One numerical method of boundary problem's ‘solution without saturation’

The numerical method of 1D boundary problem's solution for selfadjoint two-order differential operator is considered. The method presented is similar famous K. I. Babenko's algorithm and based on boundary problem's reduction to some integral equation.One's digitization is selected to aim nonsaturated algorithm for smooth functions with bounded highest derivative. The algorithm's convergence and nonsaturation for functions from classes $W^r_\infty(M;[a,b])$ are proved.