Computation of eigenvalues of fractional Sturm–Liouville problems

We consider the eigenvalues of the fractional-order Sturm--Liouville equation of the form \begin{equation*} -{}^{c}D_{0^+}^{\alpha}\circ D_{0^+}^{\alpha} y(t)+q(t)y(t)=\lambda y(t),\quad 0<\alpha\leq 1,\quad t\in[0,1], \end{equation*} with Dirichlet boundary conditions $I_{0^+}^{1-\alpha}y(t)\vert_{t=0}=0\quad\mbox{and}\quad I_{0^+}^{1-\alpha}y(t)\vert_{t=1}=0,$ where $q\in L^2(0,1)$ is a real-valued potential function. The method is used based on a Picard's iterative procedure. We show that the eigenvalues are obtained from the zeros of the Mittag-Leffler function and its derivatives.

Keywords:

Fractional Sturm–Liouville , Fractional calculus , Iterative methods , Eigenvalues