We propose JacobiGaussLobatto collocation approximation for the numerical solution of a class of fractionalinspace advectiondispersion equation with variable coefficients based on Caputo derivative. This approach has the advantage of transforming the problem into the solution of a system of ordinary differential equations in time this system is approximated through an implicit iterative method. In addition, some of the known spectral collocation approximations can be derived as special cases from our algorithm if we suitably choose the corresponding special cases of Jacobi parameters
Spectral methods have emerged as powerful techniques used in applied mathematics and scientific computing to numerically solve differential equations [
In recent years, considerable interest in fractional partial differential equations has been motivated because of their growing applications in electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science [
In numerous physical models, an equation commonly used to describe transport diffusive problems is the classical advectiondiffusion (or dispersion) equation which may be generalized to the fractional ones to cover other very interesting physical models. The advectiondispersion equation, which is based on Fick's law, is commonly used to simulate contaminant transport in porous media [
In the last few years, theory and numerical analysis of fractional partial differential equations have received an increasing attention. In this direction, Rihan [
The main purpose of the this paper is to construct the solution of a class of space fractional advectiondispersion equation with variable coefficients using JacobiGaussLobatto collocation (JGLC) approximation, based on JacobiGaussLobatto quadrature knots, combined with an implicit iterative method for treating the time discretization. More precisely, implementing the JGLC approximation to the spatial variable of the fractional advectiondispersion equation and the corresponding boundary conditions reduces the problem to the time integration of a system of ordinary differential equations in respect to the time variable. To the best of our knowledge, such algorithm has not been implemented for solving space fractional initialboundary problems.
The plan of the paper is as follows. In the next section, we introduce basic properties of Jacobi polynomials. In Section
In this section, we give some definitions and properties of the fractional calculus (see, e.g., [
For
Form (
Let
For integer
Let
With the aid of (
Next, let
The set of shifted Jacobi polynomials is a complete
Since the Jacobi spectral collocation method approximates the initialboundary problems in physical space and it is a global method, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear problems (see, for instance, [
In this section, we consider the space fractional advectiondispersion equations with space and time variable coefficients [
Now we introduce the JacobiGaussLobatto quadratures in two different intervals
We define the discrete inner product and norm as follows:
Associating with this quadrature rule, we denote by
We now derive an efficient algorithm for solving spacefractional advection diffusion equation (
The first order spatial derivative of the spectral solution can be approximated by the JGLC points
According to
The fractional derivative of order
The spatial partial fractional derivatives of order
If we apply the JacobiGaussLobatto collocation method of (
Let us denote that
The system of ordinary differential equations (
In order to check the accuracy and reliability of the proposed algorithm, we present two numerical examples using the proposed algorithm. In the first example, we compute the space fractional diffusion equation to check the accuracy, and space fractional advectiondispersion equation with variable coefficients is solved in the second example which confirms the good accuracy of our method. Comparing the results obtained by various choices of Jacobi parameters
Consider the space fractional diffusion equation (see, [
In Table
Comparing maximum absolute errors of the proposed method and [




CN [ 
Extra CN [ 
BEFD [ 

3 



—  —  — 
5 



—  —  — 
10 






15 






20 






25 



—  —  — 
We contrast our results with the corresponding results for the backward Euler finite difference scheme (BEFD [
In Figures
The comparison of the curves of analytical solutions and approximate solutions at
The comparison of the curves of analytical solutions and approximate solutions at
Consider the space fractional advectiondispersion equation with variable coefficients:
For the sake of comparison of some different values of Jacobi parameters
Comparing maximum absolute errors for different choices of







3 





6 





12 





18 





24 





In case of Chebyshev polynomials of the first kind
The spacetime graph of approximate solutions at
The spacetime graph of approximate solutions at
The comparison of the curves of analytical solutions and approximate solutions at
The comparison of the curves of analytical solutions and approximate solutions at
The obtained results of this example show that the Jacobi GaussLobatto collocation method is simple and very accurate for all values of
In this paper, we have proposed the Jacobi GaussLobatto collocation spectral approximation for tackling fractionalinspace advectiondispersion equation subject to initialboundary conditions. Applying the collocation method has reduced the problem to system of ordinary differential equations in time. This system may be solved by an implicit iterative technique. One of the main advantages of the proposed method is the Legendre GaussLobatto collocation approximation, and the four kinds of Chebyshev GaussLobatto collocation approximations may be obtained as special cases of the proposed Jacobi GaussLobatto collocation approximation by taking the corresponding special cases of the Jacobi parameters
The implementation of Jacobi GaussLobatto collocation spectral approximation for timespace fractional advectiondispersion equations may also constitute another line of our future lines of research. We also conclude that this algorithm can be useful in dealing with coupled nonlinear partial differential equations.
The author declares that there is no conflict of interests regarding the publication of this paper.
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under Grant no. (130141D1433). The author, therefore, acknowledges the DSR technical and financial support.