An ε-uniform convergent method for a general boundary-value problem for singularly perturbed differential-difference equations: Small shifts of mixed type with layer behavior

Abstract

In this paper, we continue the study of singularly perturbed differential-difference equations with small shifts, which is motivated by the problem of determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in the dendrites [1]. We consider a more general boundary-value problem which contains both convection and reaction terms with both type of shifts (negative as well as positive) than the problem discussed in paper [2,7]. We consider the case when the solution of such type of boundary-value problem exhibits boundary layer behavior.

An ε-uniform convergent scheme based on fitting operator is derived for boundary value problems for singularly perturbed differential-difference equations with small shifts. We introduce an exponential fitting parameter to the standard finite difference scheme which reflects the singularly perturbed nature of differential operator. The method is analyzed for convergence. Several numerical experiments are carried in support of theoretical results and to show the effect of small shifts on the boundary layer solution.