Singularly perturbed boundary-value problems in discrete systems

Abstract

Various types of singularly perturbed boundary-value problems arising in discrete systems are formulated. For the general type it is seen that the degenerate problem is of reduced-order and hence shows its inability to accommodate all the given boundary conditions. Those left-over boundary conditions are readmitted through appropriate corrections for the initial and final boundary layers. A perturbation method is developed to obtain an approximate solution composed of an outer series, initial boundary layer correction series and final boundary layer correction series. An important feature of the method is that the corrections for the original boundary-value problem are solved as an initial-value problem and a final-value problem, facilitating a considerable reduction in computation. An example is given to demonstrate the method.