Fractional advection‐dispersion equations for modeling transport at the Earth surface

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The physical basis for anomalous diffusion in bed load transport (2012, background): “In the case of heavy tails, a direct translation may be made between the tails of the microscale particle step/wait distributions and a fractional advection diffusion equation describing the bulk transport [Weeks et al., 1996; Schumer et al., 2009].”
A probabilistic description of the bed load sediment flux: 2. Particle activity and motions (2012, background): “…the idea of anomalous diffusion 2 considers the scaling of the mean squared displacement R with the time interval k as R ~ k , whereF for normal (Fickian) diffusion the exponent F = 1, for subdiffusion 0 < F < 1, and for superdiffusion F > 1 [e.g. Metzler and Klafter, 2000; Schumer et al., 2009].”
Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative (2016, background): “Applicabilities of PDEs with time fractional derivatives have been discussed by many researchers in wide elds as well; [11, 12, 39, 41] for instance.”
A probabilistic description of the bed load sediment flux: 1. Theory (2012, unknown): “…of a growing interest in the possibility of non-Fickian behavior in the transport of sediment and associated materials [e.g. Nikora et al., 2002; Schumer et al., 2009; Foufoula-Georgiou and Stark, 2010; Bradley et al., 2010; Ganti et al., 2010; Voller and Paola, 2010; Hill et al., 2010; Martin…”