Monte Carlo methods for solving the first boundary value problem for a polyharmonic equation

Results of solving the first boundary value problem for a polyharmonic equation are presented. The technique is based on the probabilistic representation of the solution of this problem constructed by the authors. Such a solution is shown to be a parametric derivative of the solution of a special Dirichlet problem for the Helmholtz equation. Based on this fact, new “walk-by-spheres” algorithms for a polyharmonic equation are developed. This made it possible to construct an algorithm implementing the Monte Carlo method for estimating the covariance function of the solution of a biharmonic equation with random functional parameters.