Fourth-order high-precision algorithms for one-sided tempered fractional diffusion equations

In this paper, high-order numerical algorithms for two classes of time-independent one-sided tempered fractional diffusion equations were studied. The time derivative was discretized by the backward difference formula, the space tempered fractional derivatives were discretized based on tempered weighted and shifted Grünwald difference operators combined with the quasi-compact technique, and the effective second-order numerical approximations of the left and right third-order Riemann-Liouville tempered derivatives were given, thus the detailed fourth-order numerical schemes of these two classes of equations were derived. With the energy method, we proved rigorously that the numerical schemes were stable and convergent with order $O(\tau +h^4)$ and were only related to the tempered parameter $\lambda$. Finally, some examples were given to verify the validity of the numerical schemes.