American Journal of Numerical Analysis, 2015, Vol. 3, No. 2, 39-48 Available online at http://pubs.sciepub.com/ajna/3/2/2 © Science and Education Publishing DOI:10.12691/ajna-3-2-2 A Uniformly Convergent Scheme for A System of Two Coupled Singularly Perturbed Reaction-Diffusion Robin Type Boundary Value Problems with Discontinuous Source Term Pathan Mahabub Basha * , Vembu Shanthi Department of Mathematics, National Institute of Technology, Tiruchirappalli, India *Corresponding author: pmbasha9@gmail.com Received January 1, 2015, Revised February 25, 2015; Accepted September 08, 2015 Abstract In this paper, a uniformly convergent scheme for a system of two coupled singularly perturbed reaction- diffusion Robin type mixed boundary value problems (MBVPs) with discontinuous source term is presented. A fitted mesh method has been used to obtain the difference scheme for the system of MBVPs on a piecewise uniform Shishkin mesh. A cubic spline scheme is used for Robin boundary conditions and the classical central difference scheme is used for the differential equations at the interior points. An error analysis is carried out and numerical results are provided to show that the method is uniformly convergent with respect to the singular perturbation parameter which supports the theoretical results. Keywords: singular perturbation problem, weakly coupled system, discontinuous source term, Robin boundary conditions, Shishkin mesh, fitted mesh method, uniform convergence Cite This Article: Pathan Mahabub Basha, and Vembu Shanthi, “A Uniformly Convergent Scheme for A System of Two Coupled Singularly Perturbed Reaction-Diffusion Robin Type Boundary Value Problems with Discontinuous Source Term.” American Journal of Numerical Analysis, vol. 3, no. 2 (2015): 39-48. doi: 10.12691/ajna-3-2-2. 1. Introduction Singular perturbation problems (SPPs) arise in various fields of science and engineering which include fluid mechanics, fluid dynamics, quantum mechanics, control theory, semiconductor device modeling, chemical reactor theory, elasticity, hydrodynamics, gas porous electrodes theory, etc. SPPs are characterized by the presence of a small parameter (0 1) ε <  that multiplies the highest derivative term. This leads to boundary and/or interior layers in the solution of such problems. A much attention has been drawn on these problems to obtain good approximate solutions for the past few decades. Since classical numerical methods fail to produce good approximations for these equations, it is inevitable to go for non-classical methods. There are several articles available at the literature but they are mainly based on singularly perturbed problems containing one equation. Some authors have developed robust numerical methods for a system of singularly perturbed convection-reaction- diffusion problems on smooth data. Very few researchers can be seen for problems with non-smooth data which frequently arises in electro analytic chemistry, predator- prey population dynamics, etc. as a perfect application. Oseen equations form a convection-diffusion system where as linearized Navier-Stokes equations yield a reaction-diffusion system at large Reynolds number. For a parameter-uniform methods pertaining to singular perturbation problems, one can refer the books [1,2,3]. A standard finite difference method is proved uniformly convergent on a fitted piece wise uniform Shishkin mesh for a single equation reaction-diffusion problem [2]. The same approach for coupled system of two singularly perturbed reaction-diffusion problems, with diffusion coefficients 1 2 , ε ε was originally proposed by Shishkin [4] and identified three different cases () ( ) ( ) 1 2 1 2 1 2 0 1; 0 1; 0 1. i ii iii ε ε ε ε ε ε < = < = < ≤    For case- () i , Matthews et al. [5] proved almost first order convergence using classical finite difference scheme on Shishkin mesh for a system of singularly perturbed reaction-diffusion equations subject to Dirichlet boundary conditions. Tamilselvan et al. [6] developed a numerical method using fitted piecewise uniform Shishkin mesh for the coupled system of singularly perturbed reaction- diffusion equations for case- () i with discontinuous source term subject to Dirichlet boundary conditions and obtained almost first order uniform convergence. Singularly perturbed linear second order ordinary differential equations of reaction-diffusion type with discontinuous source term subject to Dirichlet boundary conditions having diffusion parameters with different