Application von-neumann-ulam scheme to the solution of the first boundary value problem for a parabolic equation

Abstract

Statistical estimators for the solutions of boundary value problems for parabolic equations with constant coefficients are usually constructed on the random walk trajectories.The phase space of these random walk is the domain for the boundary value problem or the boundary of this domain. It is necessary to know the exact fundamental solution to construct the random walk. So, this algorithm cannot be used for heat equations when its coefficients are not constant. In this paper we construct unbiased estimators and small biased estimators for the solution of the first boundary value problem for parabolic equations in case when the coefficient of the unknown function is sufficiently smooth and other coefficients are constant. We use the random walk on balloids and von- Neumann-Ulam scheme generalization for integral equation with sub-stochastic kernel to investigate the Markov chain properties and statistical estimators. The algorithm is based on a new integral representation for the solution of the boundary value problem. Refs 0. Figs 0. Tables 0.

Keywords:

Monte Carlo methods, simulation method, heat equation, random walk on balloids