Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications

By Yong Zhang, David A. Benson, Donald M. Reeves. 2009

358 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/0b86509d289835c1f345fbe8bc678ee159486d6c

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In Civil and Structural Engineering

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Citations in the corpus

Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. (2014, methodology): “Using the above definitions, we can generalize the standard time–space FDM (Schumer et al., 2001; Zhang et al., 2009a) to the following variable-index time–space FDM: ∂α x;tð Þu x; tð Þ ∂tα x;tð Þ ¼ −υ ∂u x; tð Þ∂x þ Dþ ∂β x;tð Þu x; tð Þ ∂þxβ x;tð Þ þ D− ∂β x;tð Þu x; tð Þ ∂−xβ x;tð Þ ; ð4Þ in…”
Non-Fickian mixing: Temporal evolution of the scalar dissipation rate in heterogeneous porous media (2010, background): “The high degree of heterogeneity implies that the large scale spreading of a solute plume is non-Fickian over a large range of times [36,23, 4,3,12,34,32,37,50].”
Fourier spectral methods for fractional-in-space reaction-diffusion equations (2014, background): “The fractional Fick’s Law (1) naturally implies spatial and temporal nonlocality, and can be derived from rigorous approaches using spatial averaging theorems and measurable functions [11].”
Advanced Local Fractional Calculus and Its Applications (2012, neutral): “Zhang, Particle-tracking simulation of fractional diffusion-reaction processes, Physical Review E, 84 (6) (2011) 066704 [181] Y.”
Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain (2012, methodology): “The time, space and time–space fractional advection–dispersion equations have been used to describe important physical phenomena that arise in amorphous, colloid, glassy and porous media, in fractals and percolation clusters, comb structures, dielectrics and semiconductors, biological systems, polym”