Neumann series of Bessel functions for the solutions of the Sturm-Liouville equation in impedance form and related boundary value problems

TL;DR

This paper introduces a Neumann series of Bessel functions for solving Sturm-Liouville equations in impedance form, providing explicit coefficient construction, error bounds, and a numerical method for accurate eigenvalue computation.

Contribution

It presents a novel series representation for solutions, with explicit recursive coefficients and uniform error bounds, enhancing numerical spectral problem solving.

Findings

Explicit recursive construction of coefficients

Uniform bounds for truncation error

Numerical method with stable eigenvalue computation

Abstract

We present a Neumann series of spherical Bessel functions representation for solutions of the Sturm--Liouville equation in impedance form \[ (\kappa(x)u')' + \lambda \kappa(x)u = 0,\quad 0 < x < L, \] in the case where $\kappa \in W^{1,2}(0,L)$ and has no zeros on the interval of interest. The $x$-dependent coefficients of this representation can be constructed explicitly by means of a simple recursive integration procedure. Moreover, we derive bounds for the truncation error, which are uniform whenever the spectral parameter $\rho=\sqrt{\lambda}$ satisfies a condition of the form $|\operatorname{Im}\rho|\leq C$. Based on these representations, we develop a numerical method for solving spectral problems that enables the computation of eigenvalues with non-deteriorating accuracy.