Author Sabelfeld, Karl K
Title Spherical and Plane Integral Operators for PDEs : Construction, Analysis, and Applications
Imprint Berlin/Boston : De Gruyter, Inc., 2013
©2013
book jacket
Descript 1 online resource (328 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Note Intro -- Preface -- 1 Introduction -- 2 Scalar second-order PDEs -- 2.1 Spherical mean value relations for the Laplace equation -- 2.1.1 Direct spherical mean value relation -- 2.1.2 Converse mean value theorem -- 2.1.3 Integral equation equivalent to the Dirichlet problem -- 2.1.4 Poisson-Jensen formula -- 2.2 The diffusion and Helmholtz equations -- 2.2.1 Diffusion equation -- 2.2.2 Helmholtz equation -- 2.3 Generalized second-order elliptic equations -- 2.4 Parabolic equations -- 2.4.1 Heat equation -- 2.4.2 Parabolic equations with variable coefficients -- 2.4.3 Expansion of the parabolic means -- 2.5 Wave equation -- 3 High-order elliptic equations -- 3.1 Balayage operator -- 3.2 Biharmonic equation -- 3.2.1 Direct spherical mean value relation -- 3.2.2 Generalized Poisson formula -- 3.2.3 Rigid fixing of the boundary -- 3.2.4 Nonhomogeneous biharmonic equation -- 3.3 Fourth-order equation governing the bending of a plate -- 3.4 Metaharmonic equations -- 3.4.1 Polyharmonic equation -- 3.4.2 General case -- 4 Triangular systems of elliptic equations -- 4.1 One-component diffusion system -- 4.2 Two-component diffusion system -- 4.3 Coupled biharmonic-harmonic equation -- 5 Systems of elasticity theory -- 5.1 Lamé equation -- 5.1.1 Direct spherical mean value theorem -- 5.1.2 Converse spherical mean value theorem -- 5.2 Pseudovibration elastic equation -- 5.3 Thermoelastic equation -- 6 The generalized Poisson formula for the Lamé equation -- 6.1 Plane elasticity -- 6.1.1 Poisson formula for the displacements in rectangular coordinates -- 6.1.2 Poisson formula for displacements in polar coordinates -- 6.2 Generalized spatial Poisson formula for the Lamé equation| -- 6.3 An alternative derivation of the Poisson formula -- 7 Spherical means for the stress and strain tensors -- 7.1 Sphericalmeans for the displacements
7.2 Mean value relations for the stress and strain tensors -- 7.2.1 Mean value relation for the strain components -- 7.2.2 Mean value relation for the stress components -- 7.3 Mean value relations for the stress components -- 8 Random Walk on Spheres method -- 8.1 Sphericalmean as a mathematical expectation -- 8.2 Iterations of the spherical mean operator -- 8.3 The Random Walk on Spheres algorithm -- 8.3.1 The Random Walk on Spheres process for the Dirichlet problem -- 8.3.2 Inhomogeneous case -- 8.4 Biharmonic equation -- 8.5 Isotropic elastostatics governed by the Lamé equation -- 8.5.1 Naive generalization -- 8.5.2 Modification of the algorithm -- 8.5.3 Nonisotropic Random Walk on Spheres -- 8.5.4 Branching process -- 8.5.5 Analytical continuation with respect to the spectral parameter -- 8.6 Alternative Schwarz procedure -- 9 Random Walk on Fixed Spheres for Laplace and Lamé equations -- 9.1 Introduction -- 9.2 Laplace equation -- 9.2.1 Integral formulation of the Dirichlet problem -- 9.2.2 Approximation by linear algebraic equations -- 9.2.3 Set of overlapping disks -- 9.2.4 Estimation of the spectral radius -- 9.3 Isotropic elastostatics -- 9.4 Iteration methods -- 9.4.1 Stochastic iterative procedure with optimal random parameters -- 9.4.2 SOR method -- 9.5 Discrete Random Walk algorithms -- 9.5.1 Discrete Random Walk based on the iteration method -- 9.5.2 Discrete Random Walk method based on SOR -- 9.5.3 Sampling from discrete distribution -- 9.5.4 Variance of stochastic methods -- 9.6 Numerical simulations -- 9.6.1 Laplace equation -- 9.6.2 Lamé equation -- 9.7 Conclusion and discussion -- 10 Stochastic spectral projection method for solving PDEs -- 10.1 Introduction -- 10.2 Laplace equation -- 10.2.1 Two overlapping disks -- 10.2.2 Neumann boundary conditions -- 10.2.3 Overlapping of a half-plane with a set of disks
