SPECIAL MATHEMATICS COLLOQUIUM
Speaker: Zhi Zhou
Abstract. Anomalous diffusion has received much attention in recent years. It describes a diffusionprocess in which the mean square displacement of a particle grows faster (super-diffusion) or slower (sub-diffusion) than that in the normal diffusion process. In analogy with Brownian motion for normal diffusion, anomalous diffusion is the macroscopic counterpart of continuous time random walk. To begin with, I will introduce the anomalous diffusion and its wide applications. Next, as a typical example, the time-fractional diffusion (sub-diffusion) will be studied. The nonlocality of the fractional derivative appearing in the model changes dramatically the behavior of solution and hence leads to some computational challenges. Our aim is to develop efficient numerical schemes which are robust with respect to nonsmooth data, and to verify its convergence rate theoretically. I will present novel strategies to overcome these difficulties. Finally, a general nonlocal diffusion model will be proposed to study the crossover of various diffusive regimes that has been widely observed in practice. It has a finite-memory effect and transient behavior, and bridges the normal local diffusion and the fractional diffusion.