Impact of absorbing and reflective boundaries on fractional derivative models: Quantification, evaluation and application

Zhang, Yong; Yu, Xiangnan; Li, Xicheng; Kelly, James Floyd; Sun, HongGuang; Chen, Kewei

doi:10.1016/j.advwatres.2019.02.011

Cited by 20 publications

(9 citation statements)

References 49 publications

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“…They also pointed out that the sFADE simulates retention through negative skewness alone, without any additional retention terms. Similar results were also found in Y. Zhang et al (2019). They applied the bounded‐domain sFADE and TTFM to describe non‐Fickian transport in the Red Cedar River and proved that both sFADE with β = − 1 and TTFM could effectively model the observed data.…”

Section: Applications Of Fadessupporting

confidence: 91%

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A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water

Sun

Qiu

et al. 2020

WIREs Water

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Fractional advection–dispersion equations (FADEs) have been widely used in hydrological research to simulate the anomalous solute transport in surface and subsurface water. However, a large gap still exists between real‐world application (i.e., being a prediction tool) and theoretical FADEs. To better understand this disparity, the FADEs are firstly reviewed from the perspective of fractional‐in‐time and fractional‐in‐space, as well as the anomalous characteristics described by those functions. Then, challenges for the application of FADEs are summarized, including the theoretical gap of FADEs that needs multidisciplinary efforts to fill, extensive requirements for computation techniques and mathematical knowledge to apply the FADEs, the poor predictability for most parameters in the FADEs, and the limitations for collecting geologic information of flow fields. Then, some suggestions are given for future work, such as developing excellent code sets and mature simulation software with a friendly interface. This kind of work would alleviate the computation workload of hydrologists, especially those without coding expertise. Summarizing and determining the value ranges of the important parameters are needed (e.g., the order of the fractional derivative), through the extensive field, laboratory, and numerical experiments rather than blindly using their mathematical ranges. It is also needed to perform global sensitivity analysis for the FADEs. Meanwhile, comprehensive comparison work is necessary for suggesting a model suitable for a specific problem. Last but not least, further research is clearly needed to establish a link between nonlocal parameters and the heterogeneity property to develop more efficient fractional order partial differential equations. This article is categorized under: Science of Water > Methods Science of Water > Water Quality

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Section: Applications Of Fadessupporting

confidence: 91%

“…Similar results were also found in Y. Zhang et al (2019). They applied the bounded-domain sFADE and TTFM to describe non-Fickian transport in the Red Cedar River and proved that both sFADE with β = − 1 and TTFM could effectively model the observed data.…”

Section: Space Fractional Advection-dispersion Equationsupporting

confidence: 77%

A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water

Sun

Qiu

et al. 2020

WIREs Water

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“…There are three major kinds of the FADEs, including spatial FADE (s‐FADE), temporal FADE (t‐FADE), and tempered spatiotemporal FADE (st‐FADE) with fractional derivative in space, time, and both, respectively (Zhang et al 2019). The s‐FADE is a spatial nonlocal transport model, which is obtained from the classical ADE by replacing the second‐order spatial derivative with a fractional spatial derivative of order α ∈ (1, 2].…”

Section: Introductionmentioning

confidence: 99%

“…Stable, consistent explicit and implicit Euler approximations for two‐sided fractional diffusion equations with zero‐value Dirichlet or zero‐value fractional Neumann boundary conditions were developed by Kelly et al (2019). Zhang et al (2019) obtained explicit Euler finite difference approximations to solve the one‐sided s‐FADE with zero‐value Dirichlet, fractional Neumann, and fractional Robin boundary conditions. The analyses conducted by Zhang et al (2019) revealed that the zero‐value nonlocal (i.e., zero‐value fractional Neumann or fractional Robin) boundary conditions significantly affect the solute dynamics in the whole domain.…”

Section: Introductionmentioning

confidence: 99%

“…Zhang et al (2019) obtained explicit Euler finite difference approximations to solve the one‐sided s‐FADE with zero‐value Dirichlet, fractional Neumann, and fractional Robin boundary conditions. The analyses conducted by Zhang et al (2019) revealed that the zero‐value nonlocal (i.e., zero‐value fractional Neumann or fractional Robin) boundary conditions significantly affect the solute dynamics in the whole domain. The nonzero boundary conditions for the s‐FADE, including the nonzero‐value Dirichlet, Neumann, and Robin boundary conditions, are very important in practical applications.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Simulation of Solute Transport in Saturated Porous Media with Bounded Domains

Mohammadi

Mehdinejadiani

2021

Groundwater

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This work presents the first attempt to develop unconditionally stable, implicit finite difference solutions of one‐sided spatial fractional advection‐dispersion equation (s‐FADE) by imposing the nonzero Dirichlet boundary condition (ND BC) or the nonzero fractional Robin boundary condition (NFR BC) at inlet boundary and the zero fractional Neumann boundary condition (ZFN BC) at outlet boundary. The results of the numerical studies performed using artificial solute transport parameters demonstrated that the numerical solution with the NFR BC as the inlet boundary produced much more realistic concentration values. The numerical solution with the NFR BC at the inlet boundary was capable of correctly describing the Fickian and non‐Fickian behaviors of the solute transport at different α values, and it had the relatively same accuracy at different numbers of the spatial nodes. Also, the practical application of the numerical solution with the NFR BC as the inlet boundary was investigated by conducting tracer experiments in homogeneous and heterogeneous soil columns. According to the obtained results, this numerical solution described well solute transport in the homogenous and heterogeneous soils. The α values of the homogeneous and heterogeneous soils were obtained in the ranges of 1.849–1.999 and 1.248–1.570, respectively, which were in excellent agreement with the physical properties of the soils. In a nutshell, the numerical solution of the s‐FADE with the NFR BC as the inlet boundary can be successfully applied to describe the solute transport in the homogeneous and heterogeneous soils with bounded spatial domains.

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Convergence of consistent and inconsistent schemes for fractional diffusion problems with boundaries

Sousa

2022

Adv Comput Math

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Impact of absorbing and reflective boundaries on fractional derivative models: Quantification, evaluation and application

Cited by 20 publications

References 49 publications

A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water

A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water

Numerical Simulation of Solute Transport in Saturated Porous Media with Bounded Domains

Convergence of consistent and inconsistent schemes for fractional diffusion problems with boundaries

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