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.

Dr. Yubin Yan, born in April 8th, 1965, is a Senior Lecturer in Department of Mathematics at University of Chester, UK. He obtained his PhD degree in Mathematics  in Chalmers University of Technology in 2003 and was the research associate in University of Manchester (2003-2004) and University of Chester (2004-2007).  His research area is numerical analysis for the stochastic and deterministic (partial) differential equation, finite element method, and numerical method for fractional differential equation. He introduced a new framework in 2005 for the error estimates of the finite element method for stochastic parabolic equation which is now  regarded as the standard reference in this research area.   Up to now he published more than 50 refereed papers on  SIAM  J. Numerical Analysis,  BIT,  IMA J. Numerical Analysis, etc. (30 of them are SCI journals). He supervised 2 PhD theses and 15 MSc dissertations.  He is the regular referee for more than 10 scientific journals.