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- Open Access
Integral spectral Tchebyshev approach for solving space Riemann-Liouville and Riesz fractional advection-dispersion problems
- Eid H Doha^{1},
- Waleed M Abd-Elhameed^{1, 2},
- Nermeen A Elkot^{1} and
- Youssri H Youssri^{1}Email author
https://doi.org/10.1186/s13662-017-1336-6
© The Author(s) 2017
- Received: 5 June 2017
- Accepted: 26 August 2017
- Published: 13 September 2017
Abstract
The principal aim of this paper is to analyze and implement two numerical algorithms for solving two kinds of space fractional linear advection-dispersion problems. The proposed numerical solutions are spectral and they are built on assuming the approximate solutions to be certain double shifted Tchebyshev basis. The two typical collocation and Petrov-Galerkin spectral methods are applied to obtain the desired numerical solutions. The special feature of the two proposed methods is that their applications enable one to reduce, through integration, the fractional problem under investigation into linear systems of algebraic equations, which can be efficiently solved via any suitable solver. The convergence and error analysis of the double shifted Tchebyshev basis are carefully investigated, aiming to illustrate the correctness and feasibility of the proposed double expansion. Finally, the efficiency, applicability, and high accuracy of the suggested algorithms are demonstrated by presenting some numerical examples accompanied with comparisons with some other existing techniques discussed in the literature.
1 Introduction
Fractional calculus is a very important branch of mathematical analysis. This branch is basically interested in investigating the properties of derivatives and integrals of non-integer orders (called fractional derivatives and integrals). Fractional ordinary differential equations (FODEs) and fractional partial differential equations (FPDEs) have attracted considerable interest from a large number of researchers due to their ability to model a lot of phenomena in engineering, control theory, chemical physics, stochastic processes, anomalous diffusion, rheology, biology, and other sciences, such as medicine and neuronal dynamics. Due to the growing interest in these kinds of differential equations, obtaining approximate solutions of them is of fundamental importance and hence it is very useful to develop numerical techniques for solving these types of equations.
The approach of employing spectral methods is very effective in handling ordinary differential equations as well as fractional differential equations. This approach is basically built on assuming the approximate solutions to be linear combinations of certain basis functions. These basis functions may be orthogonal or nonorthogonal. Approximations by orthogonal basis functions occupy a considerable part in the literature. In fact, there are three commonly used spectral methods, namely the collocation, the tau, and the Galerkin method and its variants. The collocation method is very effective in a wide range of practical problems, particularly in the nonlinear ones (see for example [1]). The tau method is useful for treating boundary value problems of complicated boundary conditions (see for example [2]). In the Galerkin and Petrov-Galerkin methods, one chooses two sets of basis functions, which are called ‘trial functions’ and ‘test functions’. These two sets are identical in the Galerkin method, however, in the Petrov-Galerkin method, they are not identical, so the main advantage of employing the Petrov-Galerkin method is its flexibility in choosing test functions. The Galerkin and Petrov-Galerkin methods have been applied successfully in various situations. For example, the authors in [3, 4] have constructed efficient spectral Galerkin algorithms and Petrov-Galerkin algorithms for handling various even- and odd-order boundary value problems. One of the advantages of the Galerkin and Petrov-Galerkin methods is that they enable one to investigate carefully the algebraic systems resulting from their applications and their structures and complexities.
There are three types of FPDEs, namely space FPDEs, time FPDEs and space-time FPDEs. One of the most important fractional differential equation is the fractional advection-diffusion equation (FADE). Due to the importance of this equation (see, for example, [5]), a variety of papers with different numerical techniques have been proposed to handle it. For example, Shen et al. in [6] derived the fundamental solution for the space-time Riesz-Caputo FADE with an initial condition. The authors in [7] have derived some analytical solutions for the multi-term time-space Caputo-Riesz FADEs on a finite domain. The Adomian decomposition method is employed in [8] for solving an intermediate fractional advection-dispersion equation. The finite element method is used to handle the fractional advection-dispersion equation.
In this study, we are concerned with introducing numerical integral spectral solutions for two kinds of the FADEs. We apply the collocation and Petrov-Galerkin methods for this purpose. The main idea behind the proposed approach is to solve the integrated form of the equation. The advantage of using this approach is that its use enables one to reduce the solution of the equation with its boundary and initial conditions into a system of linear or nonlinear algebraic equations. The linear system can be efficiently solved using the Gaussian elimination solver or by any other suitable solver, while the nonlinear system can be solved with the aid of Newton’s iterative method.
