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

Mamuli Zakradze; Zaza Tabagari; Nana Koblishvili; Tinatin Davitashvili; José-María Sánchez-Sáez; Francisco Criado-Aldeanueva; Mamuli Zakradze; Zaza Tabagari; Nana Koblishvili; Tinatin Davitashvili; José-María Sánchez-Sáez; Francisco Criado-Aldeanueva

doi:10.3934/math.2025789

AIMS Mathematics

2025, Volume 10, Issue 8: 17657-17671. doi: 10.3934/math.2025789

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

1.
Department of Computational Methods, N. Muskhelishvili Institute of Computational Mathematics of the Georgian Technical University, Tbilisi, Georgia
2.
Faculty of Exact and Natural Sciences, I. Javakhishvili Tbilisi State University, Tbilisi, Georgia
3.
Department of Didactics of Mathematics, Faculty of Education, University of Malaga, 29071 Malaga, Spain
4.
Department of Applied Physics II, Polytechnic School, University of Malaga, 29071 Malaga, Spain

Received: 06 March 2025 Revised: 20 July 2025 Accepted: 28 July 2025 Published: 05 August 2025
MSC : 35J05, 35J25, 65C30, 65N75

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.
- Dirichlet generalized and classical harmonic problems,
- method of probabilistic solution,
- Wiener process,
- pyramidal domain
Citation: Mamuli Zakradze, Zaza Tabagari, Nana Koblishvili, Tinatin Davitashvili, José-María Sánchez-Sáez, Francisco Criado-Aldeanueva. The numerical solution of the Dirichlet generalized and classical harmonic problems for irregular $ n $-sided pyramidal domains by the method of probabilistic solutions[J]. AIMS Mathematics, 2025, 10(8): 17657-17671. doi: 10.3934/math.2025789

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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'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.

References

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