Finite difference scheme for multi-term variable-order fractional diffusion equation

Abstract In this paper, we consider a multi-term variable-order fractional diffusion equation on a finite domain, which involves the Caputo variable-order time fractional derivative of order α(x,t)∈(0,1) $\alpha(x,t) \in(0,1) $ and the Riesz variable-order space fractional derivatives of order β(x,t)∈(0,1) $\beta(x,t) \in (0,1)$, γ(x,t)∈(1,2) $\gamma(x,t)\in(1,2)$. Approximating the temporal direction derivative by L1-algorithm and the spatial direction derivative by the standard and shifted Grünwald method, respectively, a characteristic finite difference scheme is proposed. The stability and convergence of the difference schemes are analyzed via mathematical induction. Some numerical experiments are provided to show the efficiency of the proposed difference schemes