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

Abdujabar Rasulov and Gulnora Raimova

Mathematics and Computers in Simulation (MATCOM), 2018, vol. 146, issue C, 240-250

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.

Keywords: Monte Carlo method; Branching random process; Martingale; Unbiased estimator (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:146:y:2018:i:c:p:240-250

DOI: 10.1016/j.matcom.2017.04.003

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