Modeling anomalous heat diffusion: Comparing fractional derivative and non-linear diffusivity treatments

Abstract

In the Fourier heat conduction equation, when the flux definition is expressed as the product of a constant diffusivity and the temperature gradient, the characteristic length scale evolves as the square root of time. However, if we replace the 1st order transient and gradient terms in the Fourier equation with fractional derivatives and/or define a non-linear spatially dependent diffusivity, it is possible to generate an anomalous space-time scaling, i.e., a scaling where the time exponent differs from the expected value of 1/2. To compare and contrast the possible consequences of using fractional calculus along with a non-linear flux, we investigate a space-time fractional heat diffusion equation that involves a non-linear diffusivity. Following presentation of the governing non-linear fractional equation, we arrive at a space-time scaling that accounts for the combined anomalous contributions of memory (fractional derivative in time), non-locality (fractional derivative in space), and a non-linear diffusivity. We demonstrate how this scaling can manifest in a physical setting by considering the analytical solution of a non-linear fractional space-time diffusion equation, a limit case Stefan problem related to moisture infiltration into a porous media. A direct physically realizable simulation of this process shows how the anomalous space-time scaling is explicitly related to measures of both the memory and non-linearity in the system. Overall, the findings from this work clearly show how the definition of a non-linear diffusivity might contribute to anomalous diffusion behavior and suggests that, in modeling a particular observation, the roles of fractional derivatives and a suitably defined non-linear diffusivity are interchangeable.