High-Order Algorithms for Riesz Derivative and Their Applications (

We firstly develop the high-order numerical algorithms for the left and right Riemann-Liouville derivatives. Using these derived schemes, we can get high-order algorithms for the Riesz fractional derivative. Based on the approximate algorithm, we construct the numerical scheme for the space Riesz fractional diffusion equation, where a fourth-order scheme is proposed for the spacial Riesz derivative, and where a compact difference scheme is applied to approximating the first-order time derivative. It is shown that the difference scheme is unconditionally stable and convergent. Finally, numerical examples are provided which are in line with the theoretical analysis

An alternating direction Galerkin method for a time-fractional partial differential equation with damping in two space dimensions

Abstract In this paper, we propose an efficient alternating direction implicit (ADI) Galerkin method for solving the time-fractional partial differential equation with damping, where the fractional derivative is in the sense of Caputo with order in ( 1 , 2 ) $(1,2)$ . The presented numerical scheme is based on the L2- 1 σ $1_{\sigma}$ method in time and the Galerkin finite element method in space. The unconditional stability and convergence of the numerical scheme are both carefully proved. Numerical results are displayed for supporting the theoretical analysis

Numerical algorithm based on fast convolution for fractional calculus

In this paper, numerical algorithms based on fast convolution for the fractional integral and fractional derivative are proposed. Two examples are also included which show the efficiency of the derived method

Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative

This paper deals with the stability of nonlinear fractional differential systems equipped with the Caputo derivative. At first, a sufficient condition on asymptotical stability is established by using a Lyapunov-like function. Then, the fractional differential inequalities and comparison method are applied to the analysis of the stability of fractional differential systems. In addition, some other sufficient conditions on stability are also presented