hermite Polynomials 310
5.8 Approximation in Cartesian Product Domains 312
5.8.1 Fourier Approximations 312
5.8.2 Legendre Approximations 314
5.8.3 Mapped Operators and Scaled Estimates 316
5.8.4 Chebyshev and Other Jacobi Approximations 318
5.8.5 Blended Trigonometric and Algebraic Approximations 320
5.9 Approximation in Triangles and Related Domains 323
6. Theory of Stability and Convergence 327
6.1 Three Elementary Examples Revisited 328
6.1.1 A Fourier Galerkin Method for the Wave Equation . .. 328
6.1.2 A Chebyshev Collocation Method
for the Heat Equation 329
6.1.3 A Legendre Tau Method for the Poisson Equation .... 334
6.2 Towards a General Theory 337
6.3 General Formulation of Spectral Approximations
to Linear Steady Problems 338
6.4 Galerkin, Collocation, G NI and Tau Methods 344
6.4.1 Galerkin Methods 345
6.4.2 Collocation Methods 351
6.4.3 G NI Methods 360
6.4.4 Tau Methods 367
6.5 General Formulation of Spectral Approximations
to Linear Evolution Problems 376
6.5.1 Conditions for Stability and Convergence:
The Parabolic Case 378
6.5.2 Conditions for Stability and Convergence:
The Hyperbolic Case 384
6.6 The Error Equation 396
7. Analysis of Model Boundary Value Problems 401
7.1 The Poisson Equation 401
7.1.1 Legendre Methods 402
7.1.2 Chebyshev Methods 404
7.1.3 Other Boundary Value Problems 409
7.2 Singularly Perturbed Elliptic Equations 409
Contents XVII
7.2.1 Stabilization of Spectral Methods 413
7.3 The Eigenvalues of Some Spectral Operators 420
7.3.1 The Discrete Eigenvalues for Cu = —uxx 420
7.3.2 The Discrete Eigenvalues for Cu = —vuxx + (iux 424
7.3.3 The Discrete Eigenvalues for Cu = ux 427
7.4 The Preconditioning of Spectral Operators 430
7.5 The Heat Equation 433
7.6 Linear Hyperbolic Equations 439
7.6.1 Periodic Boundary Conditions 439
7.6.2 Nonperiodic Boundary Conditions 445
7.6.3 The Resolution of the Gibbs Phenomenon 447
7.6.4 Spectral Accuracy for Non Smooth Solutions 454
7.7 Scalar Conservation Laws 459
7.8 The Steady Burgers Equation 463
Appendix A. Basic Mathematical Concepts 471
A.I Hilbert and Banach Spaces 471
A.2 The Cauchy Schwarz Inequality 473
A.3 Linear Operators Between Banach Spaces 474
A.4 The Frechet Derivative of an Operator 475
A.5 The Lax Milgram Theorem 475
A.6 Dense Subspace of a Normed Space 476
A.7 The Spaces Cm(Q), m 0 476
A.8 Functions of Bounded Variation
and the Riemann( Stieltjes) Integral 476
A.9 The Lebesgue Integral and Lp Spaces 478
A. 10 Infinitely Differentiable Functions and Distributions 482
A.11 Sobolev Spaces and Sobolev Norms 484
A.12 The Sobolev Inequality 490
A.13 The Poincare Inequality 491
A.14 The Hardy Inequality 491
A.15 The Gronwall Lemma 492
Appendix B. Fast Fourier Transforms 493
Appendix C. Iterative Methods for Linear Systems 499
C.I A Gentle Approach to Iterative Methods 499
C.2 Descent Methods for Symmetric Problems 503
C.3 Krylov Methods for Nonsymmetric Problems 508
Appendix D. Time Discretizations 515
D.I Notation and Stability Definitions 515
D.2 Standard ODE Methods 519
D.2.1 Leap Frog Method 519
D.2.2 Adams Bashforth Methods 520
XVIII Contents
D.2.3 Adams Moulton Methods 521
D.2.4 Backwards Difference Formulas 524
D.2.5 Runge Kutta Methods 524
D.3 Integrating Factors 525
D.4 Low Storage Schemes 527
References 529
Index 553
|
| adam_txt |
Contents
1. Introduction 3
1.1 Historical Background 3
1.2 Some Examples of Spectral Methods 7
1.2.1 A Fourier Galerkin Method for the Wave Equation . 7
1.2.2 A Chebyshev Collocation Method for the Heat Equation 11
1.2.3 A Legendre Galerkin with Numerical Integration
(G NI) Method for the Advection Diffusion Reaction
Equation 16
1.2.4 A Legendre Tau Method for the Poisson Equation . 21
1.2.5 Basic Aspects of Galerkin, Collocation, G NI
and Tau Methods 24
1.3 Three Dimensional Applications in Fluids: A Look Ahead . 25
2. Polynomial Approximation 39
2.1 The Fourier System 41
2.1.1 The Continuous Fourier Expansion 41
