Random walks on boundary for solving PDEs

著者

• Sabelʹfelʹd, Karl Karlovich
• Simonov, N. A

書誌事項

Random walks on boundary for solving PDEs

K.K. Sabelfeld and N.A. Simonov

VSP, 1994

内容説明・目次

内容説明

01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information. This monograph presents new probabilistic representations for classical boundary value problems of mathematical physics and is the first book devoted to the walk on boundary algorithms. Compared to the well-known Wiener and diffusion path integrals, the trajectories of random walks in this publication are simlated on the boundary of the domain as Markov chains generated by the kernels of the boundary integral equations equivalent to the original boundary value problem. The book opens with an introduction for solving the interior and exterior boundary values for the Laplace and heat equations, which is followed by applying this method to all main boundary value problems of the potential and elasticity theories.

目次

Introduction RANDOM WALK ALGORITHMS FOR SOLVING INTEGRAL EQUATIONS Conventional Monte Carlo scheme Biased estimators Linear-fractional transformations and relations to iterative processes Asymptotically unbiased estimators based on singular approximation of the kernel Integral equation of the first kind RANDOM WALK ON BOUNDARY ALGORITHMS FOR SOLVING THELAPLACE EQUATION Newton potentials and boundary integral equations of the electrostatics The interior Dirichlet problem and isotropic Random Walk on Boundary process Solution of the Neumann problem Third boundary value problem and alternative methods of solving the Dirichlet problem Inhomogeneous problems Calculation of the derivatives near the boundary Normal derivative of a double layer potential WALK ON BOUNDARY ALGORITHMS FOR THE HEAT EQUATION Heat potential and Volterra boundary integral equations Nonstationary Walk on Boundary process The Dirichlet problem The Neumann problem Third boundary value problem Unbiasedness and variance of the Walk on Boundary algorithms The cost of the Walk on Boundary algorithms Inhomogeneous heat equation Calculation of derivatives on the boundary SPATIAL PROBLEMS OF ELASTICITY Elastopotentials and systems of boundary integral equations of the elasticity theory First boundary value problem and estimators for singular integrals Other boundary value problems for the Lame equations and regular integral equations VARIANTS OF THE RANDOM WALK ON BOUNDARY FOR SOLVING THE STATIONARY POTENTIAL PROBLEMS The Robin problem and the ergodic theorem Stationary diffusion equation with absorption Stabilization method Multiply connected domains RANDOM WALK ON BOUNDARY IN NON LINEAR PROBLEMS Nonlinear Poisson equation Boundary value problem for the Navier--Stokes equation Bibliography

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詳細情報

• NII書誌ID(NCID): BA25474257
• ISBN:
  • 9067641839
• 出版国コード: ne
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Utrecht
• ページ数/冊数: iv, 137 p.
• 大きさ: 25 cm