Fitted Numerical Scheme for Singularly Perturbed Differential Equations Having Two Small Delays

In this paper, singularly perturbed differential equations having delay on the convection and reaction terms are considered. The highest order derivative term in the equation is multiplied by a perturbation parameter \epsilon taking arbitrary values in the interval (۰; ۱]. For small \epsilon, the problem involves a boundary layer on the left or right side of the domain depending on the sign of the coefficient of the convective term. The terms involving the delay are approximated using Taylor series approximation. The resulting singularly perturbed boundary value problem is treated using exponentially fitted upwind finite difference method. The stability of the proposed scheme is analysed and investigated using maximum principle and barrier functions for solution bound. The formulated scheme converges independent of the perturbation parameter with rate of convergence O(N−۱). Richardson extrapolation technique is applied to accelerate the rate of convergence of the scheme to order O(N−۲). To validate the theoretical finding, three model examples having boundary layer behaviour are considered. The maximum absolute error and rate of convergence of the scheme are computed. The proposed scheme gives accurate and parameter uniformly convergent result.

Keywords:

Delay differential equation , exponentially fitted method , singularly perturbed problem , uniform convergence