A Mesh Free Stochastic Algorithm for Solving Diffusion–Convection–Reaction Equations on Complicated Domains

Abstract

A mesh free stochastic algorithm for solving transient diffusion–convection–reaction problems on domains with complicated structure is suggested. For the solutions of this kind of equations exact representations of the survival probabilities, the probability densities of the first passage time and position on a sphere are obtained. Based on these representations we construct a stochastic algorithm which is simple in implementaion for solving one- and three-dimensional diffusion–convection–reaction equations. The method is continuous both in space and time, and its advantages are particularly well manifested in solving problems on complicated domains, calculating fluxes to parts of the boundary, and other integral functionals, for instance, the total concentration of the particles which have been reacted to a time instant t.