Error Estimates for the Difference Method to System of Ordinary Differential Equations with Boundary Layer

This work deals with the numerical solution of singular perturbation system of ordinary differential equations with boundary layer. For the numerical solution of this problem fitted finite difference scheme on a uniform mesh is constructed and analyzed. The uniform error estimates for the approximate solution are obtained.


Introduction
We consider the initial-boundary value problem for the linear system of ordinary differential equations in the interval : where  is a small parameter, 1 A , 2 A , , are given constants.The functions certain regularity conditions will be specified whenever necessarily.
The above type initial/boundary value problems arise in many areas of mechanics and physics [1,2].Differential equations with a small parameter  multiplying the highest-order derivative terms are said to be singularly perturbed.They occur frequently in mathematical problems in the sciences and engineering for example, in fluid flow at high Reynold number, electrical networks, chemical reactions, control theory, the equations governing flow in porous media, the drift-diffusion equations of semi-conductor device physics, and other physical models.The mathematical models describing these phenomena contain a small parameter  and the influence of this parameter reveal itself in a sudden change of the dependent variable u  taking place with in a small layer.That is, the solution of this type of problem has a narrow region in which the solution changes rapidly and the outside solution changes smoothly [1][2][3].
It is well-known that standard discretization methods do not work well for these problems as they often produce oscillatory solutions which are inaccurate if the perturbed parameter  is small.To obtain robust numerical methods, it is necessary to fix the coefficients (fitted operator methods) or the mesh (fitted mesh methods) to the behavior of the exact solution [2,4].
In this present paper, we analyze the numerical solution of the initial/boundary problem (1)-( 4).The numerical method presented here comprises a fitted difference scheme on a uniform mesh.Fitted operator method is widely used to construct and analyse uniform difference methods, especially for a linear differential problems (see, e.g., [4][5][6][7]).In the Section 2, we state some important properties of the exact solution.The derivations of the difference scheme and uniform convergence analysis have been given in Section 3. Uniform convergence is proved in the discrete maximum norm.The approach to the construction of the discrete problem and the error analysis for the approximate solution are similar to those in [8,9].
Throughout the paper, C will denote a generic positive constant independent of  and of the mesh parameter.

Analytical Results
Lemma 2.1.Under the the solution of the problem (1)-( 4) satisfies   Here for any continuous functions   g x .
Proof.First we prove that for the solution of initial-value problem of the type the following estimates hold Using the function prove (10) after some manipulations we have Then from ( 9)-( 12) the following estimates hold The last inequalities show the validity of ( 5)-( 8).

The Difference Scheme and Convergence
Now we construct the difference scheme and investigate it.In what follow, we denote by  the uniform mesh in Before describing our numerical method, we introduce some notation for the mesh functions.For any mesh function g(x), we use On  we propose the following difference scheme for approximating (1)-( 4): where where Throughout the paper, we assume that   For solving of the ( 13)-( 16), we giving the following iterative procedure: .
where is arbitrary.
The iteration (18)-( 21) is suitable for the solution of the problem (13)-( 16) and the solution of the difference problem (13)-( 16) satisfies we will have From ( 23)-( 26) and , it is not difficult to get and thereby .
Lemma 2.4.For the truncation errors The limit case for leads to .H i The limit functions will be solution of scheme (1 d ( 2) , lim .16).Now we prove (21) an 2 .We have Proof.After setting and the following estimates hold Proof.We may write , .
By virtue that of (29)-(32) all terms in ri t-hand side of this inequality have the rate and hence the proof follows immediately