A Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations

TL;DR

This paper introduces a Neumann series of Bessel functions (NSBF) representation for Sturm-Liouville solutions, providing uniform error bounds across spectral parameters and enabling efficient computation for large intervals.

Contribution

It develops a novel NSBF-based representation for Sturm-Liouville solutions with uniform error estimates, suitable for large spectral parameter ranges.

Findings

The NSBF representation accurately approximates solutions with error bounds independent of spectral parameter.

The method is effective for both real and complex spectral parameters within a strip.

An algorithm based on NSBF is demonstrated on a test problem for various boundary conditions.

Abstract

A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm-Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter $\omega$ the difference between the exact solution and the approximate one (the truncated NSBF) depends on $N$ (the truncation parameter) and the coefficients of the equation and does not depend on $\omega$. A similar result is valid when $\omega\in\mathbb{C}$ belongs to a strip $|Im\omega|<C$. This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of $\omega$. Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm-Liouville equation is developed and illustrated on a test problem.