The paper is organized as follows. In the next section, some necessary definitions and mathematical preliminaries of fractional calculus are presented. Besides, some properties of shifted Tchebyshev polynomials are given. Sections 3 and 4 are devoted to solving FADEs by implementing and presenting two numerical algorithms based on the application of the collocation and Petrov-Galerkin methods. Section 5 focuses on investigating the convergence and error analysis of the suggested Tchebyshev double expansion. The numerical results and comparisons are displayed in Section 6. Finally, some conclusions are reported in Section 7.
2 Some fundamental properties of fractional calculus
This section is devoted to presenting some fundamental definitions and preliminary facts of the fractional calculus theory. For fundamentals of this branch, the reader is referred to [9].
Definition 1
Definition 2
Definition 3
Property 1
See [10]
Definition 4
2.1 Some properties of shifted Tchebyshev polynomials
3 Integral transforms for two kinds of the space fractional advection-dispersion equations
This section is devoted to transforming two kinds of space FADEs into their integral forms in order to treat them numerically by our proposed techniques in the upcoming section.
3.1 The first kind of space fractional advection-dispersion problem
Note 1
3.2 The second kind of space fractional advection-dispersion problem
Note 2
4 Numerical spectral treatment for two kinds of space fractional linear advection-dispersion problems
This section is concerned with explaining in detail two spectral algorithms for numerically solving two kinds of space fractional linear advection-dispersion problems. First, we select a unified double Tchebyshev expansion as basis functions, and then apply the two well-known spectral methods, namely the collocation and Petrov-Galerkin methods.
4.1 Choice of the basis functions
Theorem 1
Proof
Note 3
By writing \(a_{k}\lessapprox b_{k}\), we mean that there exists a generic constant C such that \(a_{k}< C\, b_{k}\) for large k.
Lemma 1
Proof
Lemma 2
Proof
The proof of this lemma is similar to the proof of Lemma 1. □
Theorem 2
Proof
Based on the two analytic forms of \(T_{i}^{\ell}(x)\) given in (2.11) and (2.12) and if we perform similar manipulations as in the proof of Theorem 1, then we get the desired result. □
Similarly we can prove the following estimates for the Riesz fractional derivatives of \(\phi_{i}(x)\).
Lemma 3
Lemma 4
Theorem 3
Proof
The proof is easily obtained by integrating relation (2.11) over the interval \([0,t]\). □
4.2 Numerical algorithms for handling equation (3.11)
This section is devoted to describing in detail two numerical algorithms for handling equation (3.11). The first algorithm depends on the application of the typical collocation method, while the second depends on the application of the Petrov-Galerkin method. The main idea behind the two proposed algorithms is based on making use of Theorems 1 and 3 along with the application of the collocation and Petrov-Galerkin methods in order to transform equation (3.11) into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients \(c_{ij}\). The linear system is solved using Gauss elimination, and the nonlinear one is solved via Newton’s iterative method.
4.2.1 The collocation approach
4.2.2 Petrov-Galerkin approach
4.3 Numerical algorithms for handling equation (3.23)
In this section, we describe how the collocation and Petrov-Galerkin procedures can be employed to handle equation (3.23). As we have done in Section 4.2.1, collocation and Petrov-Galerkin are employed along with Theorems 2 and 3 to convert equation (3.23) into a system of linear algebraic equations in the unknown expansion coefficients \(c_{ij}\).
4.3.1 The collocation approach
4.3.2 Petrov-Galerkin approach
5 Convergence and error analysis of the suggested double expansion
In this section we concentrate on investigating the convergence and error analysis of the suggested double Tchebyshev expansion. Three important theorems are stated and proved for this purpose. The first theorem shows that the double Tchebyshev expansion of a function \(u(x,t)\) converges uniformly to \(u(x,t)\). The second and third theorems discuss the error analysis of the full discretization scheme for the two problems (3.5) and (3.17).
First, the following lemma is useful.
Lemma 5
See [17], p.742
- 1.
\(u(x)\) is a continuous, positive, decreasing function for \(x\geq n\).
