Discrete Variational Method for Sturm-Liouville Problems with Fractional Order Derivatives

Abstract

In this paper we derive a discrete variational method for computing the eigenfunctions and eigenvalues of Sturm-Liouville problems involving fractional order derivatives. We formulate the problem as the variational minimization of a functional, whereby the functional is discretized in such a manner that it is exact for piecewise linear functions. The proposed discretization leads to a functional of symmetric quadratic forms. The problem is thereby reduced to a numerical eigenvalue problem subject to linear constraints to enforce the boundary conditions. The symmetry of the matrices ensures that all computed eigenvalues are real, in accordance with classic Sturm-Liouville problems. The method is demonstrated on the fractional vibrating string equation, as well as a fractional variant of a singular form of Bessel’s equation.