Fractional Governing Equations of Diffusion Wave and Kinematic Wave Open-Channel Flow in Fractional Time-Space. II. Numerical Simulations

Ercan, Ali; Kavvas, M. L.

doi:10.1061/(asce)he.1943-5584.0001081

Cited by 7 publications

(6 citation statements)

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“…Fractional time and space derivative orders can take values between 0 and 1. As stated in Ercan and Kavvas (), the fractional orders of a given practical problem closely relate to the boundary conditions and physical conditions of the problem. Moreover, the fractional time and space derivative orders may be partially explained by the physical mechanisms behind the spatial/temporal fractal scaling or long‐range dependence behaviour of groundwater flow, as fractal structures within time series of groundwater level fluctuations have been found in the literature (Joelson et al, ; Tu et al, ; Yu et al, ).…”

Section: Discussionmentioning

confidence: 99%

“…Solute transport in groundwater aquifers in fractional time–space takes place by means of an underlying groundwater flow field. Such a flow field may be used to explain the physics of the observed non‐Fickian behaviour of solute transport in subsurface media (Ercan & Kavvas, ). Moreover, although the Brownian models can describe the behaviour of groundwater level fluctuations in many cases, groundwater level fluctuations also show the presence of long‐range dependence and heavy‐tailed behaviour, which may not be explained by the Brownian models (Joelson, Golder, Beltrame, Néel, & Di Pietro, ; Z. Li & Zhang, ; Yu, Ghasemizadeh, Padilla, Kaeli, & Alshawabkeh, ).…”

Section: Introductionmentioning

confidence: 99%

“…Numerical algorithms designed for integer‐order differential equations deal with time and space derivatives locally, which cannot be directly applied for solving the fractional differential equations where time/space nonlocality is involved. Compared with the abundant numerical algorithms proposed for integer‐order differential equations, the numerical schemes developed for fractional differential equations are quite limited (Ercan & Kavvas, ; Podlubny, ). Some numerical solutions/applications of time, space, and time–space fractional partial differential equations can be found in Meerschaert and Tadjeran (); Lin and Xu (); Liu, Zhuang, Anh, Turner, and Burrage (); Murio (); Ercan and Kavvas (); Ercan and Kavvas (); and Kim et al ().…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

Ercan

Kavvas

2018

Hydrological Processes

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In order to model non‐Fickian transport behaviour in groundwater aquifers, various forms of the time–space fractional advection–dispersion equation have been developed and used by several researchers in the last decade. The solute transport in groundwater aquifers in fractional time–space takes place by means of an underlying groundwater flow field. However, the governing equations for such groundwater flow in fractional time–space are yet to be developed in a comprehensive framework. In this study, a finite difference numerical scheme based on Caputo fractional derivative is proposed to investigate the properties of a newly developed time–space fractional governing equations of transient groundwater flow in confined aquifers in terms of the time–space fractional mass conservation equation and the time–space fractional water flux equation. Here, we apply these time–space fractional governing equations numerically to transient groundwater flow in a confined aquifer for different boundary conditions to explore their behaviour in modelling groundwater flow in fractional time–space. The numerical results demonstrate that the proposed time–space fractional governing equation for groundwater flow in confined aquifers may provide a new perspective on modelling groundwater flow and on interpreting the dynamics of groundwater level fluctuations. Additionally, the numerical results may imply that the newly derived fractional groundwater governing equation may help explain the observed heavy‐tailed solute transport behaviour in groundwater flow by incorporating nonlocal or long‐range dependence of the underlying groundwater flow field.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

Ercan

Kavvas

2018

Hydrological Processes

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“…The standard integer governing equations of the river flow processes, having finite memory, are fundamentally in the Brownian domain, and cannot model the Hurst phenomenon in river flows that was documented by Nordin et al (1972). Meanwhile, as shown recently by Kavvas and Ercan (2015) and Ercan and Kavvas (2015), it is possible to simulate long river flow waves in time and space by the fractional diffusion wave and fractional kinematic wave models of open channel flow. Also, since the integer derivative powers become special cases of the fractional powers in the governing equations of the fractional diffusion wave and fractional kinematic wave models, it is possible to simulate the finite memory river flow behavior by means of the developed fractional models as well when the powers become integers.…”

