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

In this paper, a class of finite difference method (FDM) is designed for solving the time-fractional Liouville-Caputo and space-Riesz fractional diffusion equation. For this purpose, the fractional linear barycentric rational interpolation method (FLBRI) is adopted to discretize the Liouville-Caputo derivative 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(h(t)(gamma) +h(x)(2)), where h(t) and h(x) are the temporal and the spatial step sizes respectively, and 1 <= gamma <= 7 is the order of accuracy in the time direction. Finally, the presented numerical experiment confirms the theoretical analysis, the high accuracy and efficiency of the offered method.