Sabelfeld / Shalimova Spherical and Plane Integral Operators for PDEs

Construction, Analysis, and Applications
1. Auflage 2013
ISBN: 978-3-11-031533-2
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 338 Seiten

The book presents integral formulations for partial differential equations, with the focus on spherical and plane integral operators. The integral relations are obtained for different elliptic and parabolic equations, and both direct and inverse mean value relations are studied. The derived integral equations are used to construct new numerical methods for solving relevant boundary value problems, both deterministic and stochastic based on probabilistic interpretation of the spherical and plane integral operators.

Zielgruppe

Advanced readers and specialists in mathematical physics and nume

Autoren/Hrsg.

Sabelfeld, Karl K.

Shalimova, Irina A.

Fachgebiete

• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik

Inhaltsverzeichnis

1;Preface;5
2;1 Introduction;11
3;2 Scalar second-order PDEs;15
3.1;2.1 Spherical mean value relations for the Laplace equation;15
3.1.1;2.1.1 Direct spherical mean value relation;15
3.1.2;2.1.2 Converse mean value theorem;21
3.1.3;2.1.3 Integral equation equivalent to the Dirichlet problem;22
3.1.4;2.1.4 Poisson–Jensen formula;24
3.2;2.2 The diffusion and Helmholtz equations;25
3.2.1;2.2.1 Diffusion equation;25
3.2.2;2.2.2 Helmholtz equation;27
3.3;2.3 Generalized second-order elliptic equations;28
3.4;2.4 Parabolic equations;30
3.4.1;2.4.1 Heat equation;30
3.4.2;2.4.2 Parabolic equations with variable coefficients;35
3.4.3;2.4.3 Expansion of the parabolic means;37
3.5;2.5 Wave equation;39
4;3 High-order elliptic equations;42
4.1;3.1 Balayage operator;42
4.2;3.2 Biharmonic equation;44
4.2.1;3.2.1 Direct spherical mean value relation;44
4.2.2;3.2.2 Generalized Poisson formula;45
4.2.3;3.2.3 Rigid fixing of the boundary;49
4.2.4;3.2.4 Nonhomogeneous biharmonic equation;52
4.3;3.3 Fourth-order equation governing the bending of a plate;54
4.4;3.4 Metaharmonic equations;58
4.4.1;3.4.1 Polyharmonic equation;58
4.4.2;3.4.2 General case;60
5;4 Triangular systems of elliptic equations;65
5.1;4.1 One-component diffusion system;65
5.2;4.2 Two-component diffusion system;66
5.3;4.3 Coupled biharmonic–harmonic equation;68
6;5 Systems of elasticity theory;70
6.1;5.1 Lamé equation;70
6.1.1;5.1.1 Direct spherical mean value theorem;70
6.1.2;5.1.2 Converse spherical mean value theorem;74
6.2;5.2 Pseudovibration elastic equation;76
6.3;5.3 Thermoelastic equation;83
7;6 The generalized Poisson formula for the Lamé equation;84
7.1;6.1 Plane elasticity;84
7.1.1;6.1.1 Poisson formula for the displacements in rectangular coordinates;84
7.1.2;6.1.2 Poisson formula for displacements in polar coordinates;93
7.2;6.2 Generalized spatial Poisson formula for the Lamé equation|;96
7.3;6.3 An alternative derivation of the Poisson formula;108
8;7 Spherical means for the stress and strain tensors;112
8.1;7.1 Sphericalmeans for the displacements;112
8.2;7.2 Mean value relations for the stress and strain tensors;115
8.2.1;7.2.1 Mean value relation for the strain components;115
8.2.2;7.2.2 Mean value relation for the stress components;120
8.3;7.3 Mean value relations for the stress components;121
9;8 Random Walk on Spheres method;130
9.1;8.1 Sphericalmean as a mathematical expectation;130
9.2;8.2 Iterations of the spherical mean operator;131
9.3;8.3 The Random Walk on Spheres algorithm;132
9.3.1;8.3.1 The Random Walk on Spheres process for the Dirichlet problem;132
9.3.2;8.3.2 Inhomogeneous case;140
9.4;8.4 Biharmonic equation;142
9.5;8.5 Isotropic elastostatics governed by the Lamé equation;144
9.5.1;8.5.1 Naive generalization;144
9.5.2;8.5.2 Modification of the algorithm;145
9.5.3;8.5.3 Nonisotropic Random Walk on Spheres;147
9.5.4;8.5.4 Branching process;149
9.5.5;8.5.5 Analytical continuation with respect to the spectral parameter;151
9.6;8.6 Alternative Schwarz procedure;154
10;9 Random Walk on Fixed Spheres for Laplace and Lamé equations;158
10.1;9.1 Introduction;158
10.2;9.2 Laplace equation;160
10.2.1;9.2.1 Integral formulation of the Dirichlet problem;160
10.2.2;9.2.2 Approximation by linear algebraic equations;167
10.2.3;9.2.3 Set of overlapping disks;168
10.2.4;9.2.4 Estimation of the spectral radius;173
10.3;9.3 Isotropic elastostatics;175
10.4;9.4 Iteration methods;178
