The boundary domain integral method for elliptic systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
1998
|
| Series: | Lecture notes in mathematics
1683 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 163 S. graph. Darst. |
| ISBN: | 3540641637 |
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MARC
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| 100 | 1 | |a Pomp, Andreas |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The boundary domain integral method for elliptic systems |c Andreas Pomp |
| 246 | 1 | 3 | |a The boundary-domain integral for elliptic systems |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1998 | |
| 300 | |a XVI, 163 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1683 | |
| 650 | 4 | |a Coques (Ingénierie) - Modèles mathématiques | |
| 650 | 7 | |a Partiële differentiaalvergelijkingen |2 gtt | |
| 650 | 7 | |a Randwaardeproblemen |2 gtt | |
| 650 | 4 | |a Équations différentielles elliptiques | |
| 650 | 4 | |a Équations intégrales de frontière, Méthodes des | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Boundary element methods | |
| 650 | 4 | |a Differential equations, Elliptic | |
| 650 | 4 | |a Shells (Engineering) |x Mathematical models | |
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Record in the Search Index
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| adam_text | Table of Contents
Abbreviations and Symbols IX
Introduction XI
Part I. The General Theory for Elliptic Systems of Partial
Differential Equations
1. Pseudohomogeneous Distributions 3
1.1 Basic Definitions of Homogeneity and Pseudohomogeneity ... 3
1.2 Spherical Harmonics 5
1.3 Fourier Transform 7
1.4 The Canonical Continuation of Homogeneous Functions 7
1.5 Properties of the Canonical Continuation 9
1.6 The Canonical Projection 11
1.7 The Canonical Projection Applied to Pseudohomogeneous
Functions 14
1.8 Special Classes of Pseudohomogeneous Distributions 16
1.9 Mapping Properties of Basic Operators 18
1.9.1 Differentiation 18
1.9.2 Multiplication with Homogeneous Polynomials 18
1.9.3 Homogeneous Elliptic Differential Operators 19
1.10 The Right Inverse to Homogeneous Elliptic Differential Op¬
erators 20
1.11 Local Convergent Series with Pseudohomogeneous Terms .... 22
1.12 Basic Operators Acting on Series with Pseudohomogeneous
Terms 23
1.12.1 Differentiation 23
, 1.12.2 Multiplication 23
1.12.3 Homogeneous Elliptic Differential Operators 23
1.12.4 Fourier Transform 24
1.13 The Parity Condition 25
VI Table of Contents
2. Levi Functions for Elliptic Systems of Partial Differential
Equations 29
2.1 Definitions of Ellipticity in the Sense of Douglis and Nirenberg 29
2.2 Fundamental Solutions and Levi Functions 31
2.3 Existence Results for Fundamental Solutions 31
2.4 A General Algorithm for the Construction of Levi Functions . 32
2.4.1 Assumptions 32
2.4.2 The Steps of the Algorithm 32
2.4.3 Practical Realization 33
2.5 Local Smoothness Properties 34
2.6 Cutting the Higher Order Terms by Projections 36
2.7 Pseudohomogeous Series Expansions of Fundamental Solutions 37
2.8 The Example of an Ordinary Differential Operator 38
2.9 Global Smoothness Properties 40
2.10 Classical Methods to Construct Fundamental Solutions 42
2.10.1 Systems with Constant Coefficients 42
2.10.2 J. Hadamard s Method 42
2.10.3 Picard s Method of Iterated Kernels 43
2.10.4 F. John s Method 44
2.10.5 The Method of Freezing the Coefficients 44
2.10.6 Comparison of the Methods 44
3. Systems of Integral Equations, Generated by Levi Functions 47
3.1 Transformation of Boundary Value Problems 48
3.1.1 The Direct Method 48
3.1.2 The Indirect Method 49
3.1.3 Comparison of the Methods 50
