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

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's 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's value at any chosen point. Two examples are provided for illustration, and the results of the numerical experiments are presented and discussed.