Abstract
In this paper, we consider two types of singular fractional Sturm-Liouville operators. One comprises the composition of left-sided Caputo and left-sided Riemann-Liouville derivatives of order alpha in(0,1). The other one is the composition of left-sided Riemann-Liouville and right-sided Caputo derivatives. The reality of the corresponding eigenvalues and the orthogonality of the eigenfunctions are proved. Furthermore, we formulate the fractional Laguerre Strum-Liouville problems and derive the explicit eigenfunctions as the non-polynomial functions related to Laguerre polynomials. Finally, we introduce the generalized Laguerre transform and employ it to solve the unbounded space-fractional diffusion equations.
Highlights
The Sturm-Liouville problem (SLP) is a famous boundary value problem which is widely studied in pure and applied mathematics, physics, and other branches of science and engineering
Researchers have become more interested in formulating the notion of a fractional SturmLiouville Problem (FSLP)
We propose four types of fractional Sturm-Liouville operators (FSLOs), which include the composition of Caputo and Riemann-Liouville derivatives and are self-adjoint
Summary
The Sturm-Liouville problem (SLP) is a famous boundary value problem which is widely studied in pure and applied mathematics, physics, and other branches of science and engineering. Zayernouri and Karniadakis [ ] considered different FSLOs, one involving the composition of right-sided Riemann-Liouville and left-sided Caputo derivatives and the other one involving the composition of left-sided Riemann-Liouville and right-sided Caputo derivatives They obtained the analytical eigensolutions to the FSLPs and demonstrated the orthogonal completeness of the corresponding system of eigenfunctions. For a given real number α ∈ (n – , n) with n ∈ N, the left- and right-sided Riemann-Liouville fractional derivatives with order α > of the function f are given by. The infinite right-sided Riemann-Liouville fractional integral of order α > of the function f on the half-line is defined by xI∞α f (x) =. The infinite left-sided Riemann-Liouville and Caputo fractional derivatives of order α of the function f on the half-line are defined by.
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