Local discontinuous Galerkin method for time variable order fractional differential equations with sub-diffusion and super-diffusion

Abstract

This paper presents an approximate solution of time variable order fractional differential equations with sub-diffusion and super-diffusion. The aim of paper is to solve and analyze this problem by a fully discrete local discontinuous Galerkin scheme. The method is based on local discontinuous Galerkin method in space and a finite difference technique in time. The numerical stability and convergence of the proposed method are investigated then the convergence rate O(hk+1+△t2−α(tn)) in the case of sub-diffusion and O(hk+1+△t) in the case of super-diffusion are proven for the presented scheme. Finally, provided numerical examples illustrate efficiency of the method and accuracy of the theory.