Random Walks on Boundary for Solving PDEs

Karl K. Sabelfeld, Nikolai A. Simonov (Autoren)

Buch | Hardcover

141 Seiten

1994 | Reprint 2012
VSP International Science Publishers (Verlag)
978-90-6764-183-8 (ISBN)

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Presents probabilistic representations for classical boundary value problems of mathematical physics, and is devoted to the walk on boundary algorithms. The book should be of interest to specialists in the field of applied and computational mathematics and applied probability.

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.

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

Erscheint lt. Verlag	1.10.1994
Verlagsort	Zeist
Sprache	englisch
Gewicht	370 g
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
ISBN-10	90-6764-183-9 / 9067641839
ISBN-13	978-90-6764-183-8 / 9789067641838
Zustand	Neuware