10.3 Extension to the isotropic elasticity: Lamè equation -- 10.3.1 Elastic disk -- 10.3.2 Elastic half-plane -- 10.4 Extension to 3D problems -- 10.4.1 A sphere -- 10.4.2 Elastic half-space -- 10.5 Stochastic projection method for large linear systems -- 11 Stochastic boundary collocation and spectral methods -- 11.1 Introduction -- 11.2 Surface and volume potentials -- 11.3 Random Walk on Boundary Algorithm -- 11.4 General scheme of the method of fundamental solutions (MFS) -- 11.4.1 Kupradze-Aleksidze's method based on first-kind integral equation -- 11.4.2 MFS for Laplace and Helmholz equations -- 11.4.3 Biharmonic equation -- 11.5 MFS with separable Poisson kernel -- 11.5.1 Dirichlet problem for the Laplace equation -- 11.5.2 Evaluation of the Green function and solving inhomogeneous problems -- 11.5.3 Evaluation of derivatives on the boundary and construction of the Poisson integral formulae -- 11.6 Hydrodynamics friction and the capacitance of a chain of spheres -- 11.7 Lamé equation: plane elasticity problem -- 11.8 SVD and randomized versions -- 11.8.1 SVD background -- 11.8.2 Randomized SVD algorithm -- 11.8.3 Using SVD for the linear least squares solution -- 11.9 Numerical experiments -- 12 Solution of 2D elasticity problems with random loads -- 12.1 Introduction -- 12.2 Lamé equation with nonzero body forces -- 12.3 Random loads -- 12.4 Random Walk methods and Double Randomization -- 12.4.1 General description -- 12.4.2 Green-tensor integral representation for the correlations -- 12.5 Simulation results -- 12.5.1 Testing the simulation procedure for random loads -- 12.5.2 Testing the Random Walk algorithm for nonzero body forces -- 12.5.3 Calculation of correlations for the displacement vector -- 13 Boundary value problems with random boundary conditions -- 13.1 Introduction -- 13.1.1 Spectral representations
13.1.2 Karhunen-Loève expansion -- 13.2 Stochastic boundary value problems for the 2D Laplace equation -- 13.2.1 Dirichlet problem for a 2D disk: white noise excitations -- 13.2.2 General homogeneous boundary excitations -- 13.2.3 Neumann boundary conditions -- 13.2.4 Upper half-plane -- 13.3 3D Laplace equation -- 13.4 Biharmonic equation -- 13.5 Lamé equation: plane elasticity problem -- 13.5.1 White noise excitations -- 13.5.2 General case of homogeneous excitations -- 13.6 Response of an elastic 3D half-space to random excitations -- 13.6.1 Introduction -- 13.6.2 System of Lamé equations governing an elastic half-space with no tangential surface forces -- 13.6.3 Stochastic boundary value problem: correlation tensor -- 13.6.4 Spectral representations for partially homogeneous random fields -- 13.6.5 Displacement correlations for the white noise excitations -- 13.6.6 Homogeneous excitations -- 13.6.7 Conclusions and discussion -- 13.6.8 Appendix A: the Poisson formula -- 13.6.9 Appendix B: some 2D Fourier transform formulae -- 13.6.10 Appendix C: some 2D integrals -- 13.6.11 Appendix D: some further Fourier transform formulae -- Bibliography -- Index
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
Description based on publisher supplied metadata and other sources
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Sabelfeld, Karl K. Spherical and Plane Integral Operators for PDEs : Construction, Analysis, and Applications Berlin/Boston : De Gruyter, Inc.,c2013 9783110315295
Subject Differential equations, Partial.;Integral operators
Electronic books
Alt Author Shalimova, Irina A