The Monte Carlo solution of a boundary value problem for the metaharmonic equation

An algorithm of the Monte Carlo method is constructed for solving the metaharmonic equation (1). A system of integral equations of the second kind is derived for the functions $\delta^ku(x)$, $k = 0, 1,..., n-1$. It is shown that if the singularities of the kernels are included in the transition density of the simulated Markov chain, the Neumann series for this system converges, which enables the Monte Carlo method to be used. The case $n=2$, $x\in R^m$, important in the theory of plasticity is discussed in detail.