An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data

We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin, Lazarov, and Zhou [IMA J. Numer. Anal., 36 (2016), pp. 197--221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where k denotes the time step size. We show that the modified L1 scheme has a convergence rate of O(k2-α), 0< α <1 for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.