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

ABSTRACT

In this paper, we discuss a class of eigenvalue problems of fractional differential equations of order $\alpha \in (3,4]$ 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 ${\alpha }_c\in (3,4]$ at which the problem has no eigenvalue (for $\alpha <{\alpha }_c$), only one eigenvalue (at $\alpha ={\alpha }_c$), a finite or infinitely many eigenvalues (for $\alpha >{\alpha }_c$). The efficiency and accuracy of the present algorithm are demonstrated through several numerical examples.

KEYWORDS:

• Caputo derivative
• fractional Sturm–Liouville problems
• eigenvalues
• fractional series solution

AMS SUBJECT CLASSIFICATIONS:

• 26A33
• 34L15
• 34A08

Disclosure statement

No potential conflict of interest was reported by the authors.