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Entropy 2017, 19(7), 297; doi:10.3390/e19070297

Two Approaches to Obtaining the Space-Time Fractional Advection-Diffusion Equation

1
Institute of Mathematics and Computer Sciences, Faculty of Mathematical and Natural Sciences, Jan Długosz University in Czȩstochowa, al. Armii Krajowej 13/15, 42-200 Czȩstochowa, Poland
2
Institute of Law, Administration and Management, Faculty of Philology and History, Jan Długosz University in Czȩstochowa, Zbierskiego 2/4, 42-200 Czȩstochowa, Poland
*
Author to whom correspondence should be addressed.
Received: 3 May 2017 / Revised: 13 June 2017 / Accepted: 21 June 2017 / Published: 23 June 2017
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Abstract

Two approaches resulting in two different generalizations of the space-time-fractional advection-diffusion equation are discussed. The Caputo time-fractional derivative and Riesz fractional Laplacian are used. The fundamental solutions to the corresponding Cauchy and source problems in the case of one spatial variable are studied using the Laplace transform with respect to time and the Fourier transform with respect to the spatial coordinate. The numerical results are illustrated graphically. View Full-Text
Keywords: fractional calculus; advection-diffusion equation; Caputo derivative; Riesz derivative; Laplace transform; Fourier transform fractional calculus; advection-diffusion equation; Caputo derivative; Riesz derivative; Laplace transform; Fourier transform
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Povstenko, Y.; Kyrylych, T. Two Approaches to Obtaining the Space-Time Fractional Advection-Diffusion Equation. Entropy 2017, 19, 297.

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