报告题目 (Title)：Numerical Analysis of Some Singular and Nonlocal Partial Differential Equations(奇异非局部偏微分方程的数值分析)

报告人 (Speaker)：王立联教授（南洋理工大学）

报告时间 (Time)：2021年11月26日(周五) 10:00

报告地点 (Place)：腾讯会议，会议ID: 877 320 409

邀请人(Inviter)：李常品

主办部门：理学院数学系

报告摘要：In this talk, we shall present our recent attempts on numerical solutions of two types of PDEs involving singular or nonlocal operators: (i) parabolic and Schrodinger equations with logarithmic nonlinearities, and (ii) Maxwell’s equations in some dispersive media. The essential difficulty of the former resides in that the nonlinear term is not differentiable but just locally Holder continuous, and the underlying energy does not have definite sign. As a result, many existing numerical techniques and analysis tools cannot be directly applied to such problems. Here, we shall introduce new tools and techniques for the analysis of some workable schemes. The latter topic is concerned with electromagnetic wave propagations in dispersive media- materials with a frequency-dependent electric permittivity. The polarization response of the material leads to Maxwell’s equations involving nonlocal polarization relations. Here we shall discuss the Maxwell's equations in a Havriliak-Negami (H-N) medium, which include the relatively known Cole-Cole (C-C) and Davidson-Cole (D-C) models as special cases. In general, for the H-N model, the convolutional polarization relation has a complicated singular kernel in terms of the Mittag-Leffler (ML) function. However, for the C-C and D-C models, it can be formulated as a time-fractional equation involving Caputo and tempered fractional derivatives respectively. In this study, we focus on the numerical analysis of the H-N model in terms of energy dissipation and unconditionally stable discretization and also discuss an interesting differential-integral wave equation reduced from the Cole-Cole model.