Memory-dependent derivative versus fractional derivative (II): Remodelling diffusion process

Abstract

The memory-dependent derivative (MDD) is a new substitution for the fractional derivative (FD). It reflects the memory effect in a more distinct way. As an application, the representative heat diffusion process is remodeled with it. In fact, due to the existence of heat-conduction paradox, the time-space evolution mechanisms of this process are challenges to the modelers. The paradox cann’t be ascribed to the classical Fourier law, and the results show that the newly-constructed temporal MDD model is more reasonable than the Maxwell-Cattaneo, the temporal FD, the spatial FD and the common ones. Moreover, different mediums may accord with different memory times and weighted functions. This freedom of choice reflects the flexibility of MDD in modelling. It can be borrowed for exploring other diffusion problems.