Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications

Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This study proposed a hierarchical framework of fractional advection-dispersion equations (FADEs) for modeling pollutants moving in the river corridor at a full spectrum of scales. Applications showed that the fixed-index FADE could model bed sediment and manganese transport in streams at the geomorphologic unit scale, whereas the variable-index FADE well fitted bedload snapshots at the reach scale with spatially varying indices. Further analyses revealed that the selection of the FADEs depended on the scale, type of the geomedium (i.e., riverbed, aquifer, or soil), and the type of available observation dataset (i.e., the tracer snapshot or breakthrough curve (BTC)). When the pollutant BTC was used, a single-index FADE with scale-dependent parameters could fit the data by upscaling anomalous transport without mapping the sub-grid, intermediate multi-index anomalous diffusion. Pollutant transport in geomedia, therefore, may exhibit complex anomalous scaling in space (and/or time), and the identification of the FADE’s index for the reach-scale anomalous transport, which links the geomorphologic unit and watershed scales, is the core for reliable applications of fractional calculus in hydrology.

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Anomalous diffusion based on fractional calculus approach applied to drying analysis of apple slices: The effects of relative humidity and temperature

Journal of Food Process Engineering ◽

10.1111/jfpe.12549 ◽

2017 ◽

Vol 40 (5) ◽

pp. e12549 ◽

Cited By ~ 13

Author(s):

C. Ramírez ◽

V. Astorga ◽

H. Nuñez ◽

A. Jaques ◽

R. Simpson

Keyword(s):

Relative Humidity ◽

Fractional Calculus ◽

Anomalous Diffusion ◽

Apple Slices

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Fractional Calculus, Anomalous Diffusion, and Probability

Fractional Dynamics ◽

10.1142/9789814340595_0011 ◽

2011 ◽

pp. 265-284 ◽

Cited By ~ 4

Author(s):

Mark M. Meerschaert

Keyword(s):

Fractional Calculus ◽

Anomalous Diffusion

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On the self-similar, early-time, anomalous diffusion in random networks — Approach by fractional calculus

International Communications in Heat and Mass Transfer ◽

10.1016/j.icheatmasstransfer.2017.09.016 ◽

2017 ◽

Vol 89 ◽

pp. 134-138 ◽

Cited By ~ 3

Author(s):

Juan C. Padrino

Keyword(s):

Fractional Calculus ◽

Anomalous Diffusion ◽

Early Time ◽

The Self ◽

Random Networks ◽

Self Similar

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Inverse source problem based on two dimensionless dispersion-current functions in 2D evolution transport equations

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2014-0051 ◽

2016 ◽

Vol 24 (6) ◽

Cited By ~ 4

Author(s):

Adel Hamdi ◽

Imed Mahfoudhi

Keyword(s):

Velocity Field ◽

Point Source ◽

Time Dependent ◽

Inverse Source Problem ◽

Two Dimensional ◽

Reaction Equation ◽

Advection Dispersion ◽

Dispersion Tensor ◽

Spatially Varying ◽

Dependent Point

AbstractThe paper deals with the nonlinear inverse source problem of identifying an unknown time-dependent point source occurring in a two-dimensional evolution advection-dispersion-reaction equation with spatially varying velocity field and dispersion tensor. The

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FRACTIONAL DIFFUSION EQUATION, QUANTUM SUBDYNAMICS AND EINSTEIN'S THEORY OF BROWNIAN MOTION

International Journal of Modern Physics B ◽

10.1142/s0217979207045086 ◽

2007 ◽

Vol 21 (23n24) ◽

pp. 3993-3999

Author(s):

SUMIYOSHI ABE

Keyword(s):

Brownian Motion ◽

Quantum Theory ◽

Fractional Calculus ◽

Diffusion Equation ◽

Master Equation ◽

Anomalous Diffusion ◽

Fractional Diffusion ◽

Fractional Diffusion Equation ◽

Quantum Master Equation ◽

Objective System

The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.

