Probabilistic Approach to Solution of the Neumann Problem for Some Nonlinear Equation

In this article we will consider the Neumann boundary-value problem for the nonlinear Helmholtz equation ? Δ?u + a?u = gexp?(u) + f₀. We will assume that there exists the solution to our problem and this permits us to construct an unbiased estimator on the trajectories of certain branching processes. To do so, we apply Green’s formula and an elliptic mean value theorem. This allows us to derive a special integral equation that gives the value of the function u(x) at the point x, with its integral over the domain D and on boundary of the domain ?D = G. The solution of the problem in the form of a mathematical expectation of some random variable is also obtained. In accordance with the probabilistic representation, a branching process is constructed and an unbiased estimator of the solution of the problem is built on its trajectories. The derived unbiased estimator has finite variance. The proposed branching process has a finite average number of branches, and easily simulated. We provide numerical results based on numerical experiments carried out with these algorithms.