Asymptotic method for singular perturbation problem of ordinary difference equations (Q1178000)

The authors consider second order ordinary difference equations \((L_ \epsilon y)_ k=f(k,\epsilon)\), \(1\leq k\leq N-1\), with a small parameter \(\epsilon\) multiplying the highest order term and with variable coefficients which depend smoothly upon \(\epsilon\). The boundary conditions are \(B_ 1y:=-y(0)+c_ 1y(1)=\alpha\) and \(B_ 2y:=-c_ 2y(N-1)+y(N)=\beta\). By considering a simple example and the case with constant coefficients, the authors are led, in the case with variable coefficients, to express the solution as the sum of an outer series and a boundary layer correction series, \(y(k)=y_ t(k)+\epsilon^{N-k}w(k)\), where \(y_ t\) and \(w\) are integral power series in \(\epsilon\). The outer series satisfies the first boundary condition and the correction series is used to recover the second boundary condition. They show that the problem has a unique uniformly asymptotic solution for \(0\leq k\leq N\) and illustrate the theory by a numerical example.