The numerical solution for problems of turning point without resonance (Q1074319)

The uniform convergence of Il'in's scheme for solving the turning point problem without resonance \[ \begin{cases} L[y] \equiv \varepsilon y'' + p(x)y' + g(x)y = f(x) \quad (-a<x<b) \\ y(-a) = A, \quad y(b) = B \end{cases} \] is proved by means of Kellogg's method. A relation between the error order of the numerical solution of singular perturbation problem and that of the corresponding reduced problem is obtained, and it is shown, that the given estimation is the best one.