Asymptotic solution for singular perturbation problems of difference equation (Q1177983)

Consider the singular perturbation problem of the difference equation (*) \(\varepsilon Y_{k+1}+aY_ k+bY_{k-1}=0\), \(k=1,2,\ldots,N-1\), \(Y_ 0=\alpha\), \(Y_ N=\beta\) where \(a\) and \(b\) are nonzero constants. The author gives a new method to construct an asymptotic solution of equation (*) consisting of the following steps: (i) when \(\varepsilon=0\), the equation (*) is degenerated into a lower order difference equation; then he solves the reduced problem. (ii) Put the solution of the reduced problem into the lowest order term of the equation (*). Then he demands that the solution of this equation satisfies the boundary condition which is lost. Finally, the author takes the solution of this equation as an asymptotic solution of the equation (*).