Abstract
A new look at the fractional diffusion equation was done. Using the unified fractional derivative, a new formulation was proposed, and the equation was solved for three different order cases: neutral, dominant time, and dominant space. The solutions were expressed by generalizations of classic formulae used for the stable distributions. The entropy paradox problem was studied and clarified through the Rényi entropy: in the extreme wave regime the entropy is −∞. In passing, Tsallis and Rényi entropies for stable distributions are introduced and exemplified.
Highlights
The traditional fractional diffusion equation was based on the Caputo time derivative and on a space pseudo-derivative defined in the frequency domain
Concerning the space derivative, which was defined implicitly before, it was here considered as a particular case of the unified derivative, the Riesz–Feller derivative [36], which was defined both by a GL-type derivative and by a Riesz–Feller integral
A main point in the diffusion studies concerns the entropy computations and the corresponding entropy paradox. This seems to be a consequence of two facts: the incomplete entropy computations due to the inherent difficulties and a hasty application of scale invariance
Summary
It is no use to refer to the importance of the diffusion equation [1,2,3], which probably one of the most studied in applied sciences. Its fractional versions have attracted the attention of many researchers due to its relation with the alpha stable processes and some new applications [4,5,6,7,8,9,10,11,12,13] Such an equation can in general assume different forms with the introduction of non-linearities and using Rn as working space, we considered only the simpler linear case and n = 1, which is usually expressed as [14]. The entropy production paradox was studied, using a Rényi entropy expression in the frequency domain, when possible This happens with the stable distributions that are defined through the characteristic function.
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