scholarly journals Nonstationary Fronts in the Singularly Perturbed Power-Society Model

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
M. G. Dmitriev ◽  
A. A. Pavlov ◽  
A. P. Petrov
Keyword(s):  
Steady State ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Singularly Perturbed ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Boundary Problems ◽  
Applied Model ◽  
Parametric Control ◽  
Nonlinear Parabolic

The theory of contrasting structures in singularly perturbed boundary problems for nonlinear parabolic partial differential equations is applied to the research of formation of steady state distributions of power within the nonlinear “power-society” model. The interpretations of the solutions to the equation are presented in terms of applied model. The possibility theorem for the problem of getting the solution having some preassigned properties by means of parametric control is proved.

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

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.

scholarly journals Nonlinear parabolic problems in unbounded domains

1979 ◽  
Vol 82 (3-4) ◽  
pp. 201-209 ◽  
Author(s):  
Guy Mahler
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Unbounded Domains ◽  
Parabolic Problems ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Nonlinear Parabolic Problems ◽  
Existence Of Weak Solutions ◽  
Nonlinear Parabolic

We show the existence of weak solutions of nonlinear parabolic partial differential equations in unbounded domains, provided that a variant of the Leray-Lions conditions is satisfied.

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