Fourth Order Compact Finite Difference Method for Solving Singularly Perturbed 1D Reaction Diffusion Equations with Dirichlet Boundary Conditions

A numerical method based on finite difference scheme with uniform mesh is presented for solving singularly perturbed two-point boundary value problems of 1D reaction-diffusion equations. First, the derivatives of the given differential equation is replaced by the finite difference approximations and then, solved by using fourth order compact finite difference method by taking uniform mesh. To demonstrate the efficiency of the method, numerical illustrations have been given. Graphs are also depicted in support of the numerical results. Both the theoretical and computational rate of convergence of the method have been examined and found to be in agreement. As it can be observed from the numerical results presented in tables and graphs, the present method approximates the exact solution very well.


INTRODUCTION
Any differential equation in which the highest order derivative is multiplied by a small positive parameter ) 1 0 (  
Moreover, in the recent times many researchers have been trying to develop and present numerical methods for solving these problems. For instance, Padmaja et al. (2012) have presented a nonstandard explicit method involving the reduction of order for solving singularly perturbed two point boundary value problems. To apply the method, the authors have approximated the original problem by a pair of initial value problems and solved the first initial value problem as outer region problem whose solution can be required in the second initial value problem which they considered it as an inner region problem and is modified using the stretching transformation. The Differential Quadrature Method (DQM) has been applied for finding the numerical solution of singularly perturbed two point boundary value problems with mixed condition (Prasad and Reddy, 2011). DQM is based on the approximation of the derivatives of the unknown functions involved in the differential equations at the mesh point of the solution domain and is an efficient discretization technique in solving boundary value problems using a considerably small number of grid points. Geng (2011) has proposed the reproducing kernel method (RKM) for solving a class of singularly perturbed boundary value problems by transforming the original problem in to a new boundary value problem whose solution does not change rapidly. RKM has the advantage that it can produce smooth approximate solutions, but it is difficult to apply the method for singularly perturbed boundary value problems without transforming using appropriate transformation. However, most of the existing classical finite difference methods which have been used in solving singular perturbation problems give good result only when the mesh size is much less than the perturbation parameter which is very costly and time consuming.
In this paper, fourth order compact finite difference method is presented for solving singularly perturbed 1D reaction-diffusion equations. Compact finite difference method is a finite difference method which employs a linear combination of three consecutive points of derivatives to approximate a linear combination of the same three consecutive values of a function Fasika, W., Gemechis, F and Tesfaye, A (MEJS)
From Eq. (13) we get three-term recurrence relation of the form: Gemechis, F and Tesfaye, A (MEJS) Volume 8(2) The conditions for the discrete invariant imbedding algorithm to be stable are, (see Angel and Bellman, 1972;Elsgolt's and Norkin, 1973): For our method, one can easily show that Eq. (14) satisfies the conditions given above and hence Thomas Algorithm is stable for the proposed method.

CONVERGENCE ANALYSIS
Writing the tri-diagonal system in Eq. (14) above in matrix vector form, we get: is a column vector with , 12 12 1 We also have

NUMERICAL EXAMPLES
To demonstrate the applicability of the methods, two model singularly perturbed problems have been considered. These examples have been chosen because they have been widely discussed in the literature and their exact solutions were available for comparison.
Example 1: Consider the following singularly perturbed problem: The exact solution is given by: The numerical solutions in terms of maximum absolute errors are given in Tables 1.
Example 2: Consider the following singularly perturbed problem: The exact solution for this example is given by:     (Rashidinia et al., 2007;Kadalbajoo and Bawa, 1996;Surla et al., 1991). Further, the numerical solutions obtained by the proposed method for   h , for which most of the existing numerical methods fail to give good results, have been presented in graphs (Figures 1(a & b) and 2(a & b). As it can be observed from the tables and graphs, the present method approximates the exact solution very well and gives better solution than some existing methods reported in the literature.
Both the theoretical and numerical order of convergence have been investigated (Section 3; and Tables 3 & 4) and the results obtained confirmed that computational rate of convergence is in agreement with the theoretical estimates of the order convergence. In concise manner, the present method is conceptually simple, easy to use and readily adaptable for computer implementation for solving singularly perturbed reaction-diffusion equation.

CONCLUSION
Fourth order compact finite difference method has been presented for solving singularly perturbed reaction-diffusion equations. The method approximates the exact solution very well and gives better result than some existing methods reported in the literature. The rate of convergence of this method has been computed and is observed that it is in agreement with the theoretical estimates of the method which is of fourth order convergent.

ACKNOWLEDGMENTS
The authors would like to thank Jimma University for the financial and materials support as the work is the part of the MSc Thesis of Mr. Fasika Wondimu which is supported by the university.