Algorithmic Realization of an Exact Three-Point Difference Scheme for the Sturm–Liouville Problem

Abstract

We develop a new algorithmic implementation of the exact three-point difference schemes on a nonuniform grid for the Sturm-Liouville problem. It is shown that, in order to find the coefficients of exact scheme for an arbitrary node of the grid, it is necessary to solve two auxiliary Cauchy problems for 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). We prove the theorem on the coefficient stability of the exact three-point difference scheme.

Algorithmic realization of exact three‐point difference scheme for singular Sturm–Liouville problem

“…The proof of the theorem is analogous to the proof of the respective theorem in [6]. This theorem proves the continuous dependence of the solution of problem () on the coefficients, that is, the coefficient stability.…”

“…For the proof, we refer to the idea of the analogical proof in [6]. Lemma Let the conditions of Lemma 2.2 be satisfied.…”

“…\quad j=1,2,\ldots,N. \end{equation*}$$ We construct the ETDS, by analogy with [6], using the obvious integral representation of ()–() $$$$\begin{equation} \int _{x_{j-1} }^{x_{j+1} }G^{j} (x,\xi )\frac{d}{d\xi } {\left[k(\xi )\frac{du}{d\xi } \right]}d\xi -\int _{x_{j-1} }^{x_{j+1} }G^{j} (x,\xi )[q(\xi )-\lambda r(\xi )]u(\xi )\, d\xi =0,\quad j=1,2,\ldots,N. \end{equation}$$$$$$\Box$$…”

“…In this paper, we show that coefficients of the ETDS for the Sturm–Liouville problem with a singularity at the boundary, as in [6], can be expressed through solutions of the auxiliary Cauchy problems.…”

“…In [6,7], based on the ideas of [8,9], a new algorithmic realization of the exact three-point difference scheme (ETDS) through truncated three-point difference schemes (TDS) of any order of accuracy was developed and justified for the Sturm-Liouville problem. In these articles, it is shown that the coefficients of the ETDS and the right-hand side at any grid node can be expressed through the solutions of two auxiliary Cauchy problems, each of which can be solved approximately in one step, for example, by the Taylor series expansion method or the Runge-Kutta method.…”