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(k(2-alpha)), 0 < alpha < 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.