1. Introduction

Singularly perturbed singular boundary value Problems arise in many areas of science and engineering such as heat transfer problem with large Peclet numbers, Navier-Stokes flows with large Reynolds numbers, chemical reactor theory, aerodynamics, Reaction-diffusion process, quantum mechanics, optimal control etc. The numerical treatment of singular singularly perturbed boundary value problems present some major computational difficulties due to the boundary layer behavior of the solution and the presence of singularity. It is well known fact that the solution of these problems exhibits a multi scale character, that is, there are thin transition layer(s) where the solution varies rapidly, and while away from the layers (s) the solution behaves regularly and varies slowly.

In general, the classical numerical methods fail to give reliable results for these problems because of the layer behavior and also because of singularity. Detailed theory and numerical treatment of these problems is available in the Ref. [1-13]^[1]. Rasidinia, Mohammadi and Ghasemij ^[5] presented a numerical technique for a class of singularly perturbed two point singular boundary value problems on uniform mess using Polynomial cubic splines. Li ^[6] described a computational method for solving singularly perturbed two-point singular boundary value problem in which exact solution is represented in the form of series in reproducing kernel space. Kadalbajoo and Aggarwal [17] presented a Fitted mesh B-spline method for the solution of a class of singular singularly perturbed boundary value problems. Mohanty and Jha ^[10] presented a class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems. Mohanty and Arora ^[11] proposed a family of non-uniform mesh tension spline methods for the solution of singularly perturbed two-point singular boundary value problems with significant first derivatives. Mohanty et. al. ^[12] suggested a Convergent spline in tension methods for the solution of singularly perturbed two-point singular boundary value problems. Mohanty, Jha, and Evans ^[13] presented a Spline in compression method for the numerical solution of singularly perturbed two point singular boundary value problems. For a detailed analytical and numerical discussion on singularly perturbed problems one may refer to the books and high level monographs by: Bender and Orszag ^[1], Miller et. al. ^[3], Kevorkian and Cole ^[4], Hemkar et. al. ^[8] and O'Malley ^[9].

In this paper, we have presented a fifth order compact difference method for a class of singularly perturbed singular two-point boundary value problems. To avoid the singularity at zero a terminal boundary condition in the implicit form is derived. Using this condition as one of the boundary condition we have solved the singularly perturbed singular two-point boundary value problem by the fifth order compact difference scheme. Numerical results are presented to demonstrate the applicability of the proposed method and compared with exact solution. We have also presented the least square and maximum errors for the problems considered. It is observed from the tables that the present method approximates the exact solution very well.

This paper is organized as follows: Section 2 presents the way of finding terminal boundary condition in the implicit form and the description of the fifth order compact difference scheme. Numerical experiments are performed by considering four standard example problems and presented the computational results in the section 3, show the accuracy and efficiency of the method. In the section 4, based on the numerical experiments performed, and conclusions are presented.

2. Description of the Method

Consider singularly perturbed singular boundary value problems of the form:

(1)

with boundary conditions

(2a)

(2b)

where , q(x), r(x) are bounded continuous functions in (0, 1), q(x) > 0 and are finite constants. We know that, if a function is analytic at a point then the point is said to be an ordinary point. The point is a singular point if the functions fail to be analytic at Such problems are called singularly perturbed singular boundary value problems.

To avoid the singular point ‘0’, we introduce , a small positive deviating argument, where .

Using Taylor series expansion in the neighbourhood of the point , we have

(3)

Substituting in (1), we get

(4)

where

At, Eq. (4) becomes

We use this equation as the terminal boundary condition.

Then the considered boundary value problem (BVP) (1) with (2a) and (2b) over [, 1] is given by

(5)

with boundary conditions

(6)

and

(7)

Now we solve this boundary value problem by the fifth order compact difference scheme described below.: For this we consider the first order linear system corresponding to the above BVP as:

(8)

with the boundary conditions

where and are 2 x 2 matrices and are two dimensional vectors.

Now we divide the interval into equal parts with constant mesh length Let be the mesh points. Again we divide each subinterval into four equal smaller sub intervals. Let are the grids in the subinterval and corresponding values of the variables and its derivatives are .

By considering Taylor’s expansions of at the fractional grid t₃ (Peng ^[2]), we have

(9)

where and the coefficients are given by:

By taking the Taylor’s series expansions of at the grid point and substituting (9), we get

(10)

where

The variable and its derivative at grids subject to equations

(11)

where and are values of A and R at grids .

