Abstract
The basic assumption of models for the transport of contaminants through soil is that the movements of solute particles are characterized by the Brownian motion. However, the complexity of pore space in natural porous media makes the hypothesis of Brownian motion far too restrictive in some situations. Therefore, alternative models have been proposed. One of the models, many times encountered in hydrology, is based in fractional differential equations, which is a one-dimensional fractional advection diffusion equation where the usual second-order derivative gives place to a fractional derivative of order α, with 1 < α ≤ 2. When a fractional derivative replaces the second-order derivative in a diffusion or dispersion model, it leads to anomalous diffusion, also called super-diffusion. We derive analytical solutions for the fractional advection diffusion equation with different initial and boundary conditions. Additionally, we analyze how the fractional parameter α affects the behavior of the solutions.
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Research supported by CMUC and FCT (Portugal), through European program COMPETE/FEDER and by the research project UTAustin/MAT/066/2008.
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Sousa, E. (2013). Super-diffusive Transport Processes in Porous Media. In: Ferreira, J., Barbeiro, S., Pena, G., Wheeler, M. (eds) Modelling and Simulation in Fluid Dynamics in Porous Media. Springer Proceedings in Mathematics & Statistics, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5055-9_5
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DOI: https://doi.org/10.1007/978-1-4614-5055-9_5
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