Convergence of Finite Difference Method in Positive Time for Multi-Term Time Fractional Differential Equations

Abstract

A multi-time fractional-order reaction-diffusion equation with the Caputo fractional derivative is considered. On a uniform grid, the problem is discretised by using the $L$1 formula. For the problem solutions with a singularity at time $t$ = 0, the convergence order is $\mathcal{O}(τ^{α_1})$. For any subdomain bounded away from $t$ = 0, the method has the convergence rate $\mathcal{O}(τ)$, which is better than the convergence rate $\mathcal{O}(τ^{α_1})$ for the whole time-space domain. Results of theoretical analysis are illustrated by numerical experiments.