Fractional Sturm-Liouville problems for Weber fractional derivatives.

In this paper, we introduce the regular and singular fractional Sturm-Liouville problem (SLP) where the operator is the Weber fractional derivative of order α. We show then the eigenvalues of fractional SLP are real and the eigenfunctions corresponding to distinct eigenvalues are orthogonal. We also consider a Jacobi type singular fractional SLP and treat the behaviours of its eigenvalues. We further introduce the finite fractional Fourier transform and use this transform for solving the space-fractional diffusion equation problem. [ABSTRACT FROM AUTHOR]