2.1.2 The Discrete Fourier Expansion 47
2.1.3 Differentiation 52
2.1.4 The Gibbs Phenomenon 56
2.2 Orthogonal Polynomials in ( 1,1) 68
2.2.1 Sturm Liouville Problems 68
2.2.2 Orthogonal Systems of Polynomials 69
2.2.3 Gauss Type Quadratures and Discrete Polynomial
Transforms 70
2.3 Legendre Polynomials 75
2.3.1 Basic Formulas 75
2.3.2 Differentiation 77
2.3.3 Orthogonality, Diagonalization and Localization 81
2.4 Chebyshev Polynomials 84
2.4.1 Basic Formulas 84
2.4.2 Differentiation 87
2.5 Jacobi Polynomials 91
2.6 Approximation in Unbounded Domains 93
2.6.1 Laguerre Polynomials and Laguerre Functions 94
XIV Contents
2.6.2 Hermite Polynomials and Hermite Functions 95
2.7 Mappings for Unbounded Domains 96
2.7.1 Semi Infinite Intervals 96
2.7.2 The Real Line 97
2.8 Tensor Product Expansions 98
2.8.1 Multidimensional Mapping 99
2.9 Expansions on Triangles and Related Domains 103
2.9.1 Collapsed Coordinates and Warped Tensor Product
Expansions 103
2.9.2 Non Tensor Product Expansions 110
2.9.3 Mappings 114
3. Basic Approaches to Constructing Spectral Methods 117
3.1 Burgers Equation 118
3.2 Strong and Weak Formulations of Differential Equations 119
3.3 Spectral Approximation of the Burgers Equation 121
3.3.1 Fourier Galerkin 122
3.3.2 Fourier Collocation 123
3.3.3 Chebyshev Tau 127
3.3.4 Chebyshev Collocation 129
3.3.5 Legendre G NI 130
3.4 Convolution Sums 132
3.4.1 Transform Methods and Pseudospectral Methods 133
3.4.2 Aliasing Removal by Padding or Truncation 134
3.4.3 Aliasing Removal by Phase Shifts 135
3.4.4 Aliasing Removal for Orthogonal Polynomials 136
3.5 Relation Between Collocation, G NI
and Pseudospectral Methods 138
3.6 Conservation Forms 140
3.7 Scalar Hyperbolic Problems 145
3.7.1 Enforcement of Boundary Conditions 145
3.7.2 Numerical Examples 150
3.8 Matrix Construction for Galerkin and G NI Methods 154
3.8.1 Matrix Elements 157
3.8.2 An Example of Algebraic Equivalence between G NI
and Collocation Methods 160
3.9 Polar Coordinates 162
3.10 Aliasing Effects 163
4. Algebraic Systems and Solution Techniques 167
4.1 Ad hoc Direct Methods 169
4.1.1 Fourier Approximations 170
4.1.2 Chebyshev Tau Approximations 173
4.1.3 Galerkin Approximations 177
4.1.4 Schur Decomposition and Matrix Diagonalization 181
Contents XV
4.2 Direct Methods 186
4.2.1 Tensor Products of Matrices 186
4.2.2 Multidimensional Stiffness and Mass Matrices 187
4.2.3 Gaussian Elimination Techniques 192
4.3 Eigen Analysis of Spectral Derivative Matrices 195
4.3.1 Second Derivative Matrices 197
4.3.2 First Derivative Matrices 200
4.3.3 Advection Diffusion Matrices 206
4.4 Preconditioning 208
4.4.1 Fundamentals of Iterative Methods
for Spectral Discretizations 209
4.4.2 Low Order Preconditioning
of Model Spectral Operators in One Dimension 211
4.4.3 Low Order Preconditioning in Several Dimensions . 227
4.4.4 Spectral Preconditioning 238
4.5 Descent and Krylov Iterative Methods
for Spectral Equations 239
4.5.1 Multidimensional Matrix Vector Multiplication 239
4.5.2 Iterative Methods 241
4.6 Spectral Multigrid Methods 242
4.6.1 One Dimensional Fourier Multigrid Model Problem . . 243
4.6.2 General Spectral Multigrid Methods 246
4.7 Numerical Examples of Direct and Iterative Methods 251
4.7.1 Fourier Collocation Discretizations 251
4.7.2 Chebyshev Collocation Discretizations 253
4.7.3 Legendre G NI Discretizations 256
4.7.4 Preconditioners for Legendre G NI Matrices 259
4.8 Interlude 265
5. Polynomial Approximation Theory 267
5.1 Fourier Approximation 268
5.1.1 Inverse Inequalities for Trigonometric Polynomials . 268
5.1.2 Estimates for the Truncation and Best Approximation
Errors 269
5.1.3 Estimates for the Interpolation Error 272
5.2 Sturm Liouville Expansions 275
5.2.1 Regular Sturm Liouville Problems 275
5.2.2 Singular Sturm Liouville Problems 277
5.3 Discrete Norms 279
5.4 Legendre Approximations 281
5.4.1 Inverse Inequalities for Algebraic Polynomials 281
