A seventh order numerical method for singular perturbed differential-difference equations with negative shift

Abstract

In this paper, a seventh order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been used for delay. Such problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, we first use Taylor approximation to tackle terms containing small shifts which converts into a singularly perturbed boundary value problem. This two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a seventh order compact difference scheme is employed for the first order system and solved by using the boundary conditions. Several numerical examples are solved and compared with exact solution. We also present least square errors, maximum errors and observed that the present method approximates the exact solution very well.

Keywords

• singular perturbation problems
• differential-difference equations
• negative shift
• boundary layer
• seventh order numerical method