Fractional Advection-dispersion Equation Research Articles

Banded operational matrices for Bernstein polynomials and application to the fractional advection-dispersion equation

In the papers dealing with derivation and applications of operational matrices of Bernstein polynomials, a basis transformation, commonly a transformation to power basis, is used. The main disadvantage of this method is that the transformation may be ill-conditioned. Moreover, when applied to the numerical simulation of a functional differential equation, it leads to dense operational matrices and so a dense coefficient matrix is obtained. In this paper, we present a new property for Bernstein polynomials. Using this property, we build exact banded operational matrices for derivatives of Bernstein polynomials. Next, as an application, we propose a new numerical method based on a Petrov-Galerkin variational formulation and the new operational matrices utilizing the dual Bernstein basis for the time-fractional advection-dispersion equation. Finally, we show that the proposed method leads to a narrow-banded linear system and so less computational effort is required to obtain the desired accuracy for the approximate solution. Some numerical examples are provided to demonstrate the efficiency of the method.

Parameter identification of solute transport with spatial fractional advection-dispersion equation via Tikhonov regularization

This paper employs Tikhonov regularization method to estimate the parameters of solute transport in soil with spatial fractional derivative advection-dispersion equation. The estimated parameters include the fractional derivative order, the dispersion coefficient and the average pore-water velocity. Several examples are included to test the presented method, and the method presented is efficient, feasible and easy to develop for the problem whose analytical solutions are difficult to be obtained.

Numerical Method Based on Galerkin Approximation for the Fractional Advection-Dispersion Equation

In this paper, we present a numerical method for solving homogeneous as well as non-homogeneous fractional advection-dispersion equation (FADE). The numerical method is based on the Galerkin approximation. The Galerkin approximation converts the FADE into a system of fractional ordinary differential equations whose solutions is discussed. Convergence of the proposed method is shown. Numerical examples are given and numerical results are compared with exact solution to show the effectiveness and accuracy of the proposed method.

On superlinear fractional advection dispersion equation in R N $\mathbb{R}^{N}$

We consider the fractional advection dispersion equation in $\mathbb {R}^{N}$ . The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. We obtain the existence of nontrivial solutions of the equations, improving a recent result of Zhang-Sun-Li (Appl. Math. Model. 38:4062-4075, 2014).

Direct and inverse source problems for a space fractional advection dispersion equation

Abstract In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied. The inverse problem consists in determining the source term from final observations. We first derive the analytic solution to the direct problem which we use to prove the uniqueness and the unstability of the inverse source problem using final measurements. Finally, we illustrate the results with a numerical example.

An advanced numerical modeling for Riesz space fractional advection–dispersion equations by a meshfree approach

The space fractional advection–dispersion equations (SFADE) have been found to be very adequate in describing anomalous transport and dispersion phenomena. Aside from enjoying huge advantage in modeling, we have to face severe challenge presented by the non-locality of space fractional order derivatives, which is difficult to be handled by the traditional finite difference method (FDM) especially on a complex domain with irregularly distributed nodes. Therefore, it is crucial to develop a powerful numerical method to overcome this barrier.In this paper, the point interpolation method (PIM), a meshfree method, is further developed to solve SFADE, where the polynomial point-interpolation functions and their fractional derivatives with explicit expressions are substituted into Galerkin weak form of SFADE to obtain the discrete approximation system. Adopting an expanding technique, we develop an extended point interpolation method for SFADE with zero Dirichlet boundary condition. This technique avoids singular integral and turns system matrix into Toeplitz matrix. An efficient iteration algorithm is also suggested to reduce computing time and storage. Numerical experiments is presented to validate the newly developed method and to investigate accuracy and efficiency.

Backward fractional advection dispersion model for contaminant source prediction

AbstractThe forward Fractional Advection Dispersion Equation (FADE) provides a useful model for non‐Fickian transport in heterogeneous porous media. The space FADE captures the long leading tail, skewness, and fast spreading typically seen in concentration profiles from field data. This paper develops the corresponding backward FADE model, to identify source location and release time. The backward method is developed from the theory of inverse problems, and then explained from a stochastic point of view. The resultant backward FADE differs significantly from the traditional backward Advection Dispersion Equation (ADE) because the fractional derivative is not self‐adjoint and the probability density function for backward locations is highly skewed. Finally, the method is validated using tracer data from a well‐known field experiment, where the peak of the backward FADE curve predicts source release time, while the median or a range of percentiles can be used to determine the most likely source location for the observed plume. The backward ADE cannot reliably identify the source in this application, since the forward ADE does not provide an adequate fit to the concentration data.

