scholarly journals Fourth-Order Schemes of Exponential Type for Singularly Perturbed Parabolic Partial Differential Equations

2006 ◽  
Vol 36 (3) ◽  
pp. 1049-1066 ◽  
Author(s):  
A.A. Salama ◽  
H.Z. Zidan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Exponential Type ◽  
Singularly Perturbed ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential

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.

The Solution of the Variable Coefficients Fourth-Order Parabolic Partial Differential Equations by the Homotopy Perturbation Method

2009 ◽  
Vol 64 (7-8) ◽  
pp. 420-430 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Perturbation Method ◽  
Fourth Order ◽  
Homotopy Perturbation Method ◽  
Variable Coefficients ◽  
Parabolic Partial Differential Equation ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Homotopy Perturbation

AbstractIn this work, the homotopy perturbation method proposed by Ji-Huan He [1] is applied to solve both linear and nonlinear boundary value problems for fourth-order partial differential equations. The numerical results obtained with minimum amount of computation are compared with the exact solution to show the efficiency of the method. The results show that the homotopy perturbation method is of high accuracy and efficient for solving the fourth-order parabolic partial differential equation with variable coefficients. The results show also that the introduced method is a powerful tool for solving the fourth-order parabolic partial differential equations.

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