Fractional Advection-dispersion Equation Research Articles

A new method of solving the Riesz fractional advection–dispersion equation with nonsmooth solution

The Riesz fractional advection–dispersion equation with weak singularities at boundaries is solved. Our important contributions are to propose a new approach, construct successfully the fractional polynomial approximate functions with weak singularities at both endpoints in spatial direction, provide the minimum residual solution (MRS) and convergence order. Numerical examples show that the proposed method has significant advantages for dealing with weak singularities at boundaries.

Upscaling solute transport in rough single-fractured media with matrix diffusion using a time fractional advection-dispersion equation

Complex structures in fractures and the rock matrix, which are difficult to map exhaustively at the local scale, can dominate contaminant transport in fractured aquifers. An upscaled model is therefore needed to effectively characterize solute transport in a heterogeneous fracture-matrix system on a large spatiotemporal scale. Most existing upscaling methods, however, do not specifically quantify the influence of matrix diffusion and fracture surface roughness on solute transport, and their upscaling parameters are usually difficult to obtain. To fill this knowledge gap, first, a time fractional advection–dispersion equation (t-FADE) model, which can accurately describe solute transport in straight fractured media, is proposed. Next, the influence of local roughness and tortuosity of fractures on transport is comprehensively considered by introducing trend lines. The t-FADE is used to characterize solute transport in each segment of rough fractures, and an upscaled model describing solute transport in rough fractures is established by averaging the governing equations of each segment. Finally, a time-dependent attenuation function is introduced into the convolutional function of the upscaled model to overcome the error caused by overlapping diffusion regions. Model comparison and analysis reveal that the rough wall of fractures strengthens matrix retention capacity, leading to a delayed peak of the breakthrough curve (BTC) with a lower peak concentration, which can be quantitatively characterized by the ratio of the actual fracture length to its longitudinal length; however, the fracture structure does not affect the solute BTC’s trailing concentration at the constant fracture mean flow velocity without considering mechanical dispersion, so the relatively simple straight fracture model can be used to describe the trailing stage of transport in rough-walled fracture media.

Numerical method for fractional Advection–Dispersion equation using shifted Vieta–Lucas polynomials

In the pursuit of creating more precise and flexible mathematical models for complex physical phenomena, this study constructs a unique fractional model for the Advection–Dispersion equation. The Advection–Dispersion equation is a fundamental mathematical tool for analyzing fluid dynamics and mass transfer processes, but traditional integer-order models often fail to accurately capture anomalous transport behaviors. To overcome this limitation, we employ a non-singular non-integer operator based on the Heydari–Hosseininia notion. This operator allows us to transform the classical advection–dispersion equation into a more robust and flexible fractional model, which can better represent a variety of transport phenomena across different scales and mediums.The numerical approximation of the proposed fractional Advection–Dispersion equation model is achieved using a set of specific polynomials known as shifted Vieta–Lucas polynomials. Aided by their unique properties, the shifted Vieta–Lucas polynomials enable us to transform the original differential system into a more computationally tractable algebraic system via a non-integer derivative matrix. Furthermore, we undertake a detailed analysis of convergence and truncation error associated with the shifted Vieta–Lucas polynomials, underpinning the validity and stability of our numerical method. To illustrate the efficiency and robustness of the derived method, we provide several example problems which are solved using our method. The corresponding solutions are extensively presented with the aid of figures and tables, demonstrating the method’s remarkable performance in solving the fractional Advection–Dispersion equation model. The proposed method shows promising potential for further application and development in solving other fractional differential equations and related scientific problems.

Contaminated water flow modelling through the porous media by using fractional advection-dispersion equation (FADE)

ABSTRACT This study aims to evaluate the validity of Fractional Advection-Dispersion equation (FADE) in breakthrough curves in saturated homogenous soil media, i.e. clay, and to examine the three primary restraints, pore water velocity, dispersion coefficient, and order of fractional differentiation which are impacting solute transport behavior (Fickian or Non-Fickian). In this study, the FADE framework used to characterize the transport process at depth 50 cm soil (clay). FADE Main based on FORTRAN, to estimate the parameters of FADE including fractional differentiation (λ), the dispersive coefficient (D), and the average pore-water velocity (v). If the value of is equal to or greater than 2, the transport is said to be Fickian otherwise is non-Fickian. In this study, the fractional differentiation of non-Fickian behavior was found 1.85 which is less than 2. On the other hand, early long-time tailing to the soil media showed an increase in the dispersion coefficient (D) for FADE and higher values in differentiation coefficient (λ = 1.85–1.99). The results of breakthrough curves (BTCs) of relative concentration (C/C0) show that it is best fitted by using FADE. For the assessment of fitting, Root Mean Square Error (RMSE) and determination Coefficient was used to find the quality of fit. .