A two-sided fractional conservation of mass equation

A two-sided fractional conservation of mass equation is derived by using left and right fractional Mean Value Theorems. This equation extends the one-sided fractional conservation of mass equation of Wheatcraft and Meerschaert. Also, a two-sided fractional advection-dispersion equation is derived. The derivations are based on Caputo fractional derivatives.

WITHDRAWN: High-order numerical methods for the Riesz space fractional advection–dispersion equations

Abstract In this paper, we propose high-order numerical methods for the Riesz space fractional advection–dispersion equations (RSFADE) on a finite domain. The RSFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivative with the Riesz fractional derivatives of order α ∈ ( 0 , 1 ) and β ∈ ( 1 , 2 ] , respectively. Firstly, we utilize the weighted and shifted Grunwald difference operators to approximate the Riesz fractional derivative and present the finite difference method for the RSFADE. Specifically, we discuss the Crank–Nicolson scheme and solve it in matrix form. Secondly, we prove that the scheme is unconditionally stable and convergent with the accuracy of O ( τ 2 + h 2 ) . Thirdly, we use the Richardson extrapolation method (REM) to improve the convergence order which can be O ( τ 4 + h 4 ) . Finally, some numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent with theoretical analysis.

Fast Calculation of Linear Finite Element Method for the Stationary Fractional Advection Dispersion Equations

Appropriate variational formulation and detailed implementation of linear finite element for the stationary fractional advection dispersion equation(FADE) are discussed. Since fractional derivative is nonlocal operator, the stiffness matrix of finite element on traditional variational formulation for FADE is no longer sparse and the computation becomes costly. In this paper, we establish some fractional order integral and differential formulas for linear interpolation basis functions, and then design a special variational formulation which makes the stiffness matrix possess some good properties, such as quasi-symmetry, quasi-sparseness and strictly diagonally domination. These properties are very important in reducing the computational cost and guaranteeing the stability of finite element equations. Numerical examples demonstrating these properties are presented and the applications in contaminant transport in groundwater flow are given.

Variable separation for time fractional advection-dispersion equation with initial and boundary conditions

In this paper, variable separation method combined with the properties of Mittag-Leffler function is used to solve a variable-coefficient time fractional advection-dispersion equation with initial and boundary conditions. As a result, a explicit exact solution is obtained. It is shown that the variable separation method can provide a useful mathematical tool for solving the time fractional heat transfer equations.

A Comparative Study of Finite Element and Finite Difference Methods for Two-Dimensional Space-Fractional Advection-Dispersion Equation

AbstractThe paper makes a comparative study of the finite element method (FEM) and the finite difference method (FDM) for two-dimensional fractional advection-dispersion equation (FADE) which has recently been considered a promising tool in modeling non-Fickian solute transport in groundwater. Due to the non-local property of integro-differential operator of the space-fractional derivative, numerical solution of FADE is very challenging and little has been reported in literature, especially for high-dimensional case. In order to effectively apply the FEM and the FDM to the FADE on a rectangular domain, a backward-distance algorithm is presented to extend the triangular elements to generic polygon elements in the finite element analysis, and a variable-step vector Grünwald formula is proposed to improve the solution accuracy of the conventional finite difference scheme. Numerical investigation shows that the FEM compares favorably with the FDM in terms of accuracy and convergence rate whereas the latter enjoys less computational effort.

Infinitely many solutions for boundary value problems arising from the fractional advection dispersion equation

We consider the existence of infinitely many solutions to the boundary value Problem $$\frac{d} {{dt}}\left( {\frac{1} {2}0D_t^{ - \beta } (u'(t)) + \frac{1} {2}tD_T^{ - \beta } (u'(t))} \right) + \nabla F(t,u(t)) = 0a.e.t \in [0,T],u(0) = u(T) = 0.$$ Under more general assumptions on the nonlinearity, we obtain new criteria to guarantee that this boundary value problem has infinitely many solutions in the superquadratic, subquadratic and asymptotically quadratic cases by using the critical point theory.

