Algorithmic implementation of an exact three-point difference scheme for a certain class of singular Sturm–Liouville problems
In this article, we present a new algorithmic implementation of exact three-point difference schemes for a certain class of singular Sturm–Liouville problems. We demonstrate that computing the coefficients of the exact scheme at any grid node $x_j$ requires solving two auxiliary Cauchy problems for the second-order linear ordinary differential equations: one problem on the interval $[x_{j-1},x_{j}]$ (forward) and one problem on the interval $[x_{j},x_{j+1}]$ (backward).