On fractional-Legendre spectral Galerkin method for fractional Sturm–Liouville problems

In this paper, we present a numerical technique for solving fractional Sturm–Liouville problems with variable coefficients subject to mixed boundary conditions. The proposed algorithm is a spectral Galerkin method based on fractional-order Legendre functions. Tedious manipulation of the series appearing in the implementation of the method have been carried out to obtain a system of algebraic equations for the coefficients. Our findings demonstrate the possibility of having no eigenvalues, finite number of eigenvalues or infinite number of eigenvalues depending on the fractional order. The convergence and effectiveness of the present algorithm are demonstrated through several numerical examples.