A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation  Based on Chebyshev Spectral Collocation Method

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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“…T. Liu, Sun, Zhang, & Fu, 2019;X. T. Liu, Sun, Zhang, Zheng, & Yu, 2019;Wei, Chen, Zhang, Wei, & Garrard, 2018;Yi & Sun, 2019;Yu et al, 2018).…”

Section: Challenges and Suggestions For Future Workmentioning

confidence: 99%

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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An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions

Zhang

Jia

Jiang

2020

Computers & Mathematics with Applications

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A Hybrided Method for Temporal Variable-Order Fractional Partial Differential Equations with Fractional Laplace Operator

Wang,

2024

Fractal Fract

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In this paper, we present a more general approach based on a Picard integral scheme for nonlinear partial differential equations with a variable time-fractional derivative of order α(x,t)∈(1,2) and space-fractional order s∈(0,1), where v=u′(t) is introduced as the new unknown function and u is recovered using the quadrature. In order to get rid of the constraints of traditional plans considering the half-time situation, integration by parts and the regularity process are introduced on the variable v. The convergence order can reach O(τ2+h2), different from the common L1,2−α schemes with convergence rate O(τ2,3−α(x,t)) under the infinite norm. In each integer time step, the stability, solvability and convergence of this scheme are proved. Several error results and convergence rates are calculated using numerical simulations to evidence the theoretical values of the proposed method.

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A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation Based on Chebyshev Spectral Collocation Method

Cited by 5 publications

References 32 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

An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions

A Hybrided Method for Temporal Variable-Order Fractional Partial Differential Equations with Fractional Laplace Operator

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