Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion

Abstract

In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm−Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.

Keywords

• higher-order Sturm–Liouville problems
• inverse Sturm–Liouville problems
• Magnus expansion