Space-time Fractional Advection-dispersion Equation Research Articles

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.

Shehu Transform of Hilfer-Prabhakar Fractional Derivatives and Applications on some Cauchy Type Problems

In this paper, we are interested on the Shehu transform of both Prabhakar and Hilfer–Prabhakar fractional derivative and its regularized version. These results are presented in terms of Mittag-Leffler type function and also utilized to obtain the solutions of some Cauchy type problems, such as Space-time Fractional Advection-Dispersion equation and Generalized fractional Free Electron Laser (FEL) equation, at which Hilfer-Prabhakar fractional derivative of fractional order and its regularized version are involved.

Analytical Solution of Generalized Space-Time Fractional Advection-Dispersion Equation via Coupling of Sumudu and Fourier Transforms

The objective of this article is to present the computational solution of space-time advection-dispersion equation of fractional order associated with Hilfer-Prabhakar fractional derivative operator as well as fractional Laplace operator. The method followed in deriving the solution is that of joint Sumudu and Fourier transforms. The solution is derived in compact and graceful forms in terms of the generalized Mittag-Leffler function, which is suitable for numerical computation. Some illustration and special cases of main theorem are also discussed.

Solution of one-dimensional space- and time-fractional advection–dispersion equation by homotopy perturbation method

This study develops solution of one-dimensional space–time fractional advection–dispersion equation (FADE). Various forms of dispersion and velocity profiles (i.e. space dependent and both space–time dependent) are considered throughout the study. Homotopy perturbation method (HPM) is used to solve the problem semi-analytically. The advantage of HPM is that it does not require much information about the boundary of the aquifer. The initial condition may be measured for an aquifer, but sometimes it is very difficult to specify the boundary conditions. The FADE is employed for modeling the fate of contaminants in both homogeneous and heterogeneous porous formations subject to an increasing spatially dependent source condition. It is found that the contaminant concentration changes with the order of FADE as fractional-order derivative contains the memory of the system, i.e. how the system changes from one integer order to another integer order. FADEs are used to model the non-local system, hence this study helps understand the physical meaning of parameters involved in the velocity and dispersion.

A numerical scheme for space-time fractional advection-dispersion equation

In this paper, we develop a numerical resolution of the space-time fractional advection-dispersion equation. We utilize spectral-collocation method combining with a product integration technique in order to discretize the terms involving spatial fractional order derivatives that leads to a simple evaluation of the related terms. By using Bernstein polynomial basis, the problem is transformed into a linear system of algebraic equations. Matrix formulation, error analysis and order of convergence of the proposed method are also discussed. Some numerical experiments are presented to demonstrate the effectiveness of the proposed method and to confirm the analytic results.

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.

Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation

Fractional advection-dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in porous medium. A space-time fractional advection-dispersion equation (FADE) is a generalization of the classical ADE in which the first-order space derivative is replaced with Caputo or Riemann-Liouville derivative of order , and the second-order space derivative is replaced with the Caputo or the Riemann-Liouville fractional derivative of order . We derive the solution of the new equation in terms of Mittag-Leffler functions using Laplace transfrom. Some examples are given. The results from comparison let no doubt that the FADE is better in prediction than ADE.

An analytic solution for the space–time fractional advection–dispersion equation using the optimal homotopy asymptotic method

We present an analytic algorithm to solve the space–time fractional advection–dispersion equation (FADE) based on the optimal homotopy asymptotic method (OHAM), which has the advantage of controlling the region and rate of convergence of the solution series via several auxiliary parameters over the traditional homotopy analysis method (HAM) having only one auxiliary parameter. Furthermore, our proposed algorithm gives better results compared to the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) in the sense that fewer iterations are required to get a sufficiently accurate solution and the solution has a greater radius of convergence. We find that the iterations obtained by the proposed method converge to the numerical/exact solution of the ADE as the fractional orders α,β,γ tend to their integral values. Numerical examples are given to illustrate the proposed algorithm. The figures and tables show the superiority of the OHAM over the HAM.

An analytic algorithm for the space–time fractional advection–dispersion equation

Fractional advection–dispersion equation (FADE) is a generalization of the classical ADE in which the first order time derivative and first and second order space derivatives are replaced by Caputo derivatives of orders 0 < α ⩽ 1 , 0 < β ⩽ 1 and 1 < γ ⩽ 2 , respectively. We use Caputo definition to avoid (i) mass balance error, (ii) hyper-singular improper integral, (iii) non-zero derivative of constant, and (iv) fractional derivative involved in the initial condition which is often ill-defined. We present an analytic algorithm to solve FADE based on homotopy analysis method (HAM) which has the advantage of controlling the region and rate of convergence of the solution series via the auxiliary parameter ℏ over the variational iteration method (VIM) and homotopy perturbation method (HPM). We find that the proposed method converges to the numerical/exact solution of the ADE as the fractional orders α, β, γ tend to their integral values. Numerical examples are given to illustrate the proposed algorithm. Example 5 describes the intermediate process between advection and dispersion via Caputo fractional derivative.

Approximate solution of the fractional advection–dispersion equation

In this paper, we consider practical numerical method to solve a space–time fractional advection–dispersion equation with variable coefficients on a finite domain. The equation is obtained from the standard advection–dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative, and the first-order and second-order space derivatives by the Riemann–Liouville fractional derivative, respectively. Here, a new method for solving this equation is proposed in the reproducing kernel space. The representation of solution is given by the form of series and the n-term approximation solution is obtained by truncating the series. The method is easy to implement and the numerical results show the accuracy of the method.

Homotopy perturbation method for solving the space–time fractional advection–dispersion equation

In this paper we present a reliable algorithm, the homotopy perturbation method, to construct numerical solutions of the space–time fractional advection–dispersion equation in the form of a rapidly convergent series with easily computable components. Fractional advection–dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium. The fractional derivatives are described in the Caputo sense. Some examples are given. Numerical results show that the homotopy perturbation method is easy to implement and accurate when applied to space–time fractional advection–dispersion equations.

Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation

In this paper, we consider a space–time fractional advection dispersion equation (STFADE) on a finite domain. The STFADE is obtained from the standard advection dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0, 1], and the first-order and second-order space derivatives by the Riemman–Liouville fractional derivatives of order β ∈ (0, 1] and of order γ ∈ (1, 2], respectively. For the space fractional derivatives D x β u ( x , t ) and D x γ u ( x , t ) , we adopted the Grünwald formula and the shift Grünwald formula, respectively. We propose an implicit difference method (IDM) and an explicit difference method (EDM) to solve this equation. Stability and convergence of these methods are discussed. Using mathematical induction, we prove that the IDM is unconditionally stable and convergent, but the EDM is conditionally stable and convergent. Numerical results are in good agreement with theoretical analysis.

The fundamental solution of the space-time fractional advection-dispersion equation

A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order α ∈ (0, 1], and the second-order space derivative is replaced with a Riesz-Feller derivative of order β ∈ (0, 2]. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.