A Comparative Study of Efficient Numerical Schemes for Time-Fractional Subdiffusion Equation Involving Singularity

Abstract

This article compares two efficient numerical schemes for solving time-fractional subdiffusion equations. The fractional derivative is taken in the Caputo sense of order \(\alpha \in (0,1)\). The solution to this problem generally has a layer due to mild singularity near \(t=0\). As a result, the standard numerical scheme degrades the convergence rate on uniform meshes. The L1-2 and L2-\(1_{\sigma }\) techniques are used in the time direction on a graded mesh to study the layer behavior of the solution. In contrast, the spatial derivative is approximated by applying the central finite difference formula on a uniform mesh. The computational results and comparison with existing literature demonstrate the effectiveness of the proposed schemes.