Computation of eigenvalues of fractional Sturm–Liouville problems

Document Type : Research Article

Authors

• E.M. Maralani ¹
• F.D. Saei ¹
• A.A.J. Akbarfam ²
• K. Ghanbari ³

¹ Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.

² University of Tabriz, Tabriz, Iran.

³ Sahand university of Technology, Tabriz, Iran.

10.22067/ijnao.2020.11305.0

Abstract

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

20.1001.1.24236977.2021.11.1.7.4

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