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Abstract and Applied Analysis
Volume 2014, Article ID 283019, 8 pages
Research Article

Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative

Instituto de Energías Renovables, Universidad Nacional Autónoma de México (IER-UNAM). A.P. 34, Privada Xochicalco s/n, Col. Centro, 62580 Temixco, MOR 04510, Mexico

Received 27 May 2014; Revised 12 July 2014; Accepted 19 July 2014; Published 20 August 2014

Academic Editor: Cristina Marcelli

Copyright © 2014 José Francisco Gómez Aguilar and Margarita Miranda Hernández. 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.


An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the derivative is considered as , for the space and time domain, respectively. The fractional derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parameters and are introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leffler function depending on the parameters and . The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior.