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On the Fractional Diffusion-Advection Equation for Fluids and Plasmas

Dipartimento di Fisica, Universitá della Calabria, I-87036 Rende, Italy
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Fluids 2019, 4(2), 62; https://doi.org/10.3390/fluids4020062
Received: 25 February 2019 / Revised: 21 March 2019 / Accepted: 27 March 2019 / Published: 1 April 2019
(This article belongs to the Special Issue Modelling of Plasma Flow)
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Abstract

The problem of studying anomalous superdiffusive transport by means of fractional transport equations is considered. We concentrate on the case when an advection flow is present (since this corresponds to many actual plasma configurations), as well as on the case when a boundary is also present. We propose that the presence of a boundary can be taken into account by adopting the Caputo fractional derivatives for the side of the boundary (here, the left side), while the Riemann-Liouville derivative is used for the unbounded side (here, the right side). These derivatives are used to write the fractional diffusion–advection equation. We look for solutions in the steady-state case, as such solutions are of practical interest for comparison with observations both in laboratory and astrophysical plasmas. It is shown that the solutions in the completely asymmetric cases have the form of Mittag-Leffler functions in the case of the left fractional contribution, and the form of an exponential decay in the case of the right fractional contribution. Possible applications to space plasmas are discussed. View Full-Text
Keywords: anomalous transport; advection; diffusion equation; fractional derivatives; plasmas; analytical methods anomalous transport; advection; diffusion equation; fractional derivatives; plasmas; analytical methods
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Zimbardo, G.; Perri, S. On the Fractional Diffusion-Advection Equation for Fluids and Plasmas. Fluids 2019, 4, 62.

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