Wave equation for Sturm-Liouville operator with singular intermediate coefficient and potential

TL;DR

This paper investigates a wave equation involving a Sturm-Liouville operator with singular coefficients and potential, employing separation of variables and Fourier series expansion to establish existence, uniqueness, and consistency of very weak solutions.

Contribution

It introduces a method to analyze wave equations with singular Sturm-Liouville operators, including eigenfunction expansion and modified Prufer transform, proving fundamental solution properties.

Findings

Existence of very weak solutions established

Uniqueness and consistency proved for solutions

Eigenfunctions characterized via modified Prufer transform

Abstract

In this paper, we consider a wave equation on a bounded domain with a Sturm-Liouville operator with a singular intermediate coefficient and a singular potential. To obtain and evaluate the solution, the method of separation of variables is used, then the expansion in the Fourier series in terms of the eigenfunctions of the Sturm-Liouville operator is used. The Sturm-Liouville eigenfunctions are determined by such coefficients using the modified Prufer transform. Existence, uniqueness and consistency theorems are also proved for a very weak solution of the wave equation with singular coefficients.