Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

Abstract

In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm-Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm-Liouville theory. In the class of r-CFSLPs, we discuss two types of CFSLPs which include left-and right-sided CFDs, each of order ∈ (n, n + 1], and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed-point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of s-CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order ∈ (0, 1] and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs-I and-II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved. KEYWORDS conformable fractional derivatives and integrals, eigenvalues and eigenfunctions, fixed-point theorem, fractional diffusion, fractional Sturm-Liouville operators, transform methods JEL CLASSIFICATION

Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

“…Similar to Proposition 5, we formulate the equivalence result for integral and differential version of eigenvalue equations corresponding to PSLP. The proof is based on the composition rules (18) and ( 19) and on Proposition 6, which describes inverse integral operator (1 + T q ) −1 . We omit the proof as it is analogous to the proof of Proposition 5.…”

“…In the papers [15,16], a fractional Sturm-Liouville operator is built by using the left and right tempered derivatives. Next, in [17,18], an FSLO is constructed as a composition of conformable fractional derivatives. In addition, in paper [19], the authors show how to build an FSLO with composite fractional derivatives.…”

Extension of the variational method to conformable quantum mechanics

“…Ordered in increasing sequence . Any state Ψ α of the system can be expanded in terms of these eigenstates as 15–18 …”

Exact and Numerical Solution of the Fractional Sturm–Liouville Problem with Neumann Boundary Conditions

“…Fractional eigenvalue problems have also been considered within the framework of tempered fractional calculus [ 12 , 13 ] and conformable fractional calculus [ 14 , 15 , 16 , 17 ]. Recently, a fractional Sturm–Liouville operator containing composite fractional derivatives has been proposed in paper [ 18 ], and Prabhakar derivatives were applied in the construction of fractional eigenvalue problems subjected to homogeneous Dirichlet or mixed boundary conditions in papers [ 19 , 20 ].…”