Finite-difference approximation of the inverse Sturm-Liouville problem with frozen argument

TL;DR

This paper develops a numerical method for approximating the inverse Sturm-Liouville problem with frozen argument using finite-difference methods, including algorithms, theoretical proofs, and numerical validation.

Contribution

It introduces a new inverse problem framework for the discrete approximation of the Sturm-Liouville problem with frozen argument, including uniqueness theorems and reconstruction algorithms.

Findings

Established inverse problem theory for discrete systems with frozen argument

Derived correction terms for eigenvalue approximation between continuous and discrete problems

Developed and validated a numerical algorithm for potential recovery from eigenvalues

Abstract

This paper deals with the discrete system being the finite-difference approximation of the Sturm-Liouville problem with frozen argument. The inverse problem theory is developed for this discrete system. We describe the two principal cases: degenerate and non-degenerate. For these two cases, appropriate inverse problems statements are provided, uniqueness theorems are proved, and reconstruction algorithms are obtained. Moreover, the relationship between the eigenvalues of the continuous problem and its finite-difference approximation is investigated. We obtain the "correction terms" for approximation of the discrete problem eigenvalues by using the eigenvalues of the continuous problem. Relying on these results, we develop a numerical algorithm for recovering the potential of the Sturm-Liouville operator with frozen argument from a finite set of eigenvalues. The effectiveness of this algorithm is illustrated by numerical examples.