Journal of Optimization Theory and Applications
The height of a binary search tree: the limiting distribution perspective
Theoretical Computer Science
Observability of Singularly Perturbed Linear Time-Dependent Differential Systems with Small Delay
Journal of Dynamical and Control Systems
Uniform numerical method for singularly perturbed delay differential equations
Computers & Mathematics with Applications
Journal of Computational Methods in Sciences and Engineering
Neural, Parallel & Scientific Computations
Neural, Parallel & Scientific Computations
Neural, Parallel & Scientific Computations - Special issue on computational techniques for differential equations will applications
Computers & Mathematics with Applications
Neural, Parallel & Scientific Computations
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In this paper, an investigation is initiated of boundary-value problems for singularly perturbed linear second-order differential-difference equations with small shifts, 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 for neurons. In particular, this paper focuses on problems with solutions that exhibit layer behavior at one or both of the boundaries. The analyses of the layer equations using Laplace transforms lead to novel results. It is shown that the layer behavior can change its character and even be destroyed as the shifts increase but remain small. In the companion paper [SIAM J. Appi. Math., 54 (1994), pp. 273283], similar boundary-value problems with solutions that exhibit rapid oscillations are studied.