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020    |z9783110315295 
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035    (Au-PeEL)EBL1524379 
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035    (CaONFJC)MIL808134 
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040    MiAaPQ|beng|erda|epn|cMiAaPQ|dMiAaPQ 
050  4 QA374.S23 2013eb 
082 0  515/.353 
100 1  Sabelfeld, Karl K 
245 10 Spherical and Plane Integral Operators for PDEs :
       |bConstruction, Analysis, and Applications 
264  1 Berlin/Boston :|bDe Gruyter, Inc.,|c2013 
264  4 |c©2013 
300    1 online resource (328 pages) 
336    text|btxt|2rdacontent 
337    computer|bc|2rdamedia 
338    online resource|bcr|2rdacarrier 
505 0  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 
505 8  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 
505 8  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 
505 8  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 
520    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 
588    Description based on publisher supplied metadata and other
       sources 
590    Electronic reproduction. Ann Arbor, Michigan : ProQuest 
       Ebook Central, 2020. Available via World Wide Web. Access 
       may be limited to ProQuest Ebook Central affiliated 
       libraries 
650  0 Differential equations, Partial.;Integral operators 
655  4 Electronic books 
700 1  Shalimova, Irina A 
776 08 |iPrint version:|aSabelfeld, Karl K.|tSpherical and Plane 
       Integral Operators for PDEs : Construction, Analysis, and 
       Applications|dBerlin/Boston : De Gruyter, Inc.,c2013
       |z9783110315295 
856 40 |uhttps://ebookcentral.proquest.com/lib/sinciatw/
       detail.action?docID=1524379|zClick to View