Section: On the Modeling Of Hydrologic Processes Under Uncertainty Anmentioning

confidence: 99%

Some Issues and Emerging Methods in the Modeling of Hydrologic Processes

Kavvas¹

2016

J. Japan Soc. Hydrol. and Water Resour.

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In this article some fundamental issues in hydrologic modeling are discussed, and some emerging methods toward their solutions are presented. INTRODUCTION Many of the hydrologic modeling tools date back to 1960's, such as Sugawara's tank model, and, later, the Stanford watershed model, which formed the basis for the conceptual watershed hydrology models that have been used by government agencies and practitioners all around the world. These models are still being in current use as the main tools for hydrologic water balance studies, environmental modeling studies, flood forecasting, seasonal flow forecasting, etc. Meanwhile, for the risk-based analysis and design of hydraulic structures the standard methods have been the flood frequency analysis and the probable maximum flood estimation. The flood frequency analysis procedure was standardized in the USA in mid-1970s by the development of the Bulletin 17 by the US Water Resources Council (1976). This standardized flood frequency analysis method was revised in 1982 by the US Department of Interior Interagency Advisory Committee on Water Data (1982), and has been used for more than 30 years until present by all US Agencies and water resources engineering practitioners in the US and elsewhere around the world. As for the design of very large hydraulic structures and nuclear power plants, the probable maximum precipitation procedure was developed by the US Weather Bureau in mid-1950s (US Weather Bureau, 1956). While it has gone through some minor revisions through a series of Hydrometerological Reports, this approach to the estimation of probable maximum precipitation was adopted by World Meteorological Organization (WMO, 1986), and have been practiced around the world. The probable maximum flood has been estimated by rainfall-runoff models that used various-duration probable maximum precipitation estimates as their input. With the emergence of climate change as a major issue for the earth system, many standard methods in hydrologic modeling that are based on the fundamental assumption of statistical equilibrium of the hydro-climate of a study region, are now in question for their scientific basis. Furthermore, sparseness or lack of atmospheric and/or hydrologic data at many regions of the world have prevented rigorous hydrologic modeling efforts for the assessment of water balances and

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“…As Tarasov () stated, the fractional equations are utilized to describe the fractal distributions of mass, charge, and probability. However, their engineering applications, especially in the field of fluid dynamics, are limited (Miller & Ross, ; Kulish & Lage, ; Tarasov, ; Kavvas & Ercan, , ; Ercan & Kavvas, ).…”

Section: Introductionmentioning

confidence: 99%

Time–space fractional governing equations of one‐dimensional unsteady open channel flow process: Numerical solution and exploration

Ercan

Kavvas

2017

Hydrological Processes

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Although fractional integration and differentiation have found many applications in various fields of science, such as physics, finance, bioengineering, continuum mechanics, and hydrology, their engineering applications, especially in the field of fluid flow processes, are rather limited. In this study, a finite difference numerical approach is proposed to solve the time–space fractional governing equations of 1‐dimensional unsteady/non‐uniform open channel flow process. By numerical simulations, results of the proposed fractional governing equations of the open channel flow process were compared with those of the standard Saint‐Venant equations. Numerical simulations showed that flow discharge and water depth can exhibit heavier tails in downstream locations as space and time fractional derivative powers decrease from 1. The fractional governing equations under consideration are generalizations of the well‐known Saint‐Venant equations, which are written in the integer differentiation framework. The new governing equations in the fractional‐order differentiation framework have the capability of modelling nonlocal flow processes both in time and in space by taking the global correlations into consideration. Furthermore, the generalized flow process may possibly shed light on understanding the theory of the anomalous transport processes and observed heavy‐tailed distributions of particle displacements in transport processes.

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Fractional Governing Equations of Diffusion Wave and Kinematic Wave Open-Channel Flow in Fractional Time-Space. II. Numerical Simulations

Cited by 7 publications

References 58 publications

Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

Some Issues and Emerging Methods in the Modeling of Hydrologic Processes

Time–space fractional governing equations of one‐dimensional unsteady open channel flow process: Numerical solution and exploration

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