Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation

In this paper, an efficient numerical method is proposed for the solution of time fractional modified anomalous sub-diffusion equation. The proposed method is based on a finite difference scheme in time variable and Legendre spectral element method for space component. The fractional derivative of equation is described in the Riemann–Liouville sense. Firstly, for obtaining a semi-discrete scheme, the time fractional derivative of the mentioned equation has been discretized by integrating both sides of it. Secondly, we use the Legendre spectral element method for full discretization in one- and two-dimensional cases. In this approach the time fractional derivative of mentioned equation is approximated by a scheme of order O(τ1+γ)O(τ1+γ) for 0 < γ < 1. We prove the stability and convergence of time discrete scheme using energy method, and show the time discrete scheme is convergent. Also, we propose an error estimate for the full discretization scheme. Numerical examples confirm the high accuracy and efficiency of the proposed numerical scheme.