ABSTRACT

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.

1. Introduction

We consider the following singularly perturbed initial/ boundary value problem for the linear system of ordinary differential equations in the interval:

(1)

(2)

(3)

(4)

where is a small parameter A₁, A₂, B₁, B₂, are given constants. The functions 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 [3-8] (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., [4-7]). 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].

Difference schemes for singularly perturbed systems with another type of initial/boundary conditions was investigated in [9-14].

Throughout the paper, C will denote a generic positive constant independent of and of the mesh parameter.

2. 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

(5)

the problem (1.1)-(1.4) has a unique solution, which satisfies

(6)

(7)

(8)

(9)

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

(11)

(12)

To prove (2.7), after some manipulations we have

hence

with

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

which arrive at (2.8).

Further, note that, by virtue of maximum principle the problem of the form

admits the estimate

(13)

Denoting

from (1.1)-(1.4) and (2.6) we have

Next, applying here (2.7), (2.8), (2.9) we arrive at

Therefore

(14)

with

(15)

From (2.10) we have

(16)

(17)

(18)

From (2.12), (2.13), (2.14) follows that the sequences uniformly converges on

for. Replacing (2.6) by appropriate system of integral equations we conclude that for the limit functions are the solution of (1.1)-(1.4).

Now the using (2.7) and (2.8) with the function yield the following stability bounds

(19)

(20)

Next from (2.9), with it follows that

(21)

From (2.13) and (2.15) obviously

Using the last relation in (2.14) we obtain

The last three inequalities show the validity of (2.2)- (2.4). Now we prove (2.5). Since

which leads to (2.5), which completes the proof.

3. The Difference Scheme and Convergence

Now we construct the difference scheme and investigate it. In what follows, we denote by the uniform mesh in:

and. Before describing our numerical method, we introduce some notation for the mesh functions. For any mesh function, we use

On we propose the following difference scheme for approximating (1.1)-(1.4):

(22)

(23)

(24)

(25)

and

For solving of the (3.1)-(3.4) we give the following iterative procedure:

where is arbitrary.

Lemma 3.1

(26)

(27)

where implies the discrete maximum norm on;

and are defined by (2.1) and (2.11) appropriately.

Proof. Denoting we will have

(28)

(29)

(30)

(31)

From (3.7)-(3.10), it is not difficult to get

where

and thereby

In similar manner we also obtain

Hence,

Consequently,

From this follows that the, to, hence the sequences, are the Cauchy sequences and convergent: and . The limit functions will be solution of scheme (3.1)-(3.4).

Now we prove (3.5), (3.6). We have

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 (3.1)-(3.4) satisfies

(32)

(33)

where .

Proof. Using the estimates for the difference equations

and

with conditions (3.3) and (3.4) appropriately, which is being obtained analoguosly as in differential case, after setting and we will get

(34)

(35)

The using each of these into another immediately leads to (3.11) and (3.12).

Lemma 3.3 For the truncation errors  

the following estimates hold

(36)

(37)

(38)

Proof. We may write

(39)

(40)

(41)

where

We note that, and, are the solutions of following problems respectively:

The relations (3.18)-(3.20), by using also the above properties of and leads immediately to (3.15)-(3.17).

Theorem 3.1 Let

Then the solution of the difference problem (3.1)-(3.4) converges uniformly in to the solution of (1.1)-(1.4) with rate.

Proof. Let Then for the errors of the approximate solution we have

where are approximating errors from Lemma 3.3. Using Lemma 3.2, we obtain:

By virtue that of (3.15)-(3.17) all terms in right-hand side of this inequality have the rate and hence the proof follows immediately.

4. Numerical Example

Consider the particular problem with

The initial guess is chosen as

and stopping criterion is

We calculate an experimental rates of convergence using double mesh method as follows [4,5]:

where

The convergence is uniform, i.e., rate of convergence independenty of perturbation parameter. Some obtained values for

are listed in the table 

5. 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 convergence in the discrete maximum norm, independently of the perturbation parameter is obtained. Obtained in numerical example experimental rates of convergence in agreement with theoretical values.

6. Acknowledgements

The author is grateful to the anonymous referees for his comments and suggestions which helped improve the quality of manuscript.

REFERENCES