An Analysis of the Modified L1 Scheme for the Time-Fractional Partial Differential Equations with Nonsmooth Data
Speaker
Dr. Yu-Bin Yan
Department of Mathematics, University of Chester, UK
Abstract

We consider error estimates for the modified L1 scheme for solving time fractional partial differential equation. Jin et al. (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal., 36, 197-221) established an $O(k)$ convergence rate for the L1 scheme for both smooth and nonsmooth initial data. We introduce a modified L1 scheme and prove that the convergence rate is $O(k^{2- \alpha}), 0 < \alpha <1$ for both smooth and nonsmooth initial data. We first write the time-fractional partial differential equation as a Volterra integral equation which is then approximated by using the convolution quadratures with some special generating functions. The numerical schemes obtained in this way are equivalent to the standard L1 scheme and the modified L1 scheme, respectively. A Laplace transform method is used to prove the error estimates for the homogeneous time-fractional partial differential equation for both smooth and nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
This is a joint work with Monzorul Khan and Prof. Neville Ford,  published in SIAM J. on Numerical Analysis, (56)2018, 210-227.