Numerical analysis of singularly perturbed delay differential equations with layer behavior

In this paper, we present a numerical method to solve boundary-value problems for singularly perturbed differential-difference equations with negative shift. In recent papers, the term negative shift has been used for delay [SIAM J. Appl. Math. 54 (1994) 249; SIAM J. Appl. Math. 42 (1982) 502; SIAM J. Appl. Math. 45 (1985) 687; SIAM J. Appl. Math. 45 (1985) 708; SIAM J. Appl. Math. 54 (1994) 273]. Similar BVPs are associated with expected first-exit time problem of the membrane potential in models for neuron and in variational problem in control theory. The stability and convergence analysis of the method is discussed. Also the effect of small shift on the boundary layer solution in both the cases, i.e., boundary layer on the left side as well as the right side is discussed by considering numerical experiments. Several numerical examples are solved using the presented method, compared the computed result with exact solution and plotted the graphs of the solution of the problems. (C) 2003 Elsevier Inc. All rights reserved.