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Eulerian Derivation Of The Fractional AdvectionDispersion Equation
, 2001
"... Z. A fractional advectiondispersion equation ADE is a generalization of the classical ADE in which the secondorder derivative is replaced with a fractionalorder derivative. In contrast to the classical ADE, the fractional ADE has solutions that resemble the highly skewed and heavytailed breakth ..."
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Cited by 38 (16 self)
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Z. A fractional advectiondispersion equation ADE is a generalization of the classical ADE in which the secondorder derivative is replaced with a fractionalorder derivative. In contrast to the classical ADE, the fractional ADE has solutions that resemble the highly skewed and heavytailed breakthrough curves observed in field and laboratory studies. These solutions, known as astable distributions, are the result of a generalized central limit theorem which describes the behavior of sums of finite or infinitevariance random variables. We use this limit theorem in a model which sums the length of particle jumps during their random walk through a heterogeneous porous medium. If the length of solute particle jumps is not constrained to a representative elementary Z. volume REV , dispersive flux is proportional to a fractional derivative. The nature of fractional derivatives is readily visualized and their parameters are based on physical properties that are measurable. When a fractional Fick's law replaces the classical Fick's law in an Eulerian evaluation of solute transport in a porous medium, the result is a fractional ADE. Fractional ADEs are ergodic equations since they occur when a generalized central limit theorem is employed. q 2001 Elsevier Science B.V. All rights reserved.
Fractal Mobile/Immobile Solute Transport
 Water Resour. Res
, 2003
"... this paper, we develop a parsimonious MIM model with fractal retention times to describe this behavior ..."
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Cited by 36 (14 self)
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this paper, we develop a parsimonious MIM model with fractal retention times to describe this behavior
Fractional Calculus and Stable Probability Distributions
, 1998
"... Fractional calculus allows one to generalize the linear (one dimensional) diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of a fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide ..."
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Cited by 27 (3 self)
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Fractional calculus allows one to generalize the linear (one dimensional) diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of a fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide certain probability density functions, in space or time, which are related to the relevant class of stable distributions. For the space fractional diffusion a randomwalk model is also proposed. Keywords  Fractional calculus, diffusion equation, stable distributions, randomwalk. 1. Introduction The purpose of this note is to outline the role of fractional calculus in generating stable probability distributions through generalized diffusion equations of fractional order. For the standard diffusion equation it is well known that the fundamental solution of the Cauchy problem provides the spatial probability density function (pdf) for the Gaussian or normal distribution, whose variance...
Multiscaling Fractional AdvectionDispersion Equations and Their Solutions
, 2003
"... rocesses; 3250 Mathematical Geophysics: Fractals and multifractals; 5104 Physical Properties of Rocks: Fracture and flow; 5139 Physical Properties of Rocks: Transport properties; KEYWORDS: fractional, dispersion, fractal, fracture, anomalous, transport Citation: Schumer, R., D. A. Benson, M. M. M ..."
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Cited by 25 (13 self)
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rocesses; 3250 Mathematical Geophysics: Fractals and multifractals; 5104 Physical Properties of Rocks: Fracture and flow; 5139 Physical Properties of Rocks: Transport properties; KEYWORDS: fractional, dispersion, fractal, fracture, anomalous, transport Citation: Schumer, R., D. A. Benson, M. M. Meerschaert, and B. Baeumer, Multiscaling fractional advectiondispersion equations and their solutions, Water Resour. Res., 39(1), 1022, doi:10.1029/2001WR001229, 2003. 1. Introduction [2] Hundreds of studies have proposed modeling techniques to address the superFickian transport of solutes in aquifers. Among them are fractional advectiondispersion equations (ADEs), analytical equations that employ fractional derivatives in describing the growth and scaling of diffusionlike plume spreading. Fractional ADEs are the limiting equations governing continuous time random walks (CTRW) with arbitrary particle jump length distribution and finite mean waiting time distribution [Compte, 1996]. Th
Approximation of LévyFeller Diffusion by Random Walk
 J. for Analysis and its Applications
, 1999
"... . After an outline of W. Feller's inversion of the (later so called) Feller potential operators and the presentation of the semigroups thus generated, we interpret the twolevel difference scheme resulting from the GrunwaldLetnikov discretization of fractional derivatives as a random walk mode ..."
