Multiscaling fractional advection‐dispersion equations and their solutions

By R. Schumer, David A. Benson, Mark M. Meerschaert, Boris Baeumer. 2003

191 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Environmental Engineering, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/2fdf5a80c02100208b3bd4cf4315ddd3e5b74a61

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In Civil and Structural Engineering

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Citations in the corpus

A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations (2009, background): “The dispersion operator may be of fractional order for more general Lévy processes [9,20,30].”
Triangular array limits for continuous time random walks (2008, background): “Space–time fractional differential equations are important in physics, finance, and hydrology, where they are used to model anomalous diffusion; see [24,30, 42,54,56,58] for an introduction to this diverse literature.”
Perspective on theories of non-Fickian transport in heterogeneous media (2009, background): “Among various modifications and generalizations of the basic fADE (23) one finds a multidimensional version [92], time-fractional ADE [51,74], and the following three alternative fADE formulations [93] with spatially varying VðxÞ and DðxÞ: a fractional flux”
Finite di$erence approximations for fractional advection-dispersion &ow equations (2004, methodology): “Keywords: Finite di$erence approximation; Stability; Backward Euler method; Implicit Euler method; Radial dispersion; Radial advection; Fractional di$usion; Fractional derivative; Fractional advection–dispersion; Numerical fractional ADE”
A second-order accurate numerical method for the two-dimensional fractional diffusion equation (2007, methodology): “Fractional diffusion equations are used to model problems in physics (see a comprehensive review by Metzler and Klafter [25]), finance [10,16,28,31,30], and hydrology [2,4,5,32,33].”
Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications (2009, background): “The Lévy measure specifies the probability of particle jumps of a given size [129], which can be defined as the following forms if the probability follows a power-law [4]:”
Operator scaling stable random fields (2006, background): “In fractured rock, the scaling coordinates of the transmissivity field correspond to the main fracture orientations, and are usually not mutually perpendicular [19].”
Fractional vector calculus for fractional advection–dispersion (2006, background): “In fractured rock, the mixing measure places all the weight on a discrete set of fracture directions [28].”
Discovering variable fractional orders of advection-dispersion equations from field data using multi-fidelity Bayesian optimization (2017, background): “The operator ∇H M is the matrix-order fractional derivative [26] whose special case is the fractional Laplacian when the inverse of the scaling matrix H−1 equals a constant space-fractional order times an identity matrix and when the mixing Lévy measure M(θ) is uniform.”