Theoretical and computational perspectives on the eigenvalues of fourth-order fractional Sturm–Liouville problem

In this paper, we discuss a class of eigenvalue problems of fractional differential equations of order α ∈ (3, 4] with variable coefficients. The method of solution is based on utilizing the fractional series solution to find theoretical eigenfunctions. Then, the eigenvalues are determined by applying the associated boundary conditions. A notable result, for certain cases, is that the eigenfunctions are characterized in terms of the Mittag-Leffler or semi Mittag-Leffler functions. The present findings demonstrate, for certain cases, the existence of a critical value α_c ∈ (3, 4] at which the problem has no eigenvalue (for α < α_c), only one eigenvalue (at α=α_c), a finite or infinitely many eigenvalues (for α > α_c). The efficiency and accuracy of the present algorithm are demonstrated through several numerical examples.