Using high-order non-uniform grid scheme for numerical analysis of singularly perturbed Fokker-Planck equation

Mohamed Adel Bouatta, Sergey Vasilyev, Shakhmurad Kanzitdinov, Evgenii Mukaseev

Abstract


The problems of stochastic analysis have a significant place in various fields of science and technology at the present time. If we solve such problems, we must take into account fluctuation effects. The causes of these fluctuations are different in these problems. The problems, which may relate to the study such such as turbulence of gaseous and liquid substances, thermal noise in materials, noise immunity in telecommunication networks, but the methods of their theoretical research are very similar. There are mathematical methods using the theory of Brownian motion, the theory of diffusion-type processes and the theory of Markov random processes, which makes it possible to solve complex problems of these types at the present time. The aim of this work is numerical analysis of solutions of the singularly perturbed Fokker-Planck equation on a high-order non-uniform grid scheme of various problems with a small parameter. These problems can be solved on the basis of the generalized theory of Brownian motion. Numerical examples demonstrated that applied numerical scheme can be used to analyze processes in queuing theory for 5G/6G network modeling, statistical radiophysics, plasma physics, solid state theory, magnetohydrodynamics, etc.

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