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  4. Algorithmic implementation of an exact three-point difference scheme for a certain class of singular Sturm–Liouville problems

Algorithmic implementation of an exact three-point difference scheme for a certain class of singular Sturm–Liouville problems

MMC.
2024;
Volume 11, Number 1
: pp. 344–357
https://doi.org/10.23939/mmc2024.01.344
Received: September 05, 2023
Revised: March 12, 2024
Accepted: March 16, 2024

Khomenko N. V., Kutniv M. V.  Algorithmic implementation of an exact three-point difference scheme for a certain class of singular Sturm–Liouville problems.  Mathematical Modeling and Computing. Vol. 11, No. 1, pp. 344–357 (2024)

Authors:
  1. N. V. Khomenko
  2. M. V. Kutniv
1
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of the National Academy of Sciences of Ukraine; Trier University
2
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of the National Academy of Sciences of Ukraine; Rzeszow University of Technology

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). We have also proven the coefficient stability theorem for the exact three-point difference scheme.

singular Sturm–Liouville problem
exact three-point difference scheme
coefficient stability
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