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

Author

Listed:
  • Mamuli Zakradze

    (Department of Computational Methods, Muskhelishvili Institute of Computational Mathematics, Georgian Technical University, 0186 Tbilisi, Georgia)

  • Zaza Tabagari

    (Department of Computational Methods, Muskhelishvili Institute of Computational Mathematics, Georgian Technical University, 0186 Tbilisi, Georgia)

  • Nana Koblishvili

    (Department of Computational Methods, Muskhelishvili Institute of Computational Mathematics, Georgian Technical University, 0186 Tbilisi, Georgia)

  • Tinatin Davitashvili

    (Faculty of Exact and Natural Sciences, Iv. Javakhishvili Tbilisi State University, 0179 Tbilisi, Georgia)

  • Jose Maria Sanchez

    (Deparment of Didactic of Mathematics, Faculty of Education, Malaga University, 29071 Malaga, Spain)

  • Francisco Criado-Aldeanueva

    (Department of Applied Physics, II, Polytechnic School, Malaga University, 29071 Malaga, Spain)

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’s 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.

Suggested Citation

  • Mamuli Zakradze & Zaza Tabagari & Nana Koblishvili & Tinatin Davitashvili & Jose Maria Sanchez & Francisco Criado-Aldeanueva, 2023. "The Numerical Solution of the External Dirichlet Generalized Harmonic Problem for a Sphere by the Method of Probabilistic Solution," Mathematics, MDPI, vol. 11(3), pages 1-12, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:539-:d:1041256
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