# The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation

**Authors:**Bouatta M.A.^{1}, Vasilyev S.A.^{1}, Vinitsky S.I.^{1}^{,2}-
**Affiliations:**- Peoples’ Friendship University of Russia (RUDN University)
- Joint Institute for Nuclear Research

**Issue:**Vol 29, No 2 (2021)**Pages:**126-145**Section:**Articles**URL:**https://journals.rudn.ru/miph/article/view/26868**DOI:**https://doi.org/10.22363/2658-4670-2021-29-2-126-145

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## Full Text

## Abstract

The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.

## Full Text

1. Introduction It is well known that the differential operator, which is applied in the theory of measure, has such form:

## About the authors

### Mohamed A. Bouatta

Peoples’ Friendship University of Russia (RUDN University)
**Author for correspondence.**

Email: adelbouatta.rudn@mail.ru

ORCID iD: 0000-0002-5477-8710

PhD’s degree student of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation### Sergey A. Vasilyev

Peoples’ Friendship University of Russia (RUDN University)
Email: vasilyev_sa@rudn.ru

ORCID iD: 0000-0003-1562-0256

Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation### Sergey I. Vinitsky

Peoples’ Friendship University of Russia (RUDN University); Joint Institute for Nuclear Research
Email: vinitsky@theor.jinr.ru

ORCID iD: 0000-0003-3078-0047

Leading researcher of Bogolyubov Laboratory of Theoretical Physics of Joint Institute for Nuclear Research, Professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation## References

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