Prediction of reinjection effects in fault-related subsidiary structures by using fractional derivative-based mathematical models for sustainable design of geothermal reservoirs

This study provides a method to evaluate the effects of cold-water injection into an advection-dominated geothermal reservoir. A fractional advection-dispersion equation (fADE) and a fractional heat transfer equation (fHTE) are applied to fault-related structures in geothermal areas where the fracture density is described by a power-law model. Synthetic production data generated by a numerical reservoir simulator reveal that the fADE and the fHTE are in reasonable agreement with the tracer responses and temperature change in a fault zone. Tracer analysis based on the fADE has potential to elucidate fault-related structures and to predict premature thermal breakthroughs quickly and efficiently.

Using the Fractional Advection Dispersion Equation to Improve the Scale Effect in the Sampling Process of Column Tests with Lateritic Soils

Breakthrough curves (BTCs) obtained from column tests in heterogeneous soils are not satisfactorily simulated with the advection-dispersion equation (ADE) for some heavy tailed cases. Furthermore, the dispersion coefficient calculated with the ADE for heavy tailed BTCs are scale dependent when simulating columns of soil larger than the original test depth. In this paper we compare the usage of a fractional ADE (FADE) and the classical ADE to fit column tests BTCs made with Brazilian lateritic soils, discussing both contaminant transport theories and underlying stochastic models. The FADE can more accurately simulate heavy tailed BTCs, and when applying the adjusted FADE parameters to longer depths of soil, the FADE also predicts a more realistic scenario of contaminant transport through heterogeneous soil. The addition of fractional calculus in the advection-dispersion equation proves to improve contaminant transport predictions based on column tests over the classical ADE, with the use of a constant fractional dispersion coefficient that is scale independent.

Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second‐order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first‐order and second‐order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.

Numerical simulation and parameters inversion for non-symmetric two-sided fractional advection-dispersion equations

Abstract This paper deals with numerical solution and parameters inversion for a one-dimensional non-symmetric two-sided fractional advection-dispersion equation (FADE) with zero Neumann boundary condition in a finite domain. A fully discretized finite difference scheme is set forth based on Grünwald–Letnikov's definition of the fractional derivative, and its stability and convergence are proved by estimating the spectral radius of the coefficient matrix of the scheme. Furthermore, an inverse problem of simultaneously determining the fractional order and the dispersion coefficients is investigated, and numerical inversions are carried out by using the optimal perturbation regularization algorithm. The inversion results show that the fractional order and the dispersion coefficients in the FADE can be determined successfully by the final observations.

Efficient Legendre spectral tau algorithm for solving the two-sided space–time Caputo fractional advection–dispersion equation

In this paper we present the operational matrices of the left Caputo fractional derivative, right Caputo fractional derivative and Riemann–Liouville fractional integral for shifted Legendre polynomials. We develop an accurate numerical algorithm to solve the two-sided space–time fractional advection–dispersion equation (FADE) based on a spectral shifted Legendre tau (SLT) method in combination with the derived shifted Legendre operational matrices. The fractional derivatives are described in the Caputo sense. We propose a spectral SLT method, both in temporal and spatial discretizations for the two-sided space–time FADE. This technique reduces the two-sided space–time FADE to a system of algebraic equations that simplifies the problem. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithm. By selecting relatively few Legendre polynomial degrees, we are able to get very accurate approximations, demonstrating the utility of the new approach over other numerical methods.

Approximate solutions of multi-order fractional advection-dispersion equation with non-polynomial conditions

Purpose – The purpose of this paper is to apply a new modified homotopy perturbation method, which is effective to solve multi-order fractional equations with non-polynomial initial and boundary conditions. Design/methodology/approach – The proposed algorithm is tested on multi-order fractional advection-dispersion equations. The fractional derivatives described in this paper are in the Caputo sense. Findings – Approximate results explicitly reveal the complete reliability, efficiency and accuracy of the new modified technique. Originality/value – It is observed that the approach may be implemented to other multi-fractional models with non-polynomial initial and boundary conditions.

Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.