Reconstruction of Sturm–Liouville Equations in Impedance Form

• Published in: Lobachevskii journal of mathematics Vol. 45; no. 12; pp. 6121 - 6132
• Main Author: Kravchenko, V. V.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.12.2024; Springer Nature B.V
• Subjects: Algebra; Algorithms; Analysis; Bessel functions; Characteristic functions; Geometry; Impedance; Linear algebra; Liouville equations; Mathematical analysis; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Numerical analysis; Probability Theory and Stochastic Processes; Schrodinger equation; Sturm-Liouville theory; inverse problem; Sturm–Liouville equation in impedance form; Neumann series of Bessel functions; Weyl function; Webster’s horn equation; 2010 Mathematics Subject Classification: 34B24, 34A55, 65L09
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080224607495

A method for solving a general inverse coefficient problem for a Sturm–Liouville equation in impedance form , , is proposed. It is based on Neumann series of Bessel functions (NSBF) representations for solutions of a related Schrödinger equation. The whole procedure reduces to a solution of a couple of systems of linear algebraic equations for the NSBF coefficients. Solving the first one, we recover a pair of characteristic functions of two Sturm–Liouville problems, while the solution of the second one leads to the knowledge of the NSBF coefficients on , and in particular of the first NSBF coefficient . The unknown coefficient in the Sturm–Liouville equation is related with as , that means that is recovered directly from the first component of the solution vector of the second system of linear algebraic equations. The approach leads to a simple and accurate numerical algorithm. Numerical efficiency is illustrated by test examples.