Numerical estimation of the fractional advection–dispersion equation under the modified Atangana–Baleanu–Caputo derivative

The transport of contaminants is a crucial environmental issue, and accurate modeling of this phenomenon is vital for developing effective strategies for its management. In this study, We introduce a non-integer model of the advection–dispersion problem arising in the transport of contaminants. The used derivative is described in the modified Atangana–Baleanu–Caputo (MABC) sense which is a new definition based on an extension of the Atangana and Baleanu derivatives. We employ discrete Chebyshev polynomials to gain the numerical solution of the considered equation. First, we generate a new operational matrix through discrete Chebyshev polynomials properties and proposed derivative. Next, via discrete Chebyshev polynomials and the operational matrix, we gain an algebraic system whose solutions are easily obtained. Finally, we solve some examples and compare the results with those obtained from other numerical methods to confirm the practicality and accuracy of the suggested scheme.

Numerical Solution of Fractional Models of Dispersion Contaminants in the Planetary Boundary Layer

In this study, a numerical solution for degenerate space–time fractional advection–dispersion equations is proposed to simulate atmospheric dispersion in vertically inhomogeneous planetary boundary layers. The fractional derivative exists in a Caputo sense. We establish the maximum principle and a priori estimates for the solutions. Then, we construct a positivity-preserving finite-difference scheme, using monotone discretization in space and L1 approximation on the non-uniform mesh for the time derivative. We use appropriate grading techniques for the time–space mesh in order to overcome the boundary degeneration and weak singularity of the solution at the initial time. The computational results are demonstrated on the Gaussian fractional model as well on the boundary layers defined by height-dependent wind flow and diffusitivity, especially for the Monin–Obukhov model.

An RBF-FD method for the time-fractional advection–dispersion equation with nonlinear source term

Fractional advection–dispersion equations have proved to be useful for modeling a wide range of problems in environmental and engineering sciences. In this work, we adapt a Radial Basis Function-generated Finite Difference (RBF-FD) method to obtain approximated numerical solutions of the initial-boundary value problem of the time-fractional advection–dispersion equation with variable coefficients and nonlinear source. We use a strategy of minimization of the local truncation error in approximating the initial condition to find appropriate local shape parameters for the Gaussian RBF. For discretizing the fractional time derivative, in the Caputo's sense, we use a scheme of (4−α)th-order, where α∈(0,1) is the order of the fractional derivative. We evaluate the performance of the RBF-FD method for different 2-dimensional problems on domains with complex geometries in which a high precision is exhibited of the solutions found. Particularly, we test our method for approximating solutions of Fisher’s equation with fractional time derivative, where the source that depends on the solution, is approximated by a linearization with respect to time, and we obtain a rate of convergence of second order in time.

Comparative Analysis of Advection–Dispersion Equations with Atangana–Baleanu Fractional Derivative

In this study, we solve the fractional advection–dispersion equation (FADE) by applying the Laplace transform decomposition method (LTDM) and the variational iteration transform method (VITM). The Atangana–Baleanu (AB) sense is used to describe the fractional derivative. This equation is utilized to determine solute transport in groundwater and soils. The FADE is converted into a system of non-linear algebraic equations whose solution leads to the approximate solution for this equation using the techniques presented. The proposed approximate method’s convergence is examined. The suggested method’s applicability is demonstrated by testing it on several illustrative examples. The series solutions to the specified issues are obtained, and they contain components that converge more quickly to the precise solutions. The actual and estimated results are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed strategy. The innovation of the current work is in the application of an effective method that requires less calculation and achieves a greater level of accuracy. Furthermore, the proposed approaches may be implemented to prove their utility in tackling fractional-order problems in science and engineering.

A fast Alikhanov algorithm with general nonuniform time steps for a two‐dimensional distributed‐order time–space fractional advection–dispersion equation

AbstractIn this paper, we propose a fast Alikhanov algorithm with nonuniform time steps for a two dimensional distributed‐order time–space fractional advection–dispersion equation. First, an efficient fast Alikhanov algorithm on the general nonuniform time steps for the evaluation of Caputo fractional derivative is presented to sharply reduce the computational work and storage, and are applied to the distributed‐order time fractional derivative or multi‐term time fractional derivative under the nonsmooth regularity assumptions. And a generalized discrete fractional Grönwall inequality is extended to multi‐term fractional derivative or distributed‐order fractional derivative for analyzing theoretically our algorithm. Then the stability and convergence of time semi‐discrete scheme are investigated. Furthermore, we derive the corresponding fully discrete scheme by finite element method and discuss its convergence. At last, the given numerical examples adequately confirm the correctness of theoretical analysis and compare the computing effectiveness between the fast algorithm and the direct method.

