Anomalous diffusion: fractional derivative equation models and applications in environmental flows

Abstract

Heterogeneity embedded in natural media and flow field challenge the application of Ficks 1st Law in anomalous diffusion well documented in many disciplines. Anomalous diffusion is one of the major topics in theoretical physics and statistical mechanics, and it is also the fundamental physical process with good potential application in environmental and hydrologic sciences and engineering. As a novel modeling tool in mathematics and physics, the fractional-order derivative diffusion equation models characterize anomalous diffusion with history-dependence and spatial non-locality, accurately describe the tailing in breakthrough curves of solute transport. We summarize the recent progresses and discuss the key challenges of fractional derivative diffusion equation models including the existed research and current development, fractional derivative modeling, numerical algorithms, and related applications in the field of environmental fluid mechanics. Here also made some preliminary discussions on issues of fractional derivative diffusion equation model, such as statistical description, model parameter determination and dimensional analysis, which may contribute to the further study of anomalous diffusion.