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External auditory canal duplication anomalies associated with congenital aural atresia

The Journal of Laryngology & Otology ◽

10.1258/002221503321046612 ◽

2003 ◽

Vol 117 (1) ◽

pp. 32-38 ◽

Cited By ~ 4

Author(s):

Nikolas H. Blevins ◽

Seema V. Byahatti ◽

Collin S. Karmody

Keyword(s):

Broad Spectrum ◽

External Auditory Canal ◽

The Other ◽

Treacher Collins Syndrome ◽

External Ear ◽

Full Spectrum ◽

Aural Atresia ◽

Branchial Cleft Anomalies ◽

Branchial Cleft ◽

Congenital Aural Atresia

Maldevelopment of the first branchial cleft can produce a broad spectrum of anomalies in its derivative structure, the external auditory canal (EAC). Failure of the cleft to develop normally can result in either the absence of a normally patent EAC (atresia, or stenosis) or a duplication anomaly (cyst, sinus, or fistula). Despite their common origins, the coexistence of these anatomical abnormalities is quite unusual. We present four patients with both aural atresia and duplication anomalies of the EAC. Three patients had non-syndromic unilateral aural atresia and presented with periauricular lesions originating from the first branchial cleft. The other patient had a variant of Treacher Collins syndrome and presented with draining infra-auricular fistulae.The classification and management of first branchial cleft anomalies is reviewed in light of these cases. An understanding of the embryogenesis of the external ear is necessary to successfully recognize and treat this spectrum of deformities. A classification system is presented that encompasses the full spectrum of first cleft anomalies.

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Discrete fractional diffusion model with two memory terms

Thermal Science ◽

10.2298/tsci1504177q ◽

2015 ◽

Vol 19 (4) ◽

pp. 1177-1181

Author(s):

Yan-Mei Qin ◽

Hua Kong ◽

Kai-Teng Wu ◽

Xiao-Ming Zhu

Keyword(s):

Numerical Simulation ◽

Fractional Calculus ◽

Diffusion Model ◽

Discrete Time ◽

Anomalous Diffusion ◽

Fractional Diffusion ◽

Fractional Difference ◽

Linear Evolution ◽

Time Domains ◽

Diffusion Behaviors

Fractional calculus can always exactly describe anomalous diffusion. Recently the discrete fractional difference is becoming popular due to the depiction of non-linear evolution on discrete time domains. This paper proposes a diffusion model with two terms of discrete fractional order. The numerical simulation is given to reveal various diffusion behaviors.

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On Riemann-Liouville and Caputo Derivatives

Discrete Dynamics in Nature and Society ◽

10.1155/2011/562494 ◽

2011 ◽

Vol 2011 ◽

pp. 1-15 ◽

Cited By ~ 63

Author(s):

Changpin Li ◽

Deliang Qian ◽

YangQuan Chen

Keyword(s):

Signal Processing ◽

Fractional Calculus ◽

Anomalous Diffusion ◽

Caputo Derivative ◽

Fractional Derivatives ◽

Science And Engineering ◽

Caputo Derivatives ◽

Fractional Equations

Recently, many models are formulated in terms of fractional derivatives, such as in control processing, viscoelasticity, signal processing, and anomalous diffusion. In the present paper, we further study the important properties of the Riemann-Liouville (RL) derivative, one of mostly used fractional derivatives. Some important properties of the Caputo derivative which have not been discussed elsewhere are simultaneously mentioned. The partial fractional derivatives are also introduced. These discussions are beneficial in understanding fractional calculus and modeling fractional equations in science and engineering.

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General fractional calculus in non-singular power-law kernel applied to model anomalous diffusion phenomena in heat transfer problems

Thermal Science ◽

10.2298/tsci170310194g ◽

2017 ◽

Vol 21 (suppl. 1) ◽

pp. 11-18 ◽

Cited By ~ 23

Author(s):

Feng Gao

Keyword(s):

Heat Transfer ◽

Fractional Calculus ◽

Fractional Derivative ◽

Power Law ◽

Anomalous Diffusion ◽

Fourier Transforms ◽

Diffusion Phenomena ◽

Complex Phenomena ◽

Transfer Problems ◽

First Time

In this paper we address the general fractional calculus of Liouville-Weyl and Liouville-Caputo general fractional derivative types with non-singular power-law kernel for the first time. The Fourier transforms and the anomalous diffusions are discussed in detail. The formulations are adopted to describe complex phenomena of the heat transfer problems.

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