Trapped dynamics widely appears in nature, e.g., the motion of particles in viscous cytoplasm. The famous continuous time random walk (CTRW) model with power law waiting time distribution (having diverging first moment) describes this phenomenon. Because of the finite lifetime of biological particles, sometimes it is necessary to temper the power law measure such that the waiting time measure has convergent first moment. Then the time operator of the Fokker-Planck equation corresponding to the CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. This paper focus on discussing the properties of the time tempered fractional derivative, and studying the well-posedness and the Jacobi-predictor-corrector algorithm for the tempered fractional ordinary differential equation. By adjusting the parameter of the proposed algorithm, high convergence order can be obtained and the computational cost linearly increases with time. The numerical results shows that our algorithm converges with order $ N_I $, where $ N_I $ is the number of used interpolating points.

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.

A New Family of Difference Schemes for Space Fractional Advection Diffusion Equation - Volume 9 Issue 2 - Can Li, Weihua Deng