Convergence Rates of a Finite Difference Method for the Fractional Subdiffusion Equations

Abstract

We consider the convergence of an effective numerical method of the subdiffusion equation with the Caputo fractional derivative in time. We investigate an implicit difference scheme and an explicit difference scheme by using the projection method in space and a finite difference method which was proposed by Ashyralyev in time. Combining the method of functional analysis and the technique of numerical analysis, we utilize the idea of layering in temporal direction to obtain that the local truncation error is $$O(n^{-\alpha })$$ . Then we prove that the implicit and explicit numerical methods converge at a rate of $$O(\tau ^\alpha )$$ in time. Finally, a numerical experiment is given to confirm the $$\alpha $$ -th order accuracy.