A numerical approximation for generalized fractional Sturm–Liouville problem with application

Abstract

In this paper, we present a numerical scheme for the generalized fractional Sturm–Liouville problem (GFSLP) with mixed boundary conditions. The GFSLP is defined in terms of the B-operator consisting of an integral operator with a kernel and a differential operator. One of the main features of the B-operator is that for different kernels, it leads to different Sturm–Liouville Problems (SLPs), and thus the same formulation can be used to discuss different SLPs. We prove the well-posedness of the proposed GFSLP. Further, the approximated eigenvalues of the GFSLP are obtained for two different kernels namely a modified power kernel and the Prabhakar kernel in the B-operator. We obtain real eigenvalues and corresponding orthogonal eigenfunctions. Theoretical and numerical convergence orders of eigenvalues and eigenvectors are also discussed. Further, the numerically obtained eigenvalues and eigenfunctions are used to construct an approximate solution of the one-dimensional fractional diffusion equation defined in a bounded domain.