Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation

Abstract

The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag–Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In addition, two different types of subordination principles are formulated for the two regimes. The Bernstein functions technique is extensively employed in the theoretical proofs.