3.2 On the Unique Solvability of the Integral Equation System .. 51
3.2.1 Sufficient Conditions 51
3.2.2 An Example for Non Unique Solvability 51
3.3 Sobolev Spaces 52
3.3.1 Spaces on K 52
3.3.2 Spaces on Lipschitz Domains 53
3.3.3 Spaces on the Boundary of Lipschitz Domains 54
3.3.4 Spaces on the Cut Plane 54
3.4 Mellin Transform and Weighted Sobolev Spaces 55
3.5 Cauchy Singular Integral Operators on the Half Axis 56
3.5.1 A Basic Theorem 56
3.5.2 The Jump Relation 61
3.5.3 A Corollary to the Basic Theorem 62
3.6 Integral Operators Considered as Pseudodifferential Operators 63
3.7 Mapping Properties of Integral Operators 65
Table of Contents VII
Part II. Application to the Shell Model of Donnell Vlasov Type
4. The Differential Equations of the DV Model 71
4.1 General Remarks to Shell Theory 71
4.2 Assumptions and Properties of the Model 72
4.3 Geometrical Configuration 73
4.4 Shell Equations in Covariant Derivatives 74
4.5 Preliminaries from Differential Geometry 76
4.6 Splitting of the Shell Equations 77
4.6.1 The First Part 78
4.6.2 The Second Part 79
4.7 The Shell Equations in Partial Derivatives 82
4.8 Shell Equations in Isothermal Parametrization 83
4.9 Factorization with Respect to Cauchy Riemann Operators ... 83
4.10 Strain Free Deformations 84
4.11 The Energy Bilinear Form 87
5. Levi Functions for the Shell Equations 89
5.1 Ellipticity of the Operator 89
5.2 Splitting of the Operator into Principal and Remaining Part . 90
5.3 Local Isothermal Parametrization 92
5.4 Pseudohomogeneous Functions of Special Structure 93
5.5 Inversion of the Cauchy Riemann Operators 95
5.6 The Initial Terms of the Levi Function 97
5.7 Mapping Properties of the Individual Operators 100
5.8 The Structure of the Series Terms 102
5.9 Pseudohomogeneous Expansion of the Fundamental Solution . 105
5.10 Form of the Remainder 106
5.10.1 The Upper Left Block 106
5.10.2 The Lower Right Entry 107
5.10.3 The Upper Right Block 107
5.10.4 The Lower Left Block 108
6. The System of Integral Equations and its Numerical Solu¬
tion 109
6.1 Boundary Geometry and the Integral Theorem of Gauss 109
6.2 Boundary Data and Green s Formula 112
6.3 The Integral Equation System for the Dirichlet Problem 115
6.3.1 The Dirichlet Problem 115
6.3.2 Transformation into a System of Integral Equations... 115
6.3.3 The Kernel Functions 116
6.3.4 The Enhanced System of Integral Equations 117
6.4 Mapping Properties of the Operator Matrix 118
6.5 Corner Singularities 120
VIII Table of Contents
6.6 Galerkin s Method 124
6.7 Error Estimates 125
6.8 Order of Convergence for Quasi Uniform Meshes 127
6.9 Approximation of the Singular Functions on Graded Meshes . 128
6.10 Order of Convergence for Graded Meshes 132
6.11 Estimation of the Admissible Local Quadrature Error 133
7. An Example: Katenoid Shell Under Torsion 137
7.1 Geometry of the Mid Surface 137
7.2 Shell Equations and Boundary Conditions 139
7.3 Levi Function and Integral Kernels 140
7.3.1 The Levi Function Ln 140
7.3.2 The Integral Kernel Lr(x,y) 141
7.3.3 Further Notations 141
7.3.4 The Integral Kernels Va and Vr 141
7.3.5 The Integral Kernel Rn 142
7.3.6 The Integral Kernel Rr 143
7.3.7 Mapping onto the Neumann Data 144
7.4 The Shell Equations as a Singular Perturbed Problem 144
7.5 Corner Singularities and Mesh Refinement 144
7.6 Discretization of the Integral Operators 145
7.6.1 Far Field 146
7.6.2 Near Field 146
7.7 Estimations for the Number of Cubature Sample Points 148
7.8 Numerical Results 152
Index 154
References 157
|
| any_adam_object | 1 |
| author | Pomp, Andreas |
| author_facet | Pomp, Andreas |
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| dewey-search | 624.1/7762/015118 |
