Anomalous diffusion memory factorization: Characteristic timescales and application to inverse problem

Abstract

Memory effects and anomalous diffusion arising in models of transport in complex media, fractals, and viscoelastic materials can be described by fractional-differential equations, such as the time-fractional diffusion equation $D^\alpha_t u - \Delta_x u = 0$. The paper develops a decomposition of solutions of these equations into a product of spatial and temporal components, corresponding to a freeze-out at long times. For fractional diffusion with the Caputo derivative the spatial factor is the inverse-Laplacian $(-\Delta)^{-1}$ of the initial data, while for the Riemann-Liouville derivative it is the inverse bi-Laplacian $ (-\Delta)^{-2}$. In both cases, the temporal factor of the solution is a scaled negative power of time. This memory artifact explicitly encodes the initial data, which gives a simple and robust way to reconstruct the initial conditions in the backward-in-time inverse problem with solution values measured at long times. We derive characteristic timescales for Mittag-Leffler functions, which correspond to such factorization in anomalous diffusion. This also enables an accurate approximation of the number of real zeros of the Mittag-Leffler function. We apply these results to heat transfer on a comb, a model which manifests subdiffusion arising from the comb's fractal structure.