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

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  • Rasulov, Abdujabar
  • Raimova, Gulnora

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.

Suggested Citation

  • Rasulov, Abdujabar & Raimova, Gulnora, 2018. "Monte Carlo method for solution of initial–boundary value problem for nonlinear parabolic equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 146(C), pages 240-250.
  • Handle: RePEc:eee:matcom:v:146:y:2018:i:c:p:240-250
    DOI: 10.1016/j.matcom.2017.04.003
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