This talk is concerned with construction and analysis of some novel numerical methods for time-fractional parabolic equations. Due to the Caputo time derivative being involved, the solutions of equations are often singular near the initial time t = 0 even for a smooth setting. Based on a simple change of variable, an equivalent s-fractional differential equation is derived and a novel L1 finite difference method is proposed for solving the s-fractional differential equation. Under the proved regularity, we show that the proposed L1 method provides the optimal accuracy. Numerical examples for both linear and nonlinear fractional equations are presented in comparison with classical L1 methods on uniform meshes and graded meshes, respectively. Our numerical results clearly show the accuracy and efficiency of the proposed methods.

Dongfang Li obtained his bachelor degree in 2006 and PhD degree in 2011 in mathematics at the Huazhong University of Science and Technology (HUST).  He was a postdoctoral scientist in Department of mathematics of statistics at McGill University, Canada (2012-2013) and in Department of mathematics at City University of Hongkong (2015-2017). From 2011 until 2017, he was also Assistant and Associate Professor at the school of mathematics and statistics, HUST. Since 2017, he was professor at the school of mathematics and statistics, HUST. His research focuses on numerical methods and analysis for time-dependent differential equations.