A Simple and Effective Second‐Order Numerical Algorithm for Tempered Fractional Differential Equation With Time Caputo‐Tempered Fractional Derivative

This paper presents an efficient numerical scheme for the space–time tempered fractional convection–diffusion equation, where the time derivative is the Caputo‐tempered fractional derivative and the space derivatives are the normalized left and right Riemann–Liouville tempered fractional derivatives. The time Caputo‐tempered fractional derivative is transformed into time Riemann–Liouville tempered fractional derivative by the relationship between Caputo fractional derivative and Riemann–Liouville fractional derivative. Using the tempered weighted and shifted Grünwald difference operators to approximate the time‐tempered fractional derivative and the space‐tempered fractional convection–diffusion term, it is obtained that the time and space directions are both second‐order precision. The stability and convergence of the proposed numerical scheme are analyzed by using the energy method with a little different from the existing work. It is found that the proposed scheme is unconditionally stable and convergent with order O ( τ 2 + h 2 ). Finally, some numerical examples are given to verify the effectiveness of the proposed scheme.