1. Introduction

Singular perturbation problems are of common occurrence in many branches of applied mathematics, such as fluid dynamics, elasticity, chemical reactor theory, aerodynamics, plasma dynamics, magneto-hydrodynamics and other domains of the world of fluid motion. A few notable examples are boundary layer problems, WKB problems, the modeling of steady and unsteady viscous flow problems with large Reynolds number, convective heat transport problems with large peclet numbers etc. It is well known fact that the solutions of these problems exhibit a multi scale character. That is, there is a thin layer(s) where the solution varies rapidly (non-uniformly), while away from the layer the solution behaves regularly (uniformly) and varies slowly. Therefore, the numerical treatment for singularly perturbed boundary value problems gives major computational difficulties.

A wide variety of papers and books have been published in the recent years, describing various methods for solving singular perturbation problems, among these we mention Awoke ^[1], Bender and Orzag ^[2], Kadalbajoo and Reddy ^{[3, 4]}, Hemker and Miller ^[5], Kevorkian and Cole ^[6], Nayfeh ^[7], O’Malley ^[8], Y.N.Reddy ^[9] and Van Dyke ^[10].

In this paper the initial value method is extended for solving singularly perturbed two point boundary value problems with internal and terminal layers. It is distinguished by the following fact: The given singularly perturbed boundary value problem is replaced by two first order initial value problems. These first order problems are solved using Runge Kutta method. Model example for each is solved to demonstrate the applicability of the method. It is observed that the present method approximates the exact solution very well.

2. Initial Value Method

To describe the initial value method, we consider a class of singularly perturbed two point boundary value problem of the form:

(1)

(2a)

(2b)

where ε is a small positive parameter are given constants, and are assumed to be sufficiently continuously differentiable functions in Furthermore, we assume that throughout the interval where M is some positive constant. This assumption merely implies that the boundary layer will be in the neighborhood of We now present an initial value method as follows:

Step 1: We obtain the reduced problem of (1) by taking ε= 0.

(3)

(3a)

Let its solution be

Step 2: Integrate equation (1)

(4)

where is a constant and

(5)

Step 3: If R_{0 =} R₀(x) satisfies (5) we can approximate the solution of (1) – (2) by solving the initial value problem

(6)

(6a)

with a suitable choice of and K

Step 4: To determine one solution of (5) we note that satisfies (3). If we require =0 we will be able to find by solving the initial value problem

(7)

(7a)

Step 5: To find an approximate value for we impose the condition that the reduced equation of (4) satisfy the boundary condition

Remark: We have taken and found . In fact for any finite value of , will change accordingly and there will be no effect to our problem.

Step 6: Thus the two Initial value problems are (7), (6) as given below:

with = 0 and

with

Thus it is possible to approximate the solution of the given two-point boundary value problem by solving two initial value problems. It is interesting to note that the approximate solution improves as the small parameter tends to zero.

3. Right End Boundary Layer

We now describe this new initial value method for the singularly perturbed two point boundary value problems with the right end boundary layer of the underlying interval. To be specific we consider a class of linear singularly perturbed two point boundary value problems of the form

(8)

(9a)

(9b)

where is a small positive parameter are given constants are assumed to be sufficiently continuously differentiable functions in . Furthermore, we assume that throughout the interval where is some negative constant. This assumption merely implies that the boundary layer will be in the neighborhood of We have the following steps

Step 1: The reduced problem in this case would be

(10)

(10a)

Let be the solution of the reduced problem. A similar approach as in case of left end boundary layer problems leads to the following:

Step 2: The two initial value problems are given by

(11)

(11a)

and

(12)

(12a)

To find an approximate value for , we impose the condition that the reduced equation of (12) satisfy the boundary condition . This gives us

There exist several efficient methods for solving these initial value problems. For detailed discussion and numerical examples refer Awoke ^[1].

4. Internal Layer Problems

We will now extend the new initial value method to singular perturbation problems with an internal layer of the underlying interval. In this case changes sign in the domain of interest. Without loss of generality we can take =0 and the interval to be [-1,1].

To describe the method we shall consider a class of linear singularly perturbed two point boundary value problems of the form:

(13)

(14a)

(14b)

where is a small positive parameter are given constants are assumed to be sufficiently continuously differentiable functions in . Furthermore, we assume that throughout the interval where is some negative constant and in the interval [0,1] where M is some positive constant. This assumption merely implies that the boundary layer will be in the neighborhood of . We now proceed as follows:

Step 1: We first find the approximate solution at Without loss of generality we can take . At equation (13) becomes

(15)

The reduced problem of (15) gives us an approximation to .

(16)

Step 2: We now divide the interval [-1, 1] into two subintervals [-1, 0] and [0, 1] so that equation (13) has a right layer in [-1, 0] and a left layer in [0, 1].

Step 3: We now use our new initial value method for right end boundary layer as described in section (3), in the interval [-1,0]. The two initial value problems related to (13) are

(17)

and

(18)

To find the value of in (18) we impose the condition that the reduced problem of (18) satisfy the boundary condition at This yields

Step 4: We next use our new initial value method for left end boundary layer as described in section (2) in the interval [0,1]. The two initial value problems related to (13) are

(19)

(20)

To approximate in (20), we impose the condition that the reduced problem of (20) satisfy the boundary condition at i.e.

Thus in a manner of speaking we have replaced the original second order problem (13-14) with two equivalent first order problems in each of the two subintervals. We solve these initial value problems to obtain solutions over the interval [-1,0] and [0,1] respectively. There now exists a number of efficient methods for the solution of these initial value problems. We use classical Runge Kutta method for our problem.