- 2.
\(\sum a_{n}\) is convergent, and \(R_{n}=\sum_{k=n+1}^{\infty}a_{k}\).
Theorem 4
Proof
Theorem 5
Proof
Theorem 6
Proof
The steps of the proof are similar to those followed in the proof of Theorem 5, but based on Lemmas 3 and 4 instead of Lemmas 1 and 2. □
Now, the following theorem investigates the stability of the suggested double Tchebyshev expansion.
Theorem 7
Under the assumptions of Theorem 4, we have \(\|g_{M+1}(x,t)-g_{M}(x,t)\|_{\omega }\lessapprox M^{-4}\).
Proof
6 Numerical results and comparisons
This section is devoted to presenting some numerical results obtained by the application of the two proposed numerical methods. All the results are obtained using the software Mathematica 9. The results are accompanied by a comparison with results from the literature, obtained by applying some other numerical techniques.
Example 1
Maximum pointwise error of Example 1
M | γ | \(\boldsymbol {E_{C}}\) | \(\boldsymbol{E_{\mathrm{PG}}}\) | γ | \(\boldsymbol {E_{C}}\) | \(\boldsymbol{E_{\mathrm{PG}}}\) | γ | \(\boldsymbol {E_{C}}\) | \(\boldsymbol{E_{\mathrm{PG}}}\) |
---|---|---|---|---|---|---|---|---|---|
4 | 1.5 | 2.1⋅10^{−4} | 8.5⋅10^{−5} | 1.8 | 6.2⋅10^{−4} | 3.4⋅10^{−5} | 1.9 | 2.7⋅10^{−4} | 8.6⋅10^{−5} |
5 | 3.8⋅10^{−6} | 5.2⋅10^{−7} | 5.4⋅10^{−6} | 3.6⋅10^{−7} | 9.3⋅10^{−6} | 8.1⋅10^{−7} | |||
6 | 3.3⋅10^{−8} | 2.7⋅10^{−9} | 3.9⋅10^{−8} | 1.9⋅10^{−9} | 6.1⋅10^{−8} | 9.7⋅10^{−9} |
Example 2
From the results in Table 3, we conclude that our spectral method is more accurate than the fractional difference method used in [19].
Example 3
Maximum pointwise error of Example 3 ( \(\pmb{M=7}\) )
( γ , σ ) | \(\boldsymbol{E_{C}}\) | \(\boldsymbol {E_{\mathrm{PG}}}\) | ( γ , σ ) | \(\boldsymbol {E_{C}}\) | \(\boldsymbol{E_{\mathrm{PG}}}\) | ( γ , σ ) | \(\boldsymbol{E_{C}}\) | \(\boldsymbol{E_{\mathrm{PG}}}\) |
---|---|---|---|---|---|---|---|---|
(1.4,0.2) | 3.7⋅10^{−9} | 4.2⋅10^{−9} | (1.5,0.6) | 5.7⋅10^{−9} | 8.2⋅10^{−9} | (1.7,0.4) | 6.1⋅10^{−9} | 8.4⋅10^{−9} |
Comparison between different errors of Example 3 ( \(\pmb{t=2}\) )
x | \(\boldsymbol{E_{C}}\) ( M = 7) | \(\boldsymbol{E_{\mathrm{PG}}}\) ( M = 7) | [ 14 ] ( M = 7, τ = 0.002) |
---|---|---|---|
0.1 | 1.7⋅10^{−9} | 7.5⋅10^{−9} | 0.94⋅10^{−7} |
0.3 | 4.5⋅10^{−9} | 5.9⋅10^{−9} | 5.48⋅10^{−7} |
0.5 | 3.8⋅10^{−9} | 6.2⋅10^{−9} | 7.02⋅10^{−7} |
0.7 | 5.4⋅10^{−9} | 1.4⋅10^{−9} | 2.89⋅10^{−6} |
0.9 | 1.88⋅10^{−9} | 7.2⋅10^{−9} | 6.06⋅10^{−6} |
Example 4
Comparison between different errors of Example 4 for the case \(\pmb{M=9}\)
