The Numerical Solution of the External Dirichlet Generalized Harmonic Problem for a Sphere by the Method of Probabilistic Solution

TL;DR

This paper presents a novel algorithm using probabilistic methods to numerically solve the external Dirichlet generalized harmonic problem for a sphere, involving domain inversion, Kelvin theorem, and Wiener process simulation.

Contribution

It introduces a new probabilistic algorithm for solving boundary value problems with discontinuities on a sphere's surface, combining domain transformation and stochastic simulation.

Findings

Algorithm successfully computes solutions with discontinuities.

Numerical examples demonstrate accuracy and effectiveness.

Method integrates domain inversion and stochastic simulation techniques.

Abstract

In the present paper, an algorithm for the numerical solution of the external Dirichlet generalized harmonic problem for a sphere by the method of probabilistic solution (MPS) is given, where generalized indicates that a boundary function has a finite number of first kind discontinuity curves. The algorithm consists of the following main stages: (1) the transition from an infinite domain to a finite domain by an inversion; (2) the consideration of a new Dirichlet generalized harmonic problem on the basis of Kelvin theorem for the obtained finite domain; (3) the numerical solution of the new problem for the finite domain by the MPS, which in turn is based on a computer simulation of the Weiner process; (4) finding the probabilistic solution of the posed generalized problem at any fixed points of the infinite domain by the solution of the new problem. For illustration, numerical examples are considered and results are presented.