exact three-point difference scheme

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 xj requires solving two auxiliary Cauchy problems for the second-order linear ordinary differential equations: one problem on the interval [xj−1,xj] (forward) and one problem on the interval [xj,xj+1] (backward).

Exact difference scheme for system nonlinear ODEs of second order on semi-infinite intervals

We constructed and substantiated the exact three-point differential scheme for the numerical solution of boundary value problems on a semi-infinite interval for systems of second order nonlinear ordinary differential equations with non-selfadjoint operator.  The existence and uniqueness of the solution of the exact three-point difference scheme and the convergence of the method of successive approximations for its findings are proved under the conditions of existence and uniqueness of the solution of the  boundary value problem.