A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts

Abstract

A boundary-value problem for singularly perturbed second order differential-difference equation with both the negative(delay) and positive(advance) shifts is considered and a new exponentially fitted three term finite difference scheme is proposed for the numerical approximation of its solution. An approximate version of the considered problem is constructed by the use of Taylor’s series expansion procedure first, and then, a new three term recurrence relationship is derived by applying the finite difference approximation techniques on it. Furthermore, an exponential fitting factor is introduced in the derived new scheme using the theory of singular perturbations and an efficient ’discrete invariant imbedding algorithm’ is used to solve the resulting tridiagonal system of equations. Convergence of the method is analyzed. Numerical experiments performed on several test examples; presentation of the computational results in terms of maximum absolute errors and the comparison of the results with some other method results, show the applicability and efficiency of the proposed scheme. Graphs plotted for the solution with varying shifts show the effect of small shifts on the boundary layer behavior of the solution. Both the theoretical and numerical analysis of the method reveals that the method is able to produce uniformly convergent solutions with quadratic convergence rate.