5. Numerical Example

To demonstrate the applicability of the method we solve one problem

Example 5.1: Consider the following SPP

For this example we have and . Further we have an internal layer of width at (for details, see O’Malley [^[8], pp 68-172, eq8.1case (i)] and Kevorkian and Cole [^[6], pp 41-43, eqs (2.3.76) and (2.3.77)]

Step 1:

Step 2: In the interval [-1,0] we have a right layer. The problem is

with y(-1)=1 and y(0)=0. The solution of the reduced problem

with is and

The two initial value problems are

and with

Step 3: In the interval [0,1] we have a left layer. The problem is

with and

The solution of the reduced problem is and

The two initial value problems are

and with

We solve the above equations using Runge Kutta method. The numerical results are presented in Table 1(a) and 1(b) for respectively.

Table 1. (a): Computational results for Example 5.1 with ε=10^-2 and h=0.01

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Table 1. (b): Computational results for Example 5.1 with ε=10^-3 and h=0.01

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6. Two Boundary Layers Problems

The suggestions given for internal layer problems can be extended mutatis mutandis to problems with two boundary layers. To describe the method we shallconsider a class of linear singularly perturbed two point boundary value problems of the form

(21)

(22a)

(22b)

here is a small positive parameter are given constants are assumed to be sufficiently continuously differentiable functions in . Furthermore, we assume that throughout the interval and in [0,1] where is some negative constant. This assumption merely implies that the boundary layer will be in the neighborhood of and 1. Without loss of generality at since it changes sign in the domain of interest.

Step 1: We first find the approximate solution at without loss of approximate we can take

At equation (21) becomes

(23)

The reduced problem of (23) gives us an approximation to .

(24)

Step 2: We now divide the interval [-1, 1] into two subintervals [-1, 0] and [0, 1] so that equation (21) has a left layer in [-1, 0] and a right layer in [0, 1].

Step 3: We now use our new initial value method for left end boundary layer as described in section (2) in the interval [-1,0]. The two initial value problems related to (21) are

(25)

with and

(26)

With

To find the value of in (26) we impose the condition that the reduced problem of (26) satisfy the boundary condition at i.e.

Step 4 ; We next use our new initial value method for right end boundary layer as described in section (3) in the interval [0,1]. The two initial value problems related to (21) are :

(27)

with and

(28)

with

We obtain

Thus in a manner of speaking we have replaced the original second order problem (21-22) with two equivalent first order problems in each of the two subintervals. We solve these initial value problems to obtain solutions over the intervals [-1,0] and [0,1] respectively. There now exist a number of efficient methods for the solution of initial value problems. We use classical Runge Kutta method for our problem.

7. Numerical Example

To demonstrate the applicability of the method we solve one problem

Example 7.1: Consider the following SPP

For this example we have and Further we have two boundary layers one at and one at (for details, see O’Malley [^[8], pp168-173, eq 8.1 case (i)]

Step 1:

Step 2: In the interval [-1,0] we have a left layer. The problem is

with

The solution of the reduced problem

with is and .

The two initial value problems are:

with and

with

Step 3: In the interval [0,1] we have a right layer the problem is

with and

The solution of the reduced problem is with

The two initial value problems are

and

We solve the above equations using Runge Kutta method. The numerical results are presented in Table 2(a) and 2(b) for respectively.

Table 2. (a): Computational results for Example 7.1 with ε=10^-2 and h=0.01

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8. Discussion and Conclusions

We have extended the new initial value method for solving general singularly perturbed two-point boundary value problems with internal layer and two layers. In general the numerical solution of a boundary value problem will be more difficult than the numerical solution of the corresponding initial value problem. Hence we prefer to convert the given second order problem into first order problems. The solution of given singularly perturbed two-point boundary value problem is computed numerically by solving two initial value problems. We used classical fourth order Runge Kutta method. Numerical results are presented in tables. It is observed that the present method approximates the exact solution very well.

Table 2. (b): Computational results for Example 7.1 with ε=10^-3 and h=0.01

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References

[1]	Awoke A.T. Numerical Treatment of a class of singular perturbation problems – Ph. D Thesis, NITW, India, April 2008.
	In article
[2]	Bender C. M. and Orszag S.A. Advanced Mathematical Methods for Scientists and Engineers, New York: McGraw-Hill, 1978.
	In article
[3]	Kadalbajoo M.K. and Reddy Y.N. An Initial Value Technique for a class of Non-linear Singular Perturbation Problems, Journal of Optimization Theory and Applications. - 1987. - Vol. 53. 395-406.
	In article
[4]	Kadalbajoo M.K. and Reddy Y.N. Numerical solution of singular perturbation problems by Terminal Boundary Value Technique, Journal of Optimization Theory and Applications. - 1987. - Vol. 52. - pp. 243-254.
	In article
[5]	P.W. Hemker, and J.J.H Miller, (Editors), Numerical Analysis of Singular Perturbation Problems, Academic Press, New York, 1979.
	In article
[6]	J. Kevorkian, and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981.
	In article		CrossRef
[7]	A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1973.
	In article
[8]	R.E. O’Malley, Introduction to Singular Perturbations, Academic Press, New York, 1974.
	In article		PubMed
[9]	Y.N. Reddy, Numerical Treatment of Singularly Perturbed Two Point Boundary Value Problems, Ph.D. thesis, IIT, Kanpur, India, 1986.
	In article
[10]	M. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964.
	In article		PubMed