Fractional calculus in hydrologic modeling: A numerical perspective

“…This was used to analyze the data collected by Klise [15]. The details of the conditioning have only recently appeared [4], but allowed us to rigorously test the sufficiency of the classical, Fickian advection dispersion equation [16]. In that paper, we challenged the notion that, based solely on a better fit to solute breakthrough data, a temporally non-local model is necessary for transport in an advection-dominated system.…”

Section: ) Multi-scale Variabilitymentioning

confidence: 99%

Final Report

Benson¹

2012

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Objective: The objective of this project was to characterize the influence that naturally complex geologic media has on anomalous dispersion and to determine if the nature of dispersion can be estimated from the underlying heterogeneous media. Project Description:This project combines outcrop-scale heterogeneity characterization, laboratory experiments, and numerical simulations. The study is designed to test whether established dispersion theory accurately predicts the behavior of solute transport through heterogeneous media and to investigate the relationship between heterogeneity and the parameters that populate these models. The dispersion theory tested by this work is based upon the fractional advection-dispersion equation (fADE) model. Unlike most dispersion studies that develop a solute transport model by fitting the solute transport breakthrough curve, this project will explore the nature of the heterogeneous media to better understand the connection between the model parameters and the aquifer heterogeneity. RESULTS Background:Our work at the Colorado School of Mines was focused on the following questions: 1) What are the effects of multi-scale geologic variability on transport of conservative and reactive solutes? 2) Can those transport effects be accounted for by classical methods, and if not, can the nonlocal fractional-order equations provide better predictions? 3) Can the fractional-order equations be parameterized through a link to some simple observable geologic features? 4) Are the classical equations of transport and reaction sufficient? 5) What is the effect of anomalous transport on chemical reaction in groundwater systems?The work is predicated on the observation that upscaled transport is defined by loss of information, or spatio-temporal averaging. This averaging tends to make the transport laws such as Fick's 2nd-order diffusion equation similar to central limit theory (e.g., Einstein [9]). The fractional-order advection-dispersion equations rely on limit theory for heavytailed random motion that has some diverging moments. The equations predict larger tails of a plume in space and/or time than those predicted by the classical 2nd-order advectiondispersion equation. The heavy tails are often seen in plumes at field sites.

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“…It has been demonstrated that the fractional order representation provides a more realistic behavior of complex systems appearing in various fields of science and engineering [3]. Due to this fact, fractional calculus has some applications in chemistry [4], bioengineering [5], hydrologic modelling [6], pharmacokinetics [7], heat transfer modelling [8], viscoelasticity [9], etc. In classical mechanics fractional calculus finds a wide range of applications, specially problems involving Lagrangian and Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

The Motion of a Bead Sliding on a Wire in Fractional Sense

et al. 2017

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In this study, we consider the motion of a bead sliding on a wire which is bent into a parabola form. We first introduce the classical Lagrangian from the system model under consideration and obtain the classical EulerLagrange equation of motion. As the second step, we generalize the classical Lagrangian to the fractional form and derive the fractional Euler-Lagrange equation in terms of the Caputo fractional derivatives. Finally, we provide numerical solution of the latter equation for some fractional orders and initial conditions. The method we used is based on a discretization scheme using a Grünwald-Letnikov approximation for the fractional derivatives. Numerical simulations verify that the proposed approach is efficient and easy to implement.

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Fractional calculus in hydrologic modeling: A numerical perspective

Cited by 173 publications

References 87 publications

Certain fractional calculus formulas involving extended generalized Mathieu series

Certain fractional calculus formulas involving extended generalized Mathieu series

Final Report

The Motion of a Bead Sliding on a Wire in Fractional Sense

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