Computational results of differential difference equations with mixed shifts having layer structure using cubic non-polynomial spline

Abstract

A computational scheme using non-polynomial spline is suggested for solving differential - difference equations having delay and advance terms, solutions with layer structure at the left-end of the interval. First, the small shifts are tackled with Taylor's expansion and accordingly the problem is transformed to a second order singular perturbation problem. The domain is decomposed into inner and outer regions using a terminal boundary point and the problem is treated as inner region and outer region problems. Terminal boundary condition has been determined by using the reduced problem of the singular perturbation problem. In order to solve the inner and exterior region problems, a fourth order method was suggested using cubic non-polynomial spline. The method is repeated for numerous terminal point choices, until the solution profiles do not differ greatly from iteration to iteration. To illustrates the process, numerical examples were solved for specific values of the parameters of perturbation, delay and advance. The results of the computations are tabulated and compared to exact solutions. Convergence of the scheme was also studied.