Higher order class of finite difference method for time-Caputo and space-Riesz fractional diffusion equation

Abstract

In this paper, a class of finite difference method (FDM) is designed for solving the time-
Caputo and space-Riesz fractional diffusion equation. For this purpose, the fractional linear
barycentric rational interpolation method (FLBRI) is adopted to discretize the Caputo deriva-
tive in the time direction as well as the second order revised fractional backward difference
formulae 2 (RFBDF2) is employed in the space direction. The energy method is used to prove
unconditionally stability and convergence analysis of the proposed method. Eventually, it is
concluded that the proposed method is convergent with the order O(htγ + h2x), where ht and hx
are the temporal and the spatial step sizes respectively, and 1 ≤ γ ≤ 7 is the order of accu-

racy in the time direction. Finally, the presented numerical experiment confirms the theoretical
analysis, the high accuracy and efficiency of the offered method.