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| discipline | Informatik Bauingenieurwesen Mathematik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:15:34Z |
| institution | BVB |
| isbn | 3540641637 |
| language | German |
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| physical | XVI, 163 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
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| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Pomp, Andreas Verfasser aut The boundary domain integral method for elliptic systems Andreas Pomp The boundary-domain integral for elliptic systems Berlin [u.a.] Springer 1998 XVI, 163 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1683 Coques (Ingénierie) - Modèles mathématiques Partiële differentiaalvergelijkingen gtt Randwaardeproblemen gtt Équations différentielles elliptiques Équations intégrales de frontière, Méthodes des Mathematisches Modell Boundary element methods Differential equations, Elliptic Shells (Engineering) Mathematical models Parametrix-Methode (DE-588)4499605-6 gnd rswk-swf Elliptisches System (DE-588)4121184-4 gnd rswk-swf Zylinderschale (DE-588)4191386-3 gnd rswk-swf Zylinderschale (DE-588)4191386-3 s Elliptisches System (DE-588)4121184-4 s Parametrix-Methode (DE-588)4499605-6 s DE-604 Lecture notes in mathematics 1683 (DE-604)BV000676446 1683 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007947378&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pomp, Andreas The boundary domain integral method for elliptic systems Lecture notes in mathematics Coques (Ingénierie) - Modèles mathématiques Partiële differentiaalvergelijkingen gtt Randwaardeproblemen gtt Équations différentielles elliptiques Équations intégrales de frontière, Méthodes des Mathematisches Modell Boundary element methods Differential equations, Elliptic Shells (Engineering) Mathematical models Parametrix-Methode (DE-588)4499605-6 gnd Elliptisches System (DE-588)4121184-4 gnd Zylinderschale (DE-588)4191386-3 gnd |
| subject_GND | (DE-588)4499605-6 (DE-588)4121184-4 (DE-588)4191386-3 |
| title | The boundary domain integral method for elliptic systems |
| title_alt | The boundary-domain integral for elliptic systems |
| title_auth | The boundary domain integral method for elliptic systems |
| title_exact_search | The boundary domain integral method for elliptic systems |
| title_full | The boundary domain integral method for elliptic systems Andreas Pomp |
| title_fullStr | The boundary domain integral method for elliptic systems Andreas Pomp |
| title_full_unstemmed | The boundary domain integral method for elliptic systems Andreas Pomp |
| title_short | The boundary domain integral method for elliptic systems |
| title_sort | the boundary domain integral method for elliptic systems |
| topic | Coques (Ingénierie) - Modèles mathématiques Partiële differentiaalvergelijkingen gtt Randwaardeproblemen gtt Équations différentielles elliptiques Équations intégrales de frontière, Méthodes des Mathematisches Modell Boundary element methods Differential equations, Elliptic Shells (Engineering) Mathematical models Parametrix-Methode (DE-588)4499605-6 gnd Elliptisches System (DE-588)4121184-4 gnd Zylinderschale (DE-588)4191386-3 gnd |
| topic_facet | Coques (Ingénierie) - Modèles mathématiques Partiële differentiaalvergelijkingen Randwaardeproblemen Équations différentielles elliptiques Équations intégrales de frontière, Méthodes des Mathematisches Modell Boundary element methods Differential equations, Elliptic Shells (Engineering) Mathematical models Parametrix-Methode Elliptisches System Zylinderschale |
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