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

Malgorzata Klimek

doi:10.3390/sym13122265

Abstract

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

Highlights

The aim of this paper is to study the fundamental properties of fractional eigenvalue problems developed by the construction of the Sturm–Liouville operator (SLO) with left and right fractional derivatives
We shall prove equivalence results for the fractional Sturm–Liouville eigenvalue problem (FSLP)/Prabhakar Sturm–Liouville eigenvalue problem (PSLP) with an equation containing the fractional differential operators (22) and (26) and investigate the properties of the integral eigenvalue problem connected to the fractional Sturm–Liouville equation (FSLE)/Prabhakar Sturm–Liouville equation (PSLE) in the case of order α fulfilling condition 1 ≥ α > 1/2 and solutions’ space restricted by the homogeneous Dirichlet boundary conditions
The results developed in this paper describe the spectrum and eigenfunctions properties for FSLP and PSLP subjected to homogeneous Dirichlet boundary conditions

Summary

Introduction

The aim of this paper is to study the fundamental properties of fractional eigenvalue problems developed by the construction of the Sturm–Liouville operator (SLO) with left and right fractional derivatives. The resulting equations were solved using some numerical schemes [1,2,3,4] In these works, the essential properties, such as the orthogonality of the eigenfunctions of the fractional operator, were not investigated. The question of whether the associated eigenvalues are real or not is not addressed Some results concerning these properties have been obtained in papers [5,6], where the discussed equations contain a classical SLO extended by including a sum of the left and the right derivatives. The FSLO contains both the left and right derivatives and is a symmetric operator on function space restricted by fractional boundary conditions which generalize conditions (1)

Objectives

Results

Discussion

Conclusion