Efficient Difference Schemes for the Caputo-Tempered Fractional Diffusion Equations Based on Polynomial Interpolation

Abstract

The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles. The related governing equations are a series of partial differential equations with tempered fractional derivatives. Using the polynomial interpolation technique, in this paper, we present three efficient numerical formulas, namely the tempered L1 formula, the tempered L1-2 formula, and the tempered L2-$1_{\sigma }$ formula, to approximate the Caputo-tempered fractional derivative of order $\alpha \in (0,1)$. The truncation error of the tempered L1 formula is of order 2−$\alpha$, and the tempered L1-2 formula and L2-$1_{\sigma }$ formula are of order 3−$\alpha$. As an application, we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations, respectively. Furthermore, the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-$1_{\sigma }$ formulas are proved by the Fourier analysis method. Finally, we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis.

Funding

the National Natural Science Foundation of China(11426174)

the National Natural Science Foundation of China(11972286)

the Natural Science Basic Research Plan in Shaanxi Province of China(2018JM1016)