Abstract
A mesh free stochastic algorithm for solving transient diffusion–convection–reaction problems on domains with complicated structure is suggested. For the solutions of this kind of equations exact representations of the survival probabilities, the probability densities of the first passage time and position on a sphere are obtained. Based on these representations we construct a stochastic algorithm which is simple in implementaion for solving one- and three-dimensional diffusion–convection–reaction equations. The method is continuous both in space and time, and its advantages are particularly well manifested in solving problems on complicated domains, calculating fluxes to parts of the boundary, and other integral functionals, for instance, the total concentration of the particles which have been reacted to a time instant t.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Devroye, Am. J. Math. Manage. Sci. 1 (4) 359–379 (1981).
S. M. Ermakov, V. V. Nekrutkin and A. S. Sipin, Random Processes for Classical Equations of Mathematical Physics (Kluwer Academic, Dordrecht, 1989).
A. Friedman, Partial Differential Equations of Parabolic Type (Dover, Mineola, NY, 2008).
A. Haji-Sheikh and E. M. Sparrow, SIAM J. Appl. Math. 14 (2), 370–379 (1966).
K. Ito and P. Mckean, Diffusion Processes and Their Sample Paths (Springer, Berlin, 1965).
P. Kloeden, E. Platen, and H. Schurz, Numerical Solution of Stochastic Differential Equations (Springer, Heidelberg, 2012).
M. E. Muller, Ann. Math. Stat. 27 (3), 569–589 (1956).
K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems (Springer, Berlin, 1991).
K. K. Sabelfeld and N. A. Simonov, Stochastic Methods for Boundary Value Problems: Numerics for High-Dimensional PDEs and Applications (De Gruyter, Berlin, 2016).
K. K. Sabelfeld, Stat. Probab. Lett. 121, 6–11 (2017).
K. K. Sabelfeld, Monte Carlo Methods Appl. 22 (4), 265–281 (2016).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © K.K. Sabelfeld, 2018, published in Doklady Akademii Nauk, 2018, Vol. 482, No. 2.
The article was translated by the author.
Rights and permissions
About this article
Cite this article
Sabelfeld, K.K. A Mesh Free Stochastic Algorithm for Solving Diffusion–Convection–Reaction Equations on Complicated Domains. Dokl. Math. 98, 435–438 (2018). https://doi.org/10.1134/S1064562418060108
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562418060108