报告题目：Efficient and accurate numerical methods for solving fractional PDEs
报告人: 沈捷，Purdue大学数学系教授，美国数学会会士，国际著名计算数学家，富布赖特奖获得者，教育部“长江学者”讲座教授。
报告时间：2018年3月13日14:00-15:00
报告地点：理学院2楼报告厅
报告摘要:
We present efficient and accurate numerical methods for fractional Laplacian equations and for time-fractional diffusion equations.
For fractional Laplacian problem in bounded domains, we adopt theCaffarelli-Silvestre extension which transforms the fractionalLaplacian equation in d-dimension into an equivalent system with localderivatives in (d+1)-dimension. We develop an efficient numerical method based on the generalized Laguerre approximation inthe extended direction and usual (FEM or spectral) approximation in the originaldomain. Moreover, we enrich the spectral approximation space by using leading singularfunctions associated with the extended y-direction so that high-accuracycan be achieved despite the singularity of extended problem at y=0.
For time-fractional diffusion equations, we can adopt a similar approach used for the extended problem of the fractional Laplacian. However, an essential difficulty arisesas the time-fractional operator is not self-adjoint which makes the diagonalization process very ill conditioned. We shall propose a novel approach to overcome this difficulty.