ABSTRACT

In this talk, I will introduce a generalized definition of Caputo derivatives based on a convolution group, and its applications to fractional ODEs, SDEs and PDEs. In particular, for time fractional ODEs, I will talk about the generalized comparison principles under very weak conditions, the monotonicity and blowup behavior for some autonomous fractional ODEs; I will also introduce a fractional SDE model with Caputo derivative and fractional Brownian motion involved to satisfy the fluctuation-dissipation theorem; furthermore, for fractional PDEs, some compactness criteria will be introduced for the existence of weak solutions to time fractional PDEs. This talk is based on a series work with Jian-Guo Liu, Jianfeng Lu et al.