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<alphaleq 1,quad tin[0,1], end{equation*} with Dirichlet boundary conditions $$I_{0^+}^{1-alpha}y(t)vert_{t=0}=0quadmbox{and}quad I_{0^+}^{1-alpha}y(t)vert_{t=1}=0,$$ where $qin L^2(0,1)$ 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.