Application of branching processes to solve boundary value problems for the semi-linear Helmholtz equation

University of World Economy and Diplomacy, 54 Mustakillik Ave., Tashkent 100007, Uzbekistan

^* Corresponding author: asrasulov@gmail.com

Abstract

This work devoted to construct a probabilistic model for solving boundary value problems for the semi-linear Helmholtz equation. Probabilistic representations of problem solving in the form of the mathematical expectation of some random variable are obtained. In accordance with probabilistic representation, branching processes are constructed and modeling formulas are specified for each of the branching processes. It has been proven that the constructed branching processes degenerate with probability one and the average number of particles of each n-th generation is less than or equal to one. For the considered problems, unbiased estimates were constructed on the trajectories of the corresponding random processes. Unbiased estimates which constructed on the trajectories of a branching process with a limited average number of branches, easily modeled in computer and has limited variance. Based on the proposed estimates, computational experiments were carried out. The results of the computational experiment show that using the constructed algorithm it is possible to solve semi-linear problems that occur in practice.