A parameter uniform method for singularly perturbed differential-difference equations with small shifts

Abstract

This paper is devoted to the numerical study for a class of boundary value problems of singularly perturbed linear second-order differential-difference equations with small shifts (i. e., containing both terms having a negative shift and terms having a positive shift). In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter e and the shifts depend on the small parameter have been considered. To obtain a parameter-uniform convergence, a piecewise-uniform mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The parameter-uniform convergence analysis of the method has been discussed. The method is shown to have almost second order parameter-uniform convergence. The effect of small shifts on boundary layers have also been discussed. To demonstrate the efficiency of the proposed scheme several examples having boundary layers have been carried out.