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Cited by 23 (6 self)
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. After an outline of W. Feller's inversion of the (later so called) Feller potential operators and the presentation of the semigroups thus generated, we interpret the twolevel difference scheme resulting from the GrunwaldLetnikov discretization of fractional derivatives as a random walk model discrete in space and time. We show that by properly scaled transition to vanishing space and time steps this model converges to the continuous Markov process that we view as a generalized diffusion process. By reinterpretation of the proof we get a discrete probability distribution that lies in the domain of attraction of the corresponding stable L'evy distribution. By letting only the timestep tend to zero we get a random walk model discrete in space but continuous in time. Finally, we present a random walk model for the timeparametrized Cauchy probability density. Keywords: Stable probability distributions, RieszFeller potentials, pseudodifferential equations, Markov processes, random w...
Initialboundaryvalue problems for the generalized timefractional diffusion equation
 in Proceedings of 3rd IFAC Workshop on Fractional Differentiation and its Applications (FDA08
, 2008
"... Abstract: In the paper, the initialboundaryvalue problems (IBVPs) for the generalized timefractional diffusion equation (GFDE) over an open bounded domain G × (0, T), G ⊂ Rn are considered. First, a maximum principle for the GFDE is proved. The principle is then applied to show that every IBVP fo ..."
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Cited by 20 (0 self)
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Abstract: In the paper, the initialboundaryvalue problems (IBVPs) for the generalized timefractional diffusion equation (GFDE) over an open bounded domain G × (0, T), G ⊂ Rn are considered. First, a maximum principle for the GFDE is proved. The principle is then applied to show that every IBVP for a GFDE possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions. To show the existence of the solution, the Fourier method is used to construct a formal solution. This formal solution is shown to be a generalized solution of the IBVP that turns out to be a classical solution under some additional conditions.
Discrete random walk models for symmetric LévyFeller diffusion processes
, 1999
"... We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index α (0 < α ≤ 2), in the symmetric case. We show that by properly scaled transition to vanishing space and time steps our random walk models converge to the c ..."
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Cited by 17 (6 self)
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We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index α (0 < α ≤ 2), in the symmetric case. We show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous Markovian stochastic processes, that we refer to as LévyFeller diffusion processes.
ON THE SPECTRUM OF TWO DIFFERENT FRACTIONAL OPERATORS
"... Abstract. In this paper we deal with two nonlocal operators, that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. Precisely, for a fixed s ∈ (0, 1) we consider the integral definition of the fractional Laplacian given by (−∆) s u(x):= ..."
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Cited by 11 (0 self)
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Abstract. In this paper we deal with two nonlocal operators, that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. Precisely, for a fixed s ∈ (0, 1) we consider the integral definition of the fractional Laplacian given by (−∆) s u(x):= c(n, s)
BOUNDARY VALUE PROBLEMS FOR THE GENERALIZED TIMEFRACTIONAL DIFFUSION EQUATION OF DISTRIBUTED ORDER
"... on the occasion of his 75th anniversary In the paper, boundary value problems for the generalized timefractional diffusion equation of distributed order over an open bounded domain G × [0, T], G ∈ IR are considered. Both some uniqueness and existence results are presented. To show the uniqueness of ..."
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Cited by 9 (0 self)
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on the occasion of his 75th anniversary In the paper, boundary value problems for the generalized timefractional diffusion equation of distributed order over an open bounded domain G × [0, T], G ∈ IR are considered. Both some uniqueness and existence results are presented. To show the uniqueness of the solution of the problem, an appropriate maximum principle for the generalized timefractional diffusion equation of distributed order is formulated and proved. The maximum principle is based on an extremum principle for the CaputoDzherbashyan fractional derivative that was earlier introduced by the author. The existence of the solution of the problem is illustrated by constructing a formal solution using the Fourier method of variables separation. The initialboundaryproblems for the generalized timefractional diffusion equation of distributed order are shown to be from the class of the wellposed problems in the Hadamard sense.