Bounds on the truncation error by finite differences for the Goursat problem

A finite difference analogue is formulated for the boundary value problem, a finite analogue of Riemann's function is developed. It is shown that the truncation error is bounded explicitly in terms of the data of the problem, which is of 0(h squared), where h is the mesh size. Bounds of the same order are obtained for the error in approximating the derivatives. Similar results are derived for the nonlinear cas if f satisfies certain continuity and differentiability conditions.