Generalized Alikhanov's approximation and numerical treatment of generalized fractional sub-diffusion equations

Abstract

In this paper, we develop a new approximation for the generalized fractional derivative, which is characterized by a scale function and a weight function. The new approximation is then used in the numerical treatment of a class of generalized fractional sub-diffusion equations. The theoretical aspects of solvability, stability and convergence are established rigorously in maximum norm by discrete energy methodology. Due to the new approximation, the theoretical temporal convergence order of the numerical scheme improves those of earlier work. To confirm, four examples are presented to illustrate the accuracy of the proposed scheme and to compare with other methods in the literature.