报告题目 (Title)：Space-fractional PDEs: waves, patterns and numerics（空间分数阶偏微分方程：波、图像与数值计算）
报告人 (Speaker)：Zegeling, P.A. 教授（Utrecht University）
报告时间 (Time)：2025年11月19日(周三) 10:00-12:00
报告地点 (Place)：校本部GJ303
邀请人(Inviter)：李常品、蔡敏
报告摘要：The talk consists of two parts, both dealing with the fractional Laplacian. First, I will describe an adaptive moving mesh method for solving space-fractional partial differential equations of fractional order between 1 and 2. The fractional Laplacian in the PDE model is defined in terms of the Riesz-derivative. The approach extends the so-called L2 method to the non-uniform mesh case. The spatial mesh generation makes use of a moving mesh PDE, MMPDE5 with additional filtering. Numerical experiments are given for the space-fractional Gray-Scott reaction-diffusion model. They reveal a rich set of different patterns, showing interesting and surprising differences in behaviour, compared to the well-known integer order case. The adaptive method detects self-replication patterns, travelling waves, and chaotic solutions, along with two remarkable evolution processes depending on the fractional order: from self-replication to standing waves and from travelling waves back to self-replication. Secondly, I will address a PDE model with a half-Laplacian operator. The analysis of this model relies on the relationship between the Hilbert transform and the half-Laplacian. A doubling-splitting method is proposed, which results in a backward wave equation (BWE). Next, a second-order parallel boundary value method is applied over a large time scale, showing that the method is convergent and stable, even in ill-posed cases. Two special cases are discussed: an advection-dominated PDE and the space-fractional Schrödinger equation. It is shown that the solution to the BWE is equivalent to the one of the original PDE, both analytically and numerically. As an additional surprising result, we find traveling wave solutions for the linear fractional-order Schrödinger equation.