Generalised fractional diffusion equations for subdiffusion on arbitrarily growing domains

Abstract

Many physical phenomena occur on domains that grow in time. When the timescales of the phenomena and domain growth are comparable, models must include the dynamics of the domain. A widespread intrinsically slow transport process is subdiffusion. Many models of subdiffusion include a history dependence. This greatly confounds efforts to incorporate domain growth. Here we derive the fractional partial differential equations that govern subdiffusion on a growing domain, based on a Continuous Time Random Walk. This requires the introduction of a new, comoving, fractional derivative.

12 pages, 1 figure

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• Langevin picture of anomalous diffusion processes in expanding medium
• The standard and the fractional Ornstein-Uhlenbeck process on a growing domain
• Encounter-controlled coalescence and annihilation on a one-dimensional growing domain
• Comparison of magnetic diffusion and reconnection in ideal and resistive relativistic magnetohydrodynamics, ideal magnetodynamics, and resistive force-free electrodynamics
• Phase transitions in the fractional three-dimensional Navier-Stokes equations