SOLUTION OF THE NEUMANN PROBLEM AND MIXED PROBLEMS BASED ON THE HELMHOLTZ EQUATION USING MONTE CARLO METHODS

Abstract

In this article, we consider the solution of the Neumann problem and mixed problems posed in the Helmholtz equation using Monte Carlo methods. The Neumann problem is a boundary value problem represented by differential equations with boundary conditions (also called second-order boundary conditions) given by the derivative of the unknown function on the boundary of the domain. To achieve the goals set, Markov chains were constructed and the Neumann problem was solved using the "boundary walk" and "sphere walk" algorithms. In cases where the boundary condition is given in the form of a derivative of the sought-after function with respect to the external normal vector pointing towards the boundary, a mixed problem arises. It is evident that the solution of integral equations using the Monte Carlo method is closely related to the simulation of a Markov chain, and the number of final transitions should be interrupted with a probability of one. It is crucial that the mathematical expectation of M(N) is finite.