The uniform l1 long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data

The goal of this paper is to prove the theory that the L1 scheme for solving time fractional partial differential equations with nonsmooth data has the uniform l(1) optimal order error estimate. For the L1 scheme combined with the first-order convolution quadrature scheme, by using Laplace transform rules we obtain the uniform l(1) long time convergence of the L1 scheme for smooth and nonsmooth initial data with of Lubich with first-order accuracy in the homogeneous case. In earlier work, various authors studied the convergence properties of the L1 scheme for smooth and nonsmooth initial data in both the homogeneous and inhomogeneous cases. However, their convergence does not apply to the uniform l(1) long time convergence behavior. (C) 2021 Elsevier Ltd. All rights reserved.