An inverse Sturm–Liouville problem with a fractional derivative

In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm–Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order α ∈ (1, 2) of fractional derivative is sufficiently away from 2.