Non-asymptotic Disturbances Estimation for Time Fractional Advection-Dispersion Equation

In this paper, a method for disturbance estimation for a class of non-homogeneous time fractional advection dispersion equation with general boundary conditions, is proposed. The disturbances are considered to affect the boundary conditions or the measurements or both. The proposed method is based on modulating functions. Numerical simulations are presented to illustrate the performance of the approach.

Hybridizable discontinuous Galerkin methods for space-time fractional advection-dispersion equations

This work focuses on the numerical solution of the initial and boundary value problems for space-time fractional advection-diffusion equations. The well-posedness of the weak solutions is shown by Lax-Milgram lemma. Two fully discrete methods are established. The main idea is based on a hybridizable discontinuous Galerkin approach in spatial direction and two finite difference schemes in temporal direction: L1 formula, the weighted and shifted Grünwald-Letnikov formula. The stability and convergence analyses of the proposed methods are derived in detail. Several numerical experiments are provided to illustrate the theoretical results.

Upscaling Solute Transport in Rough Single-Fractured Media with Matrix Diffusion Using Time Fractional Advection-Dispersion Equation

Upscaling Solute Transport in Rough Single-Fractured Media with Matrix Diffusion Using Time Fractional Advection-Dispersion Equation

Generalized Shehu Transform to $\Psi$-Hilfer-Prabhakar Fractional Derivative and its Regularized Version

In this manuscript, athours interested on the generalized Shehu transform of $\Psi$-Riemann-Liouville, $\Psi$-Caputo, $\Psi$-Hilfer fractional derivatives. Moreover, $\Psi$-Prabhakar, $\Psi$-Hilfer-Prabhakar fractional derivatives and its regularized version presented in terms of the $\Psi$-Mittag-Leffler type function. 
 They are also utilised to solve several Cauchy type problems involving $\Psi$-Hilfer-Prabhakar fractional derivatives and their regularised form, such as the space-time fractional advection-dispersion equation and the generalized fractional free-electron laser (FEL) equation.

On numerical methods for solving Riesz space fractional advection–dispersion equations based on spline interpolants

On numerical methods for solving Riesz space fractional advection–dispersion equations based on spline interpolants

Fractional derivative modeling for sediment suspension in ice-covered channels.

Characterizing the anomalous diffusion behavior of sediment transport is a key factor in calculating sediment concentration. This study attempts to seek an equation that captures the nonlocal movement feature of the transport of an ensemble of sediment particles on ice-covered channels with a steady uniform flow field. Given that the fractional advection-dispersion equation with a noninteger order on the space term is nonlocal and able to describe the long-distance transport, a mathematical model with the Caputo fractional derivative is proposed to estimate the vertical diffusion of suspended sediment particles in the ice-covered channel. Results show that the fractional derivative model has a good predictive ability to the suspended sediment concentration as compared to the measurements. Especially in regions close to the undersurface of the ice cover, the proposed model matches better with experiments than the existing analytical model. Sensitivity analyses indicate that the strength of the turbulent diffusion effect dominates the uniformity of the sediment concentration profile. Besides, the sediment concentration is more sensitive to the variation of the boundary roughness than to the change of the sediment settling velocity. It should be noted that the sediment concentration reduces with the decrease of the order of the fractional derivative, which differs from the findings in previous studies.

Finite difference solver for simulating leaching process in bounded porous media: Development and application

This work, for the first time, developed an unconditionally stable, implicit numerical solution of a one-sided spatial fractional advection-dispersion equation (s-FADE) on a bounded domain to model contaminant transport through porous media during a leaching process. We used the zero-value Robin and Neumann conditions with a fractional-order derivative as the inlet and outlet boundaries, respectively, and a constant initial concentration of the contaminant in the whole domain as an initial condition. By conducting numerical analyses, we found that there is a good agreement between the results of the numerical solution and of its analytical counterpart, validating the accuracy of the numerical solution. In the case of the α variations meaningfully altered the heterogeneity degree of the porous medium, these variations significantly affected the results of the numerical solution, especially at intermediate times. Contrary to the α parameter, the numerical solution was insensitive to the spatial node number. The noisy observational data led to the maximum uncertainty in the estimated fractional dispersion coefficient. The application of the developed numerical solution to simulate sodium chloride (NaCl) leaching curves (LCs) at a laboratory scale revealed that the numerical solution described the leaching process in the homogeneous and heterogeneous sands better than COMSOL Multiphysics®, a commercial finite element software package. According to the fitting results of the developed numerical solution to the experimental data, the mean α values were 1.876 and 1.379 for the homogeneous and heterogeneous sands, respectively. The α values correctly represented the heterogeneity degrees of the used sands and the deviation rate of the transport process from the Fickian transport. Overall, the proposed numerical solution is a reliable numerical model to simulate the leaching process in homogeneous and heterogeneous porous media with bounded domains.

