Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations. VI. Small Shifts with Rapid Oscillations

This paper continues the authors’ study of boundary-value problems for singularly perturbed linear second-order differential-difference equations with small shifts. This study was initiated in the companion paper [SIAM J. Appl. Math., 54 (1994), pp. 249–272]. In this paper, the study is extended to problems that have solutions that exhibit rapid oscillations. Restrictions on the sizes of the shifts in terms of the small parameter are found such that, generally, the shifted terms cannot be replaced with truncated Taylor series. In particular, it is shown that, even when the shifts are small relative to the width of an oscillation, they can affect the solution to leading order. The conclusion is that oscillatory solutions are more sensitive to small delays than are layer solutions. It is shown that a suitably modified version of the standard WKB method can be used to obtain leading-order oscillatory solutions of these differential-difference equations. These preliminary studies of differential-difference equations with small shifts provide techniques for treating expected first-exit time problems associated with the membrane potential of neurons for generation of action potentials.