Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2686810.22363/2658-4670-2021-29-2-126-145Research ArticleThe asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equationBouattaMohamed A.<p>PhD’s degree student of Department of Applied Probability and Informatics</p>adelbouatta.rudn@mail.ruhttps://orcid.org/0000-0002-5477-8710VasilyevSergey A.<p>Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics</p>vasilyev_sa@rudn.ruhttps://orcid.org/0000-0003-1562-0256VinitskySergey I.<p>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)</p>vinitsky@theor.jinr.ruhttps://orcid.org/0000-0003-3078-0047Peoples’ Friendship University of Russia (RUDN University)Joint Institute for Nuclear Research2806202129212614528062021Copyright © 2021, Bouatta M.A., Vasilyev S.A., Vinitsky S.I.2021<p style="text-align: justify;">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.</p>asymptotic analysissingularly perturbed differential equationCauchy problemFokker-Planck equationасимптотический анализсингулярно возмущённое дифференциальное уравнениезадача Кошиуравнение Фоккера-Планка<p>1. Introduction It is well known that the differential operator, which is applied in the theory of measure, has such form:</p>[D. Daniel, W. T. Taitano, and L. Chacon, “A fully implicit, scalable, conservative nonlinear relativistic Fokker-Planck 0D-2P solver for runaway electrons,” Computer Physics Communications, vol. 254, p. 107-361, 2020. DOI: 10.1016/j.cpc.2020.107361.][Y. Ito, “Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense star clusters (i) accurate pre-collapse solution,” New Astronomy, vol. 83, p. 101-474, 2021. DOI: 10.1016/j.newast.2020.101474.][P. 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