Corrigendum to “Richardson extrapolation technique for singularly perturbed system of parabolic partial differential equations with exponential boundary layers” [Applied Mathematics and Computation 333 (2018) 254–275]

Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuation of solutions of boundary value problems. One of these periodic orbits contains canards, trajectory segments that follow unstable portions of a slow manifold. We identify several mechanisms that lead to the formation of these and other canards in this example.

Download Full-text

Dynamics of a singularly perturbed system of two differential equations with delay

Theoretical and Mathematical Physics ◽

10.1134/s0040577921060076 ◽

2021 ◽

Vol 207 (3) ◽

pp. 770-781

Author(s):

I. S. Kashchenko ◽

E. V. Krivets

Keyword(s):

Differential Equations ◽

Singularly Perturbed ◽

Singularly Perturbed System ◽

Perturbed System ◽

Equations With Delay

Download Full-text

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

Discrete and Continuous Models and Applied Computational Science ◽

10.22363/2658-4670-2021-29-2-126-145 ◽

2021 ◽

Vol 29 (2) ◽

pp. 126-145

Author(s):

Mohamed A. Bouatta ◽

Sergey A. Vasilyev ◽

Sergey I. Vinitsky

Keyword(s):

Cauchy Problem ◽

Partial Differential Equations ◽

Differential Equations ◽

Small Parameter ◽

Asymptotic Method ◽

Planck Equation ◽

Singularly Perturbed ◽

Parabolic Partial Differential Equations ◽

Partial Differential ◽

Fokker Planck

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.

Download Full-text