The numerical solution of the Dirichlet generalized and classical harmonic problems for irregular n-sided pyramidal domains by the method of probabilistic solutions

TL;DR

This paper introduces a probabilistic method for numerically solving Dirichlet harmonic problems on irregular pyramidal domains, effectively handling boundary discontinuities and complex geometries.

Contribution

It develops a novel algorithm using the method of probabilistic solutions with Wiener process modeling for irregular pyramids with boundary discontinuities.

Findings

Successfully applied MPS to irregular pyramids

Achieved accurate numerical solutions for boundary problems

Validated method with illustrative examples

Abstract

This paper describes the application of the method of probabilistic solutions (MPS) to numerically solve the Dirichlet generalized and classical harmonic problems for irregular n sided pyramidal domains. Here, generalized means that the boundary function has a finite number of first kind discontinuity curves, with the pyramid edges acting as these curves. The pyramid base is a convex polygon, and its vertex projection lies within the base. The proposed algorithm for solving boundary problems numerically includes the following steps: a) applying MPS, which relies on computer modeling of the Wiener process; b) determining the intersection point between the simulated Wiener process path and the pyramid surface; c) developing a code for numerical implementation and verifying the accuracy of the results; d) calculating the desired function value at any chosen point. Two examples are provided for illustration, and the results of the numerical experiments are presented and discussed.