Abstract
In this paper, we study the fractional Sturm–Liouville problem with homogeneous Neumann boundary conditions. We transform the differential problem to an equivalent integral one on a suitable function space. Next, we discretize the integral fractional Sturm–Liouville problem and discuss the orthogonality of eigenvectors. Finally, we present the numerical results for the considered problem obtained by utilizing the midpoint rectangular rule.
Highlights
The Sturm–Liouville boundary value problem is an important issue in the field of differential equations
Eigenvalue problems arise in a large number of disciplines of sciences and engineering
In [34], we proved that eigenvalues generated while solving the above Sturm–Liouville problem are real, and eigenfunctions associated to distinct eigenvalues are orthogonal with respect to the following scalar product: h f, giw :=
Summary
The Sturm–Liouville boundary value problem (eigenvalue problem) is an important issue in the field of differential equations. One of the most important problems, in the extended Sturm–Liouville theory, including fractional differential operators, is to understand how to to construct fractional analogues of a classical Sturm–Liouville operator and how its spectrum and eigenfunctions behave for various types of operator and boundary conditions (for example, for fractional Dirichlet, Neumann, Robin or mixed conditions). The motivation of our variational approach, first presented in papers [5,34], is to develop the formulation of a fractional Sturm–Liouville problem with an orthogonal system of eigenfunctions and real eigenvalues In this approach, the Sturm–Liouville operator contains both the left and right fractional derivatives [35,36], and equations containing this type of differential operators are known as the fractional Euler–Lagrange equations [37,38,39].
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