Uniform Difference Scheme on the Singularly Perturbed System

This paper is concerned with the numerical solution for singular perturbation system of two coupled second ordinary differential equations with initial and boundary conditions, respectively. Fitted finite difference scheme on a uniform mesh, whose solution converges pointwise independently of the singular perturbation parameter is constructed and analyzed.


Introduction
We consider the following singularly perturbed initial/ boundary value problem for the linear system of ordinary differential equations in the interval   0,1 :             where 0   is a small parameter , 0, are given functions satisfying certain regularity conditions which are specified whenever necessary.
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 and normally boundary layers occur in their solutions.The numerical analysis of singular perturbation cases has always been far from trivial because of the boundary layer behavior of the solution.Such problems undergo rapid changes within very thin layers near the boundary or inside the problem domain.It is well known that standard numerical methods for solving such problems are unstable and fail to give accurate results when the perturbation parameter  is small.Therefore, it is important to develop suitable numerical methods to these problems, whose accuracy does not depend on the parameter value  , i.e. methods that are  -uniformly convergent.These include fitted finite difference methods, finite element methods using special elements such as exponential elements, and methods which use a priori refined or special non-uniform grids which condense in the boundary layers in a special manner.The various approaches to the design and analysis of appropriate numerical methods for singularly perturbed differential equations can be found in [3][4][5][6][7][8] (see also references cited in them).
In this present paper, we analyze the numerical solution of the initial/boundary problems (1)-( 4).The numerical method presented here comprises a fitted difference scheme on an uniform mesh.Fitted operator method is widly 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 derivation 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
Here we give useful asymptotic estimates of the exact solution of (1.1)- (1.4), that are needed in later sections.Lemma 2.1 Under the the problem (1.1)-(1.4)has a unique solution, which satisfies where for any continuous function Proof.Consider the iterative process (10) where is an arbitrary function.
First we prove that for the solution of initial-value problem of the type the following estimates hold To prove (2.7), after some manipulations we have From here by virtue of integral inequality it follows that which leads to (2.7).Now we prove (2.8).Clearly Then by using (2.7) we get Further, note that, by virtue of maximum principle the problem of the form Denoting Next, applying here (2.7), (2.8), (2.9) we arrive at with .

The Difference Scheme and Convergence
Now we construct the difference scheme and investigate it.In what follows, we denote by  the uniform mesh in   0,1 : . Before describing our n erical method, we introduce some notation for the mesh func-um tions.For any mesh function   g x , we use , .
In similar manner we also obtain . , .
The limit case for leads to (3.5).The inequality (3.6) is being proved analoguosly.

Lemma 3.2 The solution of the difference problem
where and with conditions (3.3) and (3.4) appropriately, which is being obtained analoguosly as in differential case, after setting The using each of these into another immediately leads .11)and (3.12).
For the truncation errors 1 , Proof.We may write where  are the solutions of following pr s oblem respectively: The relations (3.18)-(3.20),by using also the above

Numerical Exampl
Consider the particular problem with The i ss is chosen as and stopping criterion is  ental rates of convergence We calculate an experim  

Conclusion
The singularly perturbed initial-boundary value problem for a linear second order differential system is considered.
To solve this problem, an exponentially fitted difference scheme on a uniform mesh is presented.First order conve numerical example experimental rates of convergence in rgence in the discrete maximum norm, independently of the perturbation parameter is obtained.Obtained in