Second-order numerical methods for the tempered fractional diffusion equations

In this paper, a class of second-order tempered difference operators for the left and right Riemann–Liouville tempered fractional derivatives is constructed. And a class of second-order numerical methods is presented for solving the space tempered fractional diffusion equations, where the space tempered fractional derivatives are evaluated by the proposed tempered difference operators, and in the time direction is discreted by the Crank–Nicolson method. Numerical schemes are proved to be unconditionally stable and convergent with order $O(h^{2}+\tau ^{2})$O(h2+τ2). Numerical experiments demonstrate the effectiveness of the numerical schemes.