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Finite Difference Approximations for Fractional AdvectionDispersion Flow Equations
, 2004
"... Fractional advectiondispersion equations are used in groundwater hydrology tomqU the transport of passive tracers carried by fluid flow in a porous mrousq In this paper we develop practical numtical mumti to solve one dimUEBDqyU fractional advectiondispersion equations with variable coefficients ..."
Abstract

Cited by 96 (10 self)
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Fractional advectiondispersion equations are used in groundwater hydrology tomqU the transport of passive tracers carried by fluid flow in a porous mrousq In this paper we develop practical numtical mumti to solve one dimUEBDqyU fractional advectiondispersion equations with variable coefficients on a finite domeqV The practical application of these results is illustrated by mqUEIB# a radial flow problem Use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site.
Radial fractionalorder dispersion through fractured rock
 Water Resources Research
, 2004
"... [1] A solute transport equation with a fractionalorder dispersion term is a model of solute movement in aquifers with very wide distributions of velocity. The equation is typically formulated in Cartesian coordinates with constant coefficients. In situations where wells may act as either sources or ..."
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Cited by 5 (1 self)
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[1] A solute transport equation with a fractionalorder dispersion term is a model of solute movement in aquifers with very wide distributions of velocity. The equation is typically formulated in Cartesian coordinates with constant coefficients. In situations where wells may act as either sources or sinks in these models, a radial coordinate system provides a more natural framework for deriving the resulting differential equations and the associated initial and boundary conditions. We provide the fractional radial flow advectiondispersion equation with nonconstant coefficients and develop a stable numerical solution using finite differences. The hallmark of a spatially fractionalorder dispersion term is the rapid transport of the leading edge of a plume compared to the classical Fickian model. The numerical solution of the fractional radial transport equation is able to reproduce the early breakthrough of bromide observed in a radial tracer test conducted in a fractured granite aquifer. The early breakthrough of bromide is underpredicted by the classical radial transport model. Another conservative, yet nonnaturally occurring solute (pentaflourobenzoate), also shows early breakthrough but does not conclusively support the bromide model due to poor detection at very low concentrations. The solution method includes, through a procedure called subordination, the effects of solute partitioning on immobile water.
Acknowledgments
, 2015
"... Fractional derivatives were invented by Leibniz soon after their integerorder cousins. Recently, they have found practical applications in many areas of science and engineering. Fractional diffusion equations replace the integer derivatives in the traditional diffusion equation with their fractio ..."
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Fractional derivatives were invented by Leibniz soon after their integerorder cousins. Recently, they have found practical applications in many areas of science and engineering. Fractional diffusion equations replace the integer derivatives in the traditional diffusion equation with their fractional counterparts. These models for anomalous diffusion govern limits of continuous time random walks, where a random waiting time separates random particle jumps. A power law probability distribution for particle jumps leads to fractional derivatives in space. Power law waiting times correspond to timefractional derivatives. Particle traces are random fractals, whose dimension relates to the orders of the fractional derivatives. Parameter estimation requires novel statistical techniques, since power law data contains many outliers, and these tail values are the most important feature of the data. Numerical methods apply nonlocal variants of standard Euler finite
Article Diffusion in Relatively Homogeneous Sand Columns: A ScaleDependent or ScaleIndependent Process?
, 2013
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"... A stochastic fractal model of the Universe related to the fractional Laplacian ..."
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A stochastic fractal model of the Universe related to the fractional Laplacian
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"... The effect of nonFickian diffusion into surrounding rocks on contaminant transport in a fractured porous aquifer ..."
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The effect of nonFickian diffusion into surrounding rocks on contaminant transport in a fractured porous aquifer