5.4.2 Estimates for the Truncation and Best Approximation
Errors 283
5.4.3 Estimates for the Interpolation Error 289
5.4.4 Scaled Estimates 290
XVI Contents
5.5 Chebyshev Approximations 292
5.5.1 Inverse Inequalities for Polynomials 292
5.5.2 Estimates for the Truncation and Best Approximation
Errors 293
5.5.3 Estimates for the Interpolation Error 296
5.6 Proofs of Some Approximation Results 298
5.7 Other Polynomial Approximations 309
5.7.1 Jacobi Polynomials 309
5.7.2 Laguerre and Hermite Polynomials 310
5.8 Approximation in Cartesian Product Domains 312
5.8.1 Fourier Approximations 312
5.8.2 Legendre Approximations 314
5.8.3 Mapped Operators and Scaled Estimates 316
5.8.4 Chebyshev and Other Jacobi Approximations 318
5.8.5 Blended Trigonometric and Algebraic Approximations 320
5.9 Approximation in Triangles and Related Domains 323
6. Theory of Stability and Convergence 327
6.1 Three Elementary Examples Revisited 328
6.1.1 A Fourier Galerkin Method for the Wave Equation . . 328
6.1.2 A Chebyshev Collocation Method
for the Heat Equation 329
6.1.3 A Legendre Tau Method for the Poisson Equation . 334
6.2 Towards a General Theory 337
6.3 General Formulation of Spectral Approximations
to Linear Steady Problems 338
6.4 Galerkin, Collocation, G NI and Tau Methods 344
6.4.1 Galerkin Methods 345
6.4.2 Collocation Methods 351
6.4.3 G NI Methods 360
6.4.4 Tau Methods 367
6.5 General Formulation of Spectral Approximations
to Linear Evolution Problems 376
6.5.1 Conditions for Stability and Convergence:
The Parabolic Case 378
6.5.2 Conditions for Stability and Convergence:
The Hyperbolic Case 384
6.6 The Error Equation 396
7. Analysis of Model Boundary Value Problems 401
7.1 The Poisson Equation 401
7.1.1 Legendre Methods 402
7.1.2 Chebyshev Methods 404
7.1.3 Other Boundary Value Problems 409
7.2 Singularly Perturbed Elliptic Equations 409
Contents XVII
7.2.1 Stabilization of Spectral Methods 413
7.3 The Eigenvalues of Some Spectral Operators 420
7.3.1 The Discrete Eigenvalues for Cu = —uxx 420
7.3.2 The Discrete Eigenvalues for Cu = —vuxx + (iux 424
7.3.3 The Discrete Eigenvalues for Cu = ux 427
7.4 The Preconditioning of Spectral Operators 430
7.5 The Heat Equation 433
7.6 Linear Hyperbolic Equations 439
7.6.1 Periodic Boundary Conditions 439
7.6.2 Nonperiodic Boundary Conditions 445
7.6.3 The Resolution of the Gibbs Phenomenon 447
7.6.4 Spectral Accuracy for Non Smooth Solutions 454
7.7 Scalar Conservation Laws 459
7.8 The Steady Burgers Equation 463
Appendix A. Basic Mathematical Concepts 471
A.I Hilbert and Banach Spaces 471
A.2 The Cauchy Schwarz Inequality 473
A.3 Linear Operators Between Banach Spaces 474
A.4 The Frechet Derivative of an Operator 475
A.5 The Lax Milgram Theorem 475
A.6 Dense Subspace of a Normed Space 476
A.7 The Spaces Cm(Q), m 0 476
A.8 Functions of Bounded Variation
and the Riemann( Stieltjes) Integral 476
A.9 The Lebesgue Integral and Lp Spaces 478
A. 10 Infinitely Differentiable Functions and Distributions 482
A.11 Sobolev Spaces and Sobolev Norms 484
A.12 The Sobolev Inequality 490
A.13 The Poincare Inequality 491
A.14 The Hardy Inequality 491
A.15 The Gronwall Lemma 492
Appendix B. Fast Fourier Transforms 493
Appendix C. Iterative Methods for Linear Systems 499
C.I A Gentle Approach to Iterative Methods 499
C.2 Descent Methods for Symmetric Problems 503
C.3 Krylov Methods for Nonsymmetric Problems 508
Appendix D. Time Discretizations 515
D.I Notation and Stability Definitions 515
D.2 Standard ODE Methods 519
D.2.1 Leap Frog Method 519
D.2.2 Adams Bashforth Methods 520
XVIII Contents
D.2.3 Adams Moulton Methods 521
D.2.4 Backwards Difference Formulas 524
D.2.5 Runge Kutta Methods 524
D.3 Integrating Factors 525
D.4 Low Storage Schemes 527
References 529
Index 553 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author_GND | (DE-588)11157434X |
| building | Verbundindex |
| bvnumber | BV021996554 |
