Sturm Liouville Equations in the frame of fractional operators with exponential kernels and their discrete versions

Abstract

In this article, we study Sturm-Liouville Equations (SLEs) in the frame of fractional operators with exponential kernels. We formulate some Fractional Sturm-Liouville Problems (F SLP s) with the differential part containing the left and right sided derivatives with exponential kernels. We investigate the self-adjointness, eigenvalue and eigenfunction properties of the corresponding Fractional SturmLiouville Operators (F SLOs) by using fractional integration by parts formulas. The nabla discrete version of our results are also established. Finally, an example is analyzed to illustrate the method of solution. 

Key words: Fractional Sturm-Liouville problem, CF R and CF C fractional derivatives, CF R and CF C fractional differences, exponential kernel.