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 (Formula presented.) 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 (Formula presented.) at which the problem has no eigenvalue (for (Formula presented.)), only one eigenvalue (at (Formula presented.)), a finite or infinitely many eigenvalues (for (Formula presented.)). The efficiency and accuracy of the present algorithm are demonstrated through several numerical examples.