Monte Carlo solution of heat conductivity problems with quadratic nonlinearity on the boundary of domain

Abstract

In this work, we study the approach connected with branching Markov processes. Branching Markov random processes are created to solve boundary-value problems (BVR) with polynomial nonlinearities. The realization of these processes creates so called “trees”. Unbiased estimators of the solution of the nonlinear problem are constructed on these random trees. We also calculated in parallel way the variance (statistical error) of the constructed unbiased estimators. We offer computational algorithms for solving some nonlinear diffusion problems which frequently appears in engineering problems, particularly in heat conductivity. The case of a quadratic nonlinearity for the boundary BVP is considered in detail. Algorithms, for diffusion BVPs with nonlinear boundary conditions, differ from proposed for linear diffusion BVPs algorithms early and we compare their efficiencies. The described algorithms applied to the computational problem of thermal engineering in the presence of a nonlinear boundary condition. Also, this problem is solved as a boundary value problem of conductive heat transfer.