A Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations (Q1742981)

In the article, as a main result, the authors obtain a Neumann series of Bessel functions (NSBF) representation for solutions of Sturm-Liouville equations and for their derivatives. To be more precise, modifying some techniques of \textit{V. V. Kravchenko} et al. [``Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions'', Appl. Math. Comput. 314, 173--192 (2017)] they derive an NSBF representation for solutions of the Sturm-Liouville equation \[ -(p(y)v')'+q(y)v=\omega^2r(y)v\,. \] The coefficients are assumed to admit the application of the Liouville transformation. For all \(\omega\in\mathbb R\) the estimate of the difference between the exact solution and the approximate one (the truncated NSBF) depends on \(N\) (the truncation parameter) and \(q\) and does not depend on \(\omega\). Furthermore, they obtain error and decay rate estimates and develop an algorithm for solving initial value, boundary value or spectral problems for equation above and illustrate on a test problem. This article is very much self-contained and presents interesting results on NSBF representations of Sturm-Liouville equations.