A finite difference method for singularly perturbed differential-difference equations with layer and oscillatory behavior

In this paper, we present a finite difference method for singularly perturbed linear second order differential-difference equations of convection–diffusion type with a small shift, i.e., where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. Here, the study focuses on the effect of shift on the boundary layer behavior or oscillatory behavior of the solution via finite difference approach. An extensive amount of computational work has been carried out to demonstrate the proposed method and to show the effect of shift parameter on the boundary layer behavior and oscillatory behavior of the solution of the problem.