Fractional Sturm–Liouville problem

In this paper, we define some Fractional Sturm–Liouville Operators (FSLOs) and introduce two classes of Fractional Sturm–Liouville Problems (FSLPs) namely regular and singular FSLP. The operators defined here are different from those defined in the literature in the sense that the operators defined here contain left and right Riemann–Liouville and left and right Caputo fractional derivatives. For both classes we investigate the eigenvalue and eigenfunction properties of the FSLOs. In the class of regular FSLPs, we discuss two types of FSLPs. As an operator for the class of singular FSLPs, we introduce a Fractional Legendre Equation (FLE) and discuss its solution. It is shown that the Legendre Polynomials resulting from an FLE are the same as those obtained from the integer order Legendre equation; however, the eigenvalues of the two equations differ. Using the Legendre integral transform we demonstrate some applications of our results by solving two fractional differential equations, one ordinary and the other partial. It is our hope that this paper will initiate new research in the area of FSLPs and many of its variations.