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

Abstract

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.

Fractional governing equations of transient groundwater flow in unconfined aquifers with multi-fractional dimensions in fractional time

“…Hence, they can modulate the memory of the unconfined aquifer flow, which, in turn, can modulate the memory of the watershed baseflow. Meanwhile, the Caputo derivative, as defined in its special form D β x i 0 f (x i ) in space in this study, was shown by Kavvas and Ercan (2016) and Ercan and Kavvas (2017) to be a nonlocal quantity where the effect of the boundary conditions on the groundwater flow within the flow domain can have long spatial memories with the decrease in the powers of the spatial fractional derivatives from unity. Similarly, it was shown by Kavvas et al (2017a) that the Caputo derivative in time, D α 0 f (t), as defined in this study, is nonlocal in time and can carry the effect of initial conditions on the groundwater flow for long time periods as the power in the time fractional derivative decreases from 1.…”

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

“…The works by Guerrini and Swartzendruber () and El Abd and Milczarek () may further support the fractional units of hydraulic conductivity related variables with some confidence as they explained the anomalous behaviour observed in the experiments with the diffusivity coefficients of fractional time units and successfully modelled various experimental data on horizontal soil water flow by the formulated fractional diffusivity coefficients. The behaviour of the groundwater flow under fractional powers of the space and time derivatives obtained in this numerical study is consistent with the nature of the fractional derivatives that the spatial/temporal nonlocality would increase as the power of space/time derivative decrease from 1 (Ercan & Kavvas, ).…”

“…The resulting fractional governing equation can model the nonlocal phenomena of the groundwater flow field by taking the global correlations of time and space into consideration. The proposed time–space fractional governing equation of groundwater flow was derived based on the Caputo fractional derivative, which allows the implementation of the traditional physically interpretable initial and boundary conditions (Ercan & Kavvas, ; Podlubny, ). The proposed governing equation of transient groundwater flow in confined aquifers in fractional time and space reduces to a standard Cartesian groundwater flow governing equation in an integer‐order differential framework, when the fractional space and time orders become one.…”

“…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 (). Li and Zeng () provide detailed descriptions of numerical methods, and finite difference methods in particular, for fractional differential equations.…”

Generalizations of incompressible and compressible Navier–Stokes equations to fractional time and multi-fractional space

“…In order to solve Eq. ( 50 ) numerically, a first-order approximation of the Caputo’s fractional time derivative 45 and a second-order accurate Caputo’s fractional space derivative 56 schemes are coupled similar to the numerical solution of the fractional open channel flow problem as reported in Ercan and Kavvas 57 . When the fractional powers of space and time derivatives of Eq.…”