報告題目🕚：Efficient and accurate numerical methods for solving fractional PDEs
報告人: 沈捷，Purdue大學數學系教授🐻，美國數學會會士，國際著名計算數學家🉐🫅🏼，富布賴特獎獲得者，教育部“長江學者”講座教授🎦。
報告時間📯：2018年3月13日14:00-15:00
報告地點：必一体育平台2樓報告廳
報告摘要:
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.