x | PGCCM | [ 20 ] | [ 21 ] | ||||||
---|---|---|---|---|---|---|---|---|---|
ν = 1.2 | ν = 1.5 | ν = 1.8 | ν = 1.2 | ν = 1.5 | ν = 1.8 | ν = 1.2 | ν = 1.5 | ν = 1.8 | |
0.2 | 1⋅10^{−12} | 2⋅10^{−12} | 1⋅10^{−12} | 2⋅10^{−12} | 4⋅10^{−11} | 4⋅10^{−11} | 2⋅10^{−3} | 1⋅10^{−3} | 7⋅10^{−4} |
0.4 | 3⋅10^{−12} | 5⋅10^{−12} | 1⋅10^{−12} | 1⋅10^{−10} | 9⋅10^{−11} | 1⋅10^{−10} | 1⋅10^{−3} | 1⋅10^{−3} | 6⋅10^{−4} |
0.6 | 4⋅10^{−12} | 6⋅10^{−12} | 1⋅10^{−12} | 2⋅10^{−10} | 2⋅10^{−10} | 1⋅10^{−10} | 9⋅10^{−4} | 7⋅10^{−4} | 4⋅10^{−4} |
0.8 | 2⋅10^{−12} | 1⋅10^{−12} | 2⋅10^{−12} | 1⋅10^{−10} | 2⋅10^{−10} | 2⋅10^{−10} | 8⋅10^{−4} | 6⋅10^{−4} | 3⋅10^{−4} |
1.0 | 7⋅10^{−12} | 3⋅10^{−12} | 2⋅10^{−12} | 2⋅10^{−10} | 2⋅10^{−10} | 2⋅10^{−10} | 7⋅10^{−4} | 6⋅10^{−4} | 3⋅10^{−4} |
1.2 | 4⋅10^{−12} | 2⋅10^{−12} | 2⋅10^{−12} | 2⋅10^{−10} | 2⋅10^{−10} | 3⋅10^{−10} | 8⋅10^{−4} | 6⋅10^{−4} | 3⋅10^{−4} |
1.4 | 2⋅10^{−12} | 3⋅10^{−12} | 2⋅10^{−12} | 2⋅10^{−10} | 2⋅10^{−10} | 2⋅10^{−10} | 9⋅10^{−4} | 7⋅10^{−4} | 4⋅10^{−4} |
1.6 | 3⋅10^{−12} | 3⋅10^{−12} | 1⋅10^{−12} | 5⋅10^{−11} | 1⋅10^{−10} | 6⋅10^{−11} | 1⋅10^{−3} | 1⋅10^{−3} | 6⋅10^{−4} |
1.8 | 5⋅10^{−12} | 4⋅10^{−12} | 3⋅10^{−12} | 3⋅10^{−11} | 4⋅10^{−11} | 3⋅10^{−11} | 2⋅10^{−3} | 1⋅10^{−3} | 7⋅10^{−4} |
Max | 7⋅10^{−12} | 6⋅10^{−12} | 3⋅10^{−12} | 2⋅10^{−10} | 2⋅10^{−10} | 3⋅10^{−10} | - | - | - |
Example 5
Pointwise error of Example 5 for the case \(\pmb{M=14}\) at \(\pmb{t=0.3}\)
x | β = 0.7, γ = 1.7 | β = 0.8, γ = 1.8 | β = 0.9, γ = 1.9 |
---|---|---|---|
0.0 | 0 | 0 | 0 |
0.2 | 2.2⋅10^{−12} | 7.8⋅10^{−13} | 3.5⋅10^{−15} |
0.4 | 2.3⋅10^{−12} | 2.5⋅10^{−13} | 5.7⋅10^{−15} |
0.6 | 4.7⋅10^{−12} | 3.1⋅10^{−13} | 7.4⋅10^{−15} |
0.8 | 4.1⋅10^{−12} | 2.4⋅10^{−13} | 4.2⋅10^{−15} |
1.0 | 0 | 0 | 0 |
7 Concluding remarks
FADEs are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium. In this research article, two efficient spectral methods are presented and analyzed to solve two kinds of space fractional linear advection-dispersion problems. The spectral collocation and Petrov-Galerkin methods are employed to obtain semi-analytic solutions for the FADE. Efficient and highly accurate solutions are obtained with a small number of retained modes.
Declarations
Acknowledgements
The authors would like to thank the referees and the editor for their constructive comments, which helped substantially to improve the manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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