Investigation of Approximate Solution of Mathematical Model of Singular Perturbation Problem of Including Second Order Linear Equation with Variable Coefficients and Dirichlet Boundary Conditions

Sarakhsi, Ali Reza; Jahanshahi, Mohammad; Sarakhsi, Mojtaba

Investigation of Approximate Solution of Mathematical Model of Singular Perturbation Problem of Including Second Order Linear Equation with Variable Coefficients and Dirichlet Boundary Conditions

Authors

Ali Reza Sarakhsi ¹

Mohammad Jahanshahi ¹

Mojtaba Sarakhsi ²

¹ Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

² Department of Mathematics, Urmia University, Urmia, Iran

Abstract

The main subject of this paper is to consider the solution of singular perturbation problems which these problems appear in physical and engineering problems, such as fluid mechanics, chemical reactions, electronic circuitry, civil and fluid dynamics. In fact, a singular perturbation problem is in the form of either ordinary differential equations (O.D.E) or partial differential equations (P.D.E) in which the highest derivative is multiplied by some powers of as a positive small parameter. The purpose of the theory of singular perturbations is to solve a differential equation with some initial or boundary conditions with a small parameter. In this paper, the structure of solutions of these problems for second order ordinary differential equations with variable coefficients is considered. The next goal of this paper is to verify formation and non-formation of boundary layers in boundary points. Finally, the asymptotic expansions and uniform approximate solutions are obtained in five steps by using asymptotic matching condition.

Keywords

Boundary Layer Problem

Asymptotic Expansion of Solution

Matching Conditions