Spherical and Plane Integral Operators for PDEs (eBook)

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

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