Variable mesh non polynomial spline method for singular perturbation problems exhibiting twin layers

In this paper, we descend a variable mesh finite difference scheme based on non polynomial spline approximation for the solution of singular perturbation problems with twin boundary layers. We develop the discretization equation for the problem using the condition of continuity for the first order derivatives of the variable mesh non polynomial spline at the interior nodes. The discrete invariant imbedding algorithm is utilized to solve the tridiagonal system obtained by the method. Endeavor examples are illustrated and maximum absolute errors in comparison to the other methods in the literature are shown to vindicate the method.


Introduction
We consider a second order linear singularly perturbed two -point boundary value problem of the form: x q x y x p x y + = ′ ′ ε (1) with boundary conditions 2 1 1 0 γ ) , y( γ ) y( = = (2) where 1 γ , 2 γ are given constants, ε is a small positive parameter such that 1 0 << < ε and p(x), q(x) are bounded continuous functions. It is known that the above problem exhibits boundary layers at both ends of the interval depending upon the properties of p(x). These problems arise in many areas of engineering and applied mathematics. Examples of these are heat transport problem with Peclet numbers and Navier Stokes flows with large Reynold number. Because of the presence of boundary layers, difficulties are experienced in solving these types of problems using numerical methods with uniform mesh. In order to get a good approximation, a fine mesh is required in the boundary layer region. In this paper we derived a variable mesh finite difference method based on non polynomial spline approximation that gives third order approximation to the solution of (1) - (2).
A wide variety of splines are described in the text books [1,2]. Several numerical methods have been developed based on splines for the numerical solution of singular perturbed boundary value problems, in particular to the problems having boundary layers at one or both the ends of the interval. Kadalbajoo and Rajesh Bava [3,4] used variable mesh Splines of the third and second order convergence methods for singularly perturbed boundary value problems. Tariq Aziz, Arshad Khan [5] used uniform mesh spline methods and Khan et. Al. [6] used variable mesh approximation method using cubic spline in tension for solving such type of problems. Also the application of splines for the numerical solution of singularly perturbed boundary value problems has been described in many papers [7,8,9,10,11,12,13,14,15]. In this present paper, we have derived a uniformly convergent variable mesh finite difference scheme using non polynomial spline for the solution of above problem. The main idea is to use the condition of continuity of the first order derivatives of the variable mesh non polynomial spline at the interior nodes as a discretization equation for the problem. The advantage of our method is higher accuracy with the same computational effort and easy to implement in computer. The paper is organized as follows: In section 2, we define the non polynomial spline method. In section 3, we describe the numerical method for solving singular perturbed singular two -point boundary value problem, in Section 4, the truncation error and classification of various orders of the proposed method are given. In section 5 we discuss convergence analysis of the method. Finally, numerical results and comparison with other methods are given in final section.  Using the continuity of the first derivative at
[ ] Note that, the nonpolynomial spline relation (4) is consistent with the usual polynomial cubic spline if

Description of the Numerical method
At the grid points i x , the differential equation (1) may be discretized by ) Substituting the above equations in equation (4), we get the following tridiagonal finite difference scheme: We solve the tridiagonal system (6) This is well known fourth order Numerov method for the regular problem

Truncation error
The local truncation error are associated vectors of equation (7).
Hence the method (6)  The maximum absolute errors are presented in Table 1 and Table 2.

Example 2. Consider the singular perturbation problem
The exact solution is given by The maximum absolute errors are presented in Table 3 7

. Conclusion
We possess presented a variable mesh non-polynomial spline method for singular perturbation problems exhibiting twin layers. We have implemented the present method on standard test problems because they have been widely discussed in literature and exact solutions are available for comparison. We have presented maximum absolute errors and compared the results with other methods to support the method. The convergence analysis of the proposed method has been discussed. It is observed from the results that the present method approximate the accurate solution really advantageously for smaller value of ε also.