Solution of boundary value problems for nonlinear elliptic equations by the Monte Carlo method

In this paper we construct a new algorithm for solving the third boundary value problem with a nonlinear equation of the form Delta u = u + u(2) by the Monte Carlo method. The estimate of the Monte Carlo method is constructed for a special chain of 'random walk by spheres and balls' with branching. In addition, we obtain new estimates of the Monte Carlo method for the first and second partial derivatives of the solution of Dirichlet'problem for a nonlinear equation of the form Delta u(c) + u(c)(n) = 0, u(c/Gamma) = c psi. Using the algorithms obtained, we construct an estimate for the Laplacian of the solution au. Then we study the hypothesis that the mathematical expectation of the estimate of the solution to Dirichlet's problem yields an exact solution as the parameter c increases. The calculations carried out for the case n = 2 confirm this hypothesis.