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

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 $ω$ 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 $ω$. A similar result is valid when $ω\in\mathbb{C}$ belongs to a strip $|Imω|<C$. This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of $ω$. 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.