A class of singularly perturbed two point Boundary Value Problems (BVPs) of reaction-diffusion type for third order Ordinary Differential Equations (ODEs) with a small positive parameter $(\varepsilon)$ multiplying the highest derivative and a discontinuous source term is considered. The BVP is reduced to a weakly coupled system consisting of one first order ordinary differential equation with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested. First, in this method, we find the zero order asymptotic expansion approximation of the solution of the weakly coupled system. Then, the system is decoupled by replacing the first component of the solution by its zero order asymptotic expansion approximation of the solution in the second equation. After that the second equation is solved by a finite difference method on Shishkin mesh (a fitted mesh method). Examples are provided to illustrate the method.