A New Monte Carlo Approach for the Solution of Semi-Linear Neumann Boundary Value Problem
Abstract
In this paper we will consider the Neumann boundary-value problem for the Laplace operator with a polynomial nonlinearity on the right-hand side. We utilize Greens formula and an elliptic mean-value theorem. This allows us to derive a probabilistic representation as special integral equation which equates the value of the function u(x) at the centre-point x, with its integral over the domain D, and on boundary of the domain G. According to this probabilistic representation, a Markov branching process is constructed and an unbiased estimator of the solution of the nonlinear problem is formed by taking the mathematical expectation over these branching processes. The unbiased estimator which we derive has a finite variance. The complexity of the proposed algorithms is investigated, and the ways decreasing of the computational work are proposed. Some particular cases are considered in detail. The effectiveness of two estimators “by absorption” and “by collision” is studied in the nonlinear case.