Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator

Erdal Bas

doi:10.1155/2013/915830

Abstract

We give the theory of spectral properties for eigenvalues and eigenfunctions of Bessel type of fractional singular Sturm-Liouville problem. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively. Furthermore, we prove new approximations about the topic.

Highlights

Sturm-Liouville problem was first developed in a number of papers that were published by these authors in 1836 and 1837
A Sturm-Liouville boundary value problem consists of a second order linear ordinary differential equation
The SturmLiouville problems are important in many areas of science, engineering and mathematics

Summary

Introduction

Sturm-Liouville problem was first developed in a number of papers that were published by these authors in 1836 and 1837. Charles-Francois Sturm (1803–1855), Professor of Mechanics at the Sorbonne, had been interested, since about 1833, in the problem of heat flow in bars, so he was well aware of eigenvalue-type problems He worked closely with his friend Joseph Liouville (1809–1882), Professor of Mathematics at the College de France, on the general properties of second-order differential equations. A Sturm-Liouville boundary value problem consists of a second order linear ordinary differential equation. Fractional calculus has increasing importance for the last years because fractional calculus has been applied to almost every field of science They are viscoelasticity, electrical engineering, electrochemistry, biology, biophysics and bioengineering, signal and image processing, mechanics, mechatronics, physics, and control theory. Our purpose is to introduce singular fractional SturmLiouville problem having Bessel type and prove spectral properties of spectral data for the operator. Let us give the boundary value problem for Bessel equation and necessary data as follows

Preliminaries

Main Results

Conclusion