| callnumber-first | Q - Science |
| callnumber-label | QA320 |
| callnumber-raw | QA320 |
| callnumber-search | QA320 |
| callnumber-sort | QA 3320 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 920 VG 8900 |
| ctrlnum | (OCoLC)68629287 (DE-599)BVBBV021996554 |
| dewey-full | 515/.7222 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.7222 |
| dewey-search | 515/.7222 |
| dewey-sort | 3515 47222 |
| dewey-tens | 510 - Mathematics |
| discipline | Chemie / Pharmazie Mathematik |
| discipline_str_mv | Chemie / Pharmazie Mathematik |
| format | Book |
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| id | DE-604.BV021996554 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:10:53Z |
| indexdate | 2024-07-09T20:48:59Z |
| institution | BVB |
| isbn | 3540307257 9783540307259 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015211231 |
| oclc_num | 68629287 |
| open_access_boolean | |
| owner | DE-706 DE-634 DE-11 DE-188 DE-91G DE-BY-TUM DE-29T |
| owner_facet | DE-706 DE-634 DE-11 DE-188 DE-91G DE-BY-TUM DE-29T |
| physical | XXII, 563 S. Ill., graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Scientific computation |
| spelling | Spectral methods fundamentals in single domains ; with ... 19 tables Berlin [u.a.] Springer 2006 XXII, 563 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Scientific computation Literaturverz. S. [529] - 552 Análise espectral (análise funcional) larpcal Análise numérica larpcal Equações diferenciais parciais larpcal Spectraaltheorie gtt Differential equations, Partial Numerical solutions Numerical analysis Spectral theory (Mathematics) Spektralmethode (DE-588)4224817-6 gnd rswk-swf Spektralmethode (DE-588)4224817-6 s DE-604 Canuto, Claudio 1952- Sonstige (DE-588)11157434X oth http://deposit.dnb.de/cgi-bin/dokserv?id=2749010&prov=M&dok_var=1&dok_ext=htm Beschreibung für Leser HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015211231&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Spectral methods fundamentals in single domains ; with ... 19 tables Análise espectral (análise funcional) larpcal Análise numérica larpcal Equações diferenciais parciais larpcal Spectraaltheorie gtt Differential equations, Partial Numerical solutions Numerical analysis Spectral theory (Mathematics) Spektralmethode (DE-588)4224817-6 gnd |
| subject_GND | (DE-588)4224817-6 |
| title | Spectral methods fundamentals in single domains ; with ... 19 tables |
| title_auth | Spectral methods fundamentals in single domains ; with ... 19 tables |
| title_exact_search | Spectral methods fundamentals in single domains ; with ... 19 tables |
| title_exact_search_txtP | Spectral methods fundamentals in single domains ; with ... 19 tables |
| title_full | Spectral methods fundamentals in single domains ; with ... 19 tables |
| title_fullStr | Spectral methods fundamentals in single domains ; with ... 19 tables |
| title_full_unstemmed | Spectral methods fundamentals in single domains ; with ... 19 tables |
| title_short | Spectral methods |
| title_sort | spectral methods fundamentals in single domains with 19 tables |
| title_sub | fundamentals in single domains ; with ... 19 tables |
| topic | Análise espectral (análise funcional) larpcal Análise numérica larpcal Equações diferenciais parciais larpcal Spectraaltheorie gtt Differential equations, Partial Numerical solutions Numerical analysis Spectral theory (Mathematics) Spektralmethode (DE-588)4224817-6 gnd |
| topic_facet | Análise espectral (análise funcional) Análise numérica Equações diferenciais parciais Spectraaltheorie Differential equations, Partial Numerical solutions Numerical analysis Spectral theory (Mathematics) Spektralmethode |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2749010&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015211231&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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