Monte Carlo method for solution of initial–boundary value problem for nonlinear parabolic equations

Abstract

In this paper we will consider the initial–boundary value problem for a parabolic equation with a polynomial non-linearity relative to the unknown function. First we will derive a probabilistic representation of our problem. The representation of the solution of this problem is given in the form of a mathematical expectation, which is determined based on trajectories of branching processes. Under the assumption of the existence of the solution, an unbiased estimator is built using trajectories of a branching process. We will use a mean value theorem to write out a special integral equation, that equates the value of the unknown function u(x,t) with its integral over a spheroid and balloid with center at the point (x,t). A probabilistic representation of the solution to the problem in the form of mathematical expectation of some random variables is obtained. This probabilistic representation uses a branching process whose trajectories are used in the contraction of an unbiased estimator for the solution. The derived unbiased estimator has a finite variance, and is built up from trajectories of branching processes with a finite average number of branches. Finally, the results of numerical experiments and application to the practical problems are discussed.