On a hybrid spectral exponential Chebyshev method for time-fractional coupled Burgers equations on a semi-infinite domain
- Basim Albuohimad^{1, 2} and
- Hojatollah Adibi^{2}Email author
https://doi.org/10.1186/s13662-017-1141-2
© The Author(s) 2017
Received: 8 December 2016
Accepted: 6 March 2017
Published: 20 March 2017
Abstract
In this study we propose a hybrid spectral exponential Chebyshev method (HSECM) for solving time-fractional coupled Burgers equations (TFCBEs). The method is based upon a spectral collection method, utilizing exponential Chebyshev functions in space and trapezoidal quadrature formula (TQF), and also a finite difference method (FDM) for time-fractional derivative. Some test examples are included to demonstrate the efficiency and validity of the proposed method.
Keywords
1 Introduction
Several computational problems in various research areas such as mathematics, fluid dynamics, chemistry, biology, viscoelasticity, engineering and physics have arisen in semi-infinite domains [1–9]. Subsequently, many researchers have utilized various transformations on orthogonal polynomials to map \([ -1, 1]\) into \([0, \infty)\) maintaining their orthogonal property [10–15].
Spectral methods provide a computational approach that has become better known over the last decade and has become the topic of study for many researchers [16–26], especially when linked with the fractional calculus [9, 27–38] which is an important branch of applied mathematics. This type of differentiation and integration could be considered as a generalization of the usual definition of differentiation and integration to non-integer order.
The coupled Burgers equations have recently been applied to different areas of science, in particular in physical problems such as the phenomena of turbulence flow through a shock wave traveling in a viscous fluid (see [39, 40]).
The study of coupled Burgers equations is very important because the system is a basic model of sedimentation or evolution of scaled volume concentrations of two sorts of particles in liquid suspensions or colloids under the impact of gravity [40]. It has been studied by many authors using various techniques [41–46].
In this paper, we introduce the exponential Chebyshev functions collocation method based upon orthogonal Chebyshev polynomials to solve a time-fractional coupled Burgers equation. The fractional derivative is defined in the Caputo sense for time variable which is discretized utilizing a trapezoidal quadrature formula (TQF) and a finite difference method (FDM).
The justification of this paper is to apply the Chebyshev exponential method for efficient applicable in unbounded domains with steady state property (\(u(\infty)=\mathit{constant}\)), i.e., the solution to be regular at ∞. In fact, many problems in mathematical physics and astrophysics which occur on a semi-infinite interval are related to the diffusion equations such as Burgers, KdV and heat equations. Furthermore, many methods based on polynomials basis, such as Legendre, Chebyshev, Laguerre spectral methods and also semi-analytic methods such as Adomian decomposition, variational iteration and differential transform methods, can not justify the steady state property of fluid \(u(\infty )=\mathit{constant}\). In this study we will show that such difficulty can be surmounted by our proposed method.
The error analysis of exponential Chebyshev functions expansion has also been investigated, which confirms the efficiency of the method.
2 Definitions and basic properties
In this section, we give some definitions and basic properties of fractional calculus and Chebyshev polynomials which are required for our subsequent development.
2.1 Definition of fractional calculus
Here we recall definition and basic results of fractional calculus; for more details, we refer to [32].
Definition 1
A real function \(u(t)\), \(t>0\) is said to be in the space \(C_{\mu}\), \(\mu\in\mathbb{R}\) if there exists a real number \(p>\mu\) such that \(u(t)=t^{p}u_{1}(t)\), where \(u_{1}(t)\in C(0,\infty)\), and it is said to be in the space \(C_{\mu}^{n}\) if and only if \(u^{(n)}\in C_{\mu}\), \(n\in\mathbb{N}\).
Definition 2
Definition 3
2.2 Exponential Chebyshev functions
Definition 4
Exponential Chebyshev functions
3 Function approximation
Lemma 1
Proof
This lemma shows that the convergence of exponential Chebyshev functions approximation is involved with the function \(u(x)\). Now, by knowing that the function \(u(x)\in L^{2}_{\rho}(\Lambda)\) has some good properties, we can present an upper bound for estimating the error of function approximation by these basis functions.
Theorem 1
Proof
From the previous theorem, any real function defined in \(L^{2}_{\rho}(\Lambda)\), whose mapping under the transformation \(-L\ln(\frac{1-s}{2})\) is analytic, has a convergence series solution in the form (18). Furthermore, we can show that the error defined in (19) has superlinear convergence defined below.
Definition 5
Theorem 2
In Theorem 1, let \(M\geq M_{i}\) for any integer i, then the error is superlinear convergence to zero.
Proof
According to Theorem 2, any function \(u(x)\in L^{2}_{\rho}(\Lambda )\) that is analytic under the transformation \(x=-L\ln(\frac{1-s}{2})\) has a superlinear convergence series in the form (16).