Substituting (11) in (10), we get six linear algebraic equations with respect to five unknown variables .

By eliminating from the above equations a relation between can be obtained as follows:

(12)

where is a two dimensional vector. The relation (12) is a fifth order compact difference scheme of Eq. (8) in the th subinterval. By assuming

We get

Now the system (12) can be written in matrix form as:

Solving the above system together with the given boundary conditions (6) and (7), we will get the solution.

In the boundary condition (6), we replace the by the following fifth order approximation which is obtained by the expansion

(13)

We calculate the required derivatives from the differential equation and at we write Eq. (13), so that we have in terms of . Substitute this in Eq. (6) so that we have the boundary condition for .

3. Numerical Experiments

To demonstrate the applicability of fifth order compact difference method computationally, we consider four singularly perturbed two-point singular boundary value problems. These problems have been chosen because they have been widely discussed in the literature and because exact solutions are available for comparison.

Example 1. Consider the singularly perturbed singular boundary value problem

The exact solution of this problem is The numerical results are shown in Table 1 and Table 2 for and respectively.

Example 2. Consider the following singularly perturbed singular boundary value problem:

With boundary conditions The uniform solution of this problem is

The numerical results are shown in Table 3 and Table 4 for and respectively.

Example 3. Consider the following singularly perturbed singular boundary value problem where q(x) is also not continuous at x = 0

subject to boundary conditions . The exact solution of this problem is .

The numerical results are shown in Table 5 and Table 6 for and respectively.

Example 4. Consider the following singularly perturbed singular boundary value problem

with boundary conditions whose exact solution is not known. This problem has regular singularity at x = 0 and boundary layer also at x = 0. However, the condition on y (0) is so weak that the solution does not exhibit a boundary layer at x = 0 as , even though for x > 1. The numerical results are shown in Table 7 for and respectively.

Table 1. Numerical solution of example 1 with ε=0.01

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Table 2. Numerical solution of example 3 with ε=0.001

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Table 3. Numerical solution of example 2 with ε=0.01

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Table 4. Numerical solution of example 2 with ε=0.001

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Table 5. Numerical solution of example 3 with ε=0.01

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Table 6. Numerical solution of example 3 with ε=0.001

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Table 7. Numerical solution of example 4 with ε=0.01 and ε=0.001

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4. Discussions and Conclusions

We have described and demonstrated the applicability of the fifth order compact difference scheme for a class of singularly perturbed singular two-point boundary value problems. To avoid the singularity at zero a terminal boundary condition in the implicit form is derived. Using this condition as one of the boundary condition we solve the singularly perturbed singular two-point boundary value problem by the fifth order compact difference scheme. We have implemented this method on four examples and tabulated the computational results obtained by present method as well as the exact solutions. We have also presented the least square and maximum errors for the problems considered. It can be observed from the tables that the present method approximates the exact solution very well.

References

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[2]	D. Peng, High-Order numerical method for two-point boundary value problems, Journal of Computational Physics, 120, (1995), 253-259.
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[5]	J. Rashidinia, R. Mohammadi, M. Ghasemi, Cubic spline solution of singularly perturbed boundary value problems with significant first derivatives, Appl. Math. Comput. 190 (2007) 1762-1766.
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[6]	Jian Li, A computational method for solving singularly perturbed two-point singular boundary value problem, Int. Journal of Math. Analysis, 2 (2008) 1089-1096.
	In article
[7]	M.K. Kadalbajoo, V.K. Aggarwal, Fitted mesh B-spline method for solving a class of singular singularly perturbed boundary value problems, Int. J. Comput. Math. 82 (2005) 67-76.
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	In article
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	In article		PubMed
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	In article		CrossRef
[11]	R.K. Mohanty, Urvashi Arora, A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives, Appl. Math. Comput. 172 (2006) 531-544.
	In article		CrossRef
[12]	R.K. Mohanty, D.J. Evans, U. Arora, Convergent spline in tension methods for singularly perturbed two point singular boundary value problems, Int. J. Comput. Math. 82 (2005) 55-66.
	In article		CrossRef
[13]	R.K. Mohanty, N. Jha, D.J. Evans, Spline in compression method for the numerical solutionof singularly perturbed two point singular boundary value problems, Int. J. Comput. Math. 81(2004) 615-627.
	In article		CrossRef