Numerical approximation of tempered fractional Sturm-Liouville problem with application in fractional diffusion equation

In this paper, we discuss the numerical approximation to solve regular tempered fractional Sturm-Liouville problem (TFSLP) using finite difference method. The tempered fractional differential operators considered here are of Caputo type. The numerically obtained eigenvalues are real, and the corresponding eigenfunctions are orthogonal. The obtained eigenfunctions work as basis functions of weighted Lebesgue integrable function spaceLw2(a,b). Further, the obtained eigenvalues and corresponding eigenfunctions are used to provide weak solution of the tempered fractional diffusion equation. Approximation and error bounds of the solution of the tempered fractional diffusion equation are provided.