Spherical and Plane Integral Operators for PDEs

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