From continuous-time random walks to the fractional Jeffreys equation: Solution and properties

Abstract

Jeffreys equation provides an increasingly popular extension of the diffusive laws of Fourier and Fick for heat and particle transport. Similar to generalisations of the diffusion equation, we here investigate the connection between a time-fractional generalisation of the Jeffreys equation and a continuous-time random walk process based on a generalised waiting time density with diverging mean. We demonstrate that the mean squared displacement exhibits a variety of anomalous behaviors, such as retarding and accelerating subdiffusion, as well as a crossover from superdiffusion to subdiffusion. Moreover, we provide two alternative approaches, namely, a fractional Taylor series and distributed-order derivatives, that transform Fourier’s or Fick’s law into the time-fractional Jeffreys equation. Our discussion provides physics-based support for the fractional Jeffreys equation and shows its versatility for practical applications.