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

Abstract

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