A numerical method for fractional Sturm–Liouville problems involving the Cauchy–Euler operators

Abstract

The objective of this paper is to establish a numerical scheme for the solutions of fractional Sturm–Liouville problems involving Cauchy–Euler operators. The proposed scheme is based on the normalized Φ-Legendre functions. The Sturm–Liouville operators in this work are chosen to be a combination of the left Caputo and right Riemann–Liouville type fractional Cauchy–Euler operators. The choice of fractional operators is inspired by the structure of the classical Sturm–Liouville operator. Some important properties of eigenvalues and eigenfunctions corresponding to a class of generalized fractional Sturm–Liouville operators are also investigated. In addition to the fractional Sturm–Liouville problems, we propose schemes for solutions of some terminal and boundary value problems. Furthermore, we derive the upper bounds of the errors in approximations of derivatives of the unknown functions in terms of normalized Φ-Legendre functions. The accuracy of the proposed schemes is illustrated by numerical examples.