Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

Mohammed Al-Refai,Thabet Abdeljawad

doi:10.1155/2017/3720471

Mohammed Al-Refai, Thabet Abdeljawad

Open Access

https://doi.org/10.1155/2017/3720471

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Abstract

We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

Highlights

Introduction and PreliminariesFractional calculus is old as the Newtonian calculus [1,2,3]
The arbitrary order Riemann-Liouville results from the integrating measure dt and the Hadamard fractional integral results from the integrating measure dt/t
Hundreds of researchers did their best to develop the theory of fractional calculus and generalize it, either by obtaining more general fractional derivatives with different kernels or by defining the fractional operator on different time scales such as the discrete fractional difference operators and qfractional operators

Summary

Introduction and Preliminaries

Fractional calculus is old as the Newtonian calculus [1,2,3]. The name fractional was given to express the integration and differentiation up to arbitrary order. The word fractional there was used to express the derivative of arbitrary order no memory effect exists inside the corresponding integral inverse operator. This conformable (fractional) derivative seems to be kind of local derivative without memory. As mentioned above, with the need of new fractional derivatives with nice properties we study in this article the eigenvalue problems of Sturm-Liouville into conformable (fractional) calculus. In these studies some of the well-known results of the Sturm-Liouville problems are extended to the fractional ones with left- and right-sided fractional derivatives of Riemann-Liouville and Caputo and Riesz derivatives These results include orthogonality and completeness of eigenfunctions and countability of the real eigenvalues. For the higher order case and other details such as the product rule, chain rule, and integration by parts, we refer the reader to [9, 11]

Main Results

Illustrative Examples

Conclusion