An exponentially fitted finite difference scheme for solving boundary-value problems for singularly-perturbed differential-difference equations: Small shifts of mixed type with layer behavior

摘要

In this paper, we study a numerical approach to find the solution of the boundary-value problems for singularly perturbed differential-difference equations with small shifts. Similar boundary-value problems axe associated with expected first-exit time problems of the membrane potential in models for activity of neuron [2-6] and in variational problems in control theory. Here we propose an exponentially fitted method based on finite difference to solve boundary-value problem for a singularly perturbed differential-difference equation with small shifts of mixed type, i.e., which contains both type of terms having negative shift as well as positive shift and consider the case in which the solution of the problem exhibits layer behavior. We calculate the fitting parameter for the exponentially fitted finite difference scheme corresponding to the problem and establish the error estimate which shows that the method converges to the solution of the problem. The effect of small shifts on the boundary layer solution is shown by considering the numerical experiments. The numerical results for several test examples demonstrate the efficiency of the method.