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.
Highlights
Introduction and Model FormulationThe need to go beyond the Fourier heat conduction equation is experimentally proved under different conditions since decades
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 opposite behavior is observed in the propagation regime: the mean squared displacement (MSD) increases slower near the origin than for large times
Summary
The need to go beyond the Fourier heat conduction equation is experimentally proved under different conditions since decades. Various time-fractional generalizations of the integer-order non-Fourier heat conduction models are studied. A time-fractional generalization of the Jeffreys-type heat conduction Equation (1) is proposed in [22]. Where q is the heat flux vector, τq and τT are generalized relaxation times, k is the thermal conductivity, Dtα denotes the Riemann–Liouville fractional time-derivative of order α ∈ (0, 1), and ∇ denotes the gradient operator, acting with respect to the space variables. Properties of multi-dimensional fundamental solutions with the emphasis on their positivity, are studied in the context of time-fractional and space-time fractional diffusion-wave equations in [31,32,33]. Motivated by the aforementioned developments, in the present work we revisit the fractional Jeffreys-type heat Equation (4) with the main emphasis on the differences in behavior in the two cases: τq < τT and τq > τT.
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