General Backward Model to Identify the Source for Contaminants Undergoing Non-Fickian Diffusion in Water.

Pollutant source identification (PSI) has been conducted for four decades for tracking Fickian diffusive pollutants, while PSI for non-Fickian diffusion, well-documented in aquifers and rivers, requires novel, predictive models. To enable PSI for non-Fickian diffusive pollutants, this study derived a general backward model using the fractional-adjoint approach in sensitivity analysis for dissolved contaminants with transport governed by the spatiotemporal fractional advection-dispersion equation (fADE). The backward fADE contains a self-adjoint time-fractional term for subdiffusion and direction-dependent, non-self-adjoint space-fractional terms for superdiffusion. Field applications showed that the resultant backward location probability density function identified the point source location in all three test cases, one alluvial aquifer and two rivers. The backward model and boundary conditions derived in this study made it possible to reliably and efficiently backtrack pollutants (and may include other constituents, such as bedload) undergoing mixed sub- and superdiffusion in natural aquatic systems. The classical PSI model, however, underestimated the source location since it did not account for solute retention and preferential flow. In addition, the measured tracer snapshots (if available before PSI) can enhance the parameter predictability and improve the applicability of backward fADE PSI. Most importantly, a spatially variable dispersion coefficient is needed in the backward fADE since PSI is most likely scale dependent in natural hydrologic systems.

A Comparative Analysis of Fractional Space-Time Advection-Dispersion Equation via Semi-Analytical Methods

The approximate solutions of the time fractional advection-dispersion equation are presented in this article. The nonlocal nature of solute movement and the nonuniformity of fluid flow velocity in the advection-dispersion process lead to the formation of a heterogeneous system, which can be modeled using a fractional advection-dispersion equation, which generalizes the classical advection-dispersion equation and replaces the time derivative with the fractional Caputo derivative. Researchers use a variety of numerical techniques to study such fractional models, but the nonlocality of the derivative having fractional order leads to high computation complexity and complex calculations, so the task is to find an efficient technique that requires less computation and provides greater accuracy when numerically solving such models. A innovative techniques, homotopy perturbation method and new iteration method, are used in connection with the Elzaki transform to solve the “fractional advection-dispersion equation” which provides the solution in the convergent series form. When the homotopy perturbation method is used with the Elzaki transform, fast convergent series solutions can be obtained with less computation. By solving some cases of time-fractional advection-dispersion equation with varied initial conditions with the help of new iterative transform method and homotopy perturbation transform method demonstrates the usefulness of the proposed methods.

Applicability of the Time Fractional Derivative Model on Fickian and Non-Fickian Transport in the Single Fracture: An Experimental Investigation

The understanding of solute transport in rock fractures is of great importance in many engineering fields. In this study, two groups of experiments on artificial and natural single fractures with different fracture apertures and roughness were conducted to investigate the changes of solute transport regimes. The time fractional advection–dispersion equation (tFADE) as a promising model was applied to describe the anomalous transport. The performance of the classical advection-dispersion equation (ADE) and tFADE was compared according to the fitting precision of breakthrough curves (BTCs). The responses of the fitting parameters in the tFADE to the experimental conditions were also discussed. The results indicated that the non-Fickian transport more likely occurred in the short distance transport, and the larger Peclet number ( Pe ) led to the increase of the exponent of the power-law function in the phase of concentration decline. The tFADE was superior to ADE in capturing the non-Fickian transport especially the tailing behavior. The fractional order of time α in the tFADE was the key parameter to describe the anomalous transport process, and its responding mechanisms of were revealed: the best-fit α decreased with the increase of flow velocity and the decrease of the fracture aperture. The roughness of the single fracture which leads to a complex flow field had a significant effect on the best-fit α . The findings of this study can help for better understanding the effectiveness and physical significance of the tFADE.

Numerical solution of unsteady state fractional advection–dispersion equation

In this article, we find the numerical solution of unsteady state fractional order advection–dispersion equation. For an approximate solution of the proposed problem, we apply Laguerre spectral collocation method and the idea of finite difference method. The proposed method assisted us in reducing the fractional order unsteady state advection–dispersion equation into systems of ordinary differential equations, which we then expressed numerically using finite difference method. Finally, we used examples to test the efficacy of our method, and all of the solutions were compared to the exact solution.