10.4.1;9.4.1 Stochastic iterative procedure with optimal random parameters;178
10.4.2;9.4.2 SOR method;183
10.5;9.5 Discrete Random Walk algorithms;186
10.5.1;9.5.1 Discrete Random Walk based on the iteration method;186
10.5.2;9.5.2 Discrete Random Walk method based on SOR;187
10.5.3;9.5.3 Sampling from discrete distribution;188
10.5.4;9.5.4 Variance of stochastic methods;189
10.6;9.6 Numerical simulations;191
10.6.1;9.6.1 Laplace equation;191
10.6.2;9.6.2 Lamé equation;192
10.7;9.7 Conclusion and discussion;194
11;10 Stochastic spectral projection method for solving PDEs;196
11.1;10.1 Introduction;196
11.2;10.2 Laplace equation;197
11.2.1;10.2.1 Two overlapping disks;197
11.2.2;10.2.2 Neumann boundary conditions;202
11.2.3;10.2.3 Overlapping of a half-plane with a set of disks;204
11.3;10.3 Extension to the isotropic elasticity: Lamè equation;207
11.3.1;10.3.1 Elastic disk;207
11.3.2;10.3.2 Elastic half-plane;209
11.4;10.4 Extension to 3D problems;210
11.4.1;10.4.1 A sphere;210
11.4.2;10.4.2 Elastic half-space;211
11.5;10.5 Stochastic projection method for large linear systems;213
12;11 Stochastic boundary collocation and spectral methods;215
12.1;11.1 Introduction;215
12.2;11.2 Surface and volume potentials;216
12.3;11.3 Random Walk on Boundary Algorithm;218
12.4;11.4 General scheme of the method of fundamental solutions (MFS);220
12.4.1;11.4.1 Kupradze–Aleksidze’s method based on first-kind integral equation;222
12.4.2;11.4.2 MFS for Laplace and Helmholz equations;223
12.4.3;11.4.3 Biharmonic equation;224
12.5;11.5 MFS with separable Poisson kernel;224
12.5.1;11.5.1 Dirichlet problem for the Laplace equation;225
12.5.2;11.5.2 Evaluation of the Green function and solving inhomogeneous problems;227
12.5.3;11.5.3 Evaluation of derivatives on the boundary and construction of the Poisson integral formulae;229
12.6;11.6 Hydrodynamics friction and the capacitance of a chain of spheres;230
12.7;11.7 Lamé equation: plane elasticity problem;235
12.8;11.8 SVD and randomized versions;239
12.8.1;11.8.1 SVD background;239
12.8.2;11.8.2 Randomized SVD algorithm;240
12.8.3;11.8.3 Using SVD for the linear least squares solution;242
12.9;11.9 Numerical experiments;243
13;12 Solution of 2D elasticity problems with random loads;251
13.1;12.1 Introduction;251
13.2;12.2 Lamé equation with nonzero body forces;254
13.3;12.3 Random loads;259
13.4;12.4 Random Walk methods and Double Randomization;261
13.4.1;12.4.1 General description;261
13.4.2;12.4.2 Green-tensor integral representation for the correlations;262
13.5;12.5 Simulation results;264
13.5.1;12.5.1 Testing the simulation procedure for random loads;264
13.5.2;12.5.2 Testing the Random Walk algorithm for nonzero body forces;264
13.5.3;12.5.3 Calculation of correlations for the displacement vector;265
14;13 Boundary value problems with random boundary conditions;270
14.1;13.1 Introduction;270
14.1.1;13.1.1 Spectral representations;271
14.1.2;13.1.2 Karhunen–Loève expansion;273
14.2;13.2 Stochastic boundary value problems for the 2D Laplace equation;275
14.2.1;13.2.1 Dirichlet problem for a 2D disk: white noise excitations;277
14.2.2;13.2.2 General homogeneous boundary excitations;283
14.2.3;13.2.3 Neumann boundary conditions;284
14.2.4;13.2.4 Upper half-plane;286
15;13.3 3D Laplace equation;289
15.1;13.4 Biharmonic equation;292
15.2;13.5 Lamé equation: plane elasticity problem;295
15.2.1;13.5.1 White noise excitations;295
15.2.2;13.5.2 General case of homogeneous excitations;303
15.3;13.6 Response of an elastic 3D half-space to random excitations;307
15.3.1;13.6.1 Introduction;307
15.3.2;13.6.2 System of Lamé equations governing an elastic half-space with no tangential surface forces;308
15.3.3;13.6.3 Stochastic boundary value problem: correlation tensor;309
15.3.4;13.6.4 Spectral representations for partially homogeneous random fields;311
15.3.5;13.6.5 Displacement correlations for the white noise excitations;313
15.3.6;13.6.6 Homogeneous excitations;315
15.3.7;13.6.7 Conclusions and discussion;318
15.3.8;13.6.8 Appendix A: the Poisson formula;319
15.3.9;13.6.9 Appendix B: some 2D Fourier transform formulae;321
15.3.10;13.6.10 Appendix C: some 2D integrals;322
15.3.11;13.6.11 Appendix D: some further Fourier transform formulae;324
16;Bibliography;327
17;Index;337

Über Autor(innen)

Karl Sabelfeld and Irina Shalimova, Russian Academy of Sciences and Novosibirsk University, Novosibirsk, Russia.