4 Spectral collection method to solve TFCBEs
4.1 Trapezoidal quadrature formula
4.2 Finite difference approximations for time-fractional derivative
In this section, a fractional order finite difference approximation [27, 49] for the time-fractional partial differential equations is proposed.
5 Numerical experiments
In this section, we present four examples to illustrate the numerical results.
Example 1
TQF implementation
FDM implementation
Example 2
Example 2: Maximum absolute errors \(\pmb{e_{m,\infty}}\) with \(\pmb{m=5 }\) , \(\pmb{\alpha=0.4 }\) , \(\pmb{\beta =0.4}\) and \(\pmb{L=3}\)
τ | TQF | \(O(\tau^{2})\) | FDM | \(O(\tau^{2-\alpha})\) |
---|---|---|---|---|
\(e_{5,\infty}(u)=e_{5,\infty}(v)\) | \(e_{5,\infty}(u)=e_{5,\infty }(v)\) | |||
0.015625 | 1.62969572 × 10^{−3} | 1.58537183 × 10^{−2} | ||
0.0078125 | 4.07935306 × 10^{−4} | 1.99 | 5.42638453 × 10^{−3} | 1.54 |
0.00390625 | 9.40924121 × 10^{−5} | 2.01 | 1.82754471 × 10^{−3} | 1.57 |
0.001953125 | 1.58381076 × 10^{−5} | 2.07 | 6.04886113 × 10^{−4} | 1.55 |
Example 2: Maximum absolute errors \(\pmb{e_{m,\infty}}\) with \(\pmb{\tau=1/128}\) , \(\pmb{\alpha=0.4 }\) , \(\pmb{\beta=0.4}\) and \(\pmb{L=3}\)
m | TQF | FDM |
---|---|---|
\(e_{m,\infty}(u)=e_{m,\infty}(v)\) | \(e_{m,\infty}(u)=e_{m,\infty }(v)\) | |
3 | 3.21997018 × 10^{−4} | 3.02999632 × 10^{−3} |
4 | 2.32880457 × 10^{−4} | 2.48938916 × 10^{−3} |
5 | 1.24955778 × 10^{−4} | 1.51170703 × 10^{−3} |
Example 3
Example 3: Maximum absolute errors \(\pmb{e_{m,\infty}}\) with \(\pmb{m=5 }\) , \(\pmb{\alpha=0.5 }\) , \(\pmb{\beta =0.5}\) and \(\pmb{L=3}\)
τ | TQF | \(O(\tau^{2})\) | FDM | \(O(\tau^{2-\alpha})\) |
---|---|---|---|---|
\(e_{5,\infty}(u)=e_{5,\infty}(v)\) | \(e_{5,\infty}(u)=e_{5,\infty }(v)\) | |||
0.015625 | 1.21322402 × 10^{−4} | 1.23475331 × 10^{−3} | ||
0.0078125 | 3.17697252 × 10^{−5} | 1.95 | 4.67009965 × 10^{−4} | 1.46 |
0.00390625 | 8.15383818 × 10^{−6} | 1.96 | 1.71676167 × 10^{−4} | 1.45 |
0.001953125 | 2.07020075 × 10^{−6} | 1.97 | 6.21304589 × 10^{−5} | 1.48 |
Example 3: Maximum absolute errors \(\pmb{e_{m,\infty}}\) with \(\pmb{\tau=1/128}\) , \(\pmb{\alpha=0.5 }\) , \(\pmb{\beta=0.5}\) and \(\pmb{L=3}\)
m | TQF | FDM |
---|---|---|
\(e_{m,\infty}(u)=e_{m,\infty}(v)\) | \(e_{m,\infty}(u)=e_{m,\infty }(v)\) | |
3 | 4.04277852 × 10^{−5} | 5.75567882 × 10^{−4} |
4 | 2.66167043 × 10^{−5} | 4.15010485 × 10^{−4} |
5 | 1.85584712 × 10^{−5} | 4.08775869 × 10^{−4} |
Example 4
Example 4: Maximum absolute errors \(\pmb{e_{m,\infty}}\) with \(\pmb{\tau=1/64}\) , \(\pmb{\alpha=\beta=1}\) and \(\pmb{L=3}\)
m | TQF | FDM | ||
---|---|---|---|---|
u(x,t) | v(x,t) | u(x,t) | v(x,t) | |
5 | 2.56389687 × 10^{−3} | 2.56389687 × 10^{−3} | 2.12780543 × 10^{−3} | 2.12780543 × 10^{−3} |
7 | 1.19107232 × 10^{−3} | 1.19107232 × 10^{−3} | 1.06246855 × 10^{−4} | 1.06246855 × 10^{−4} |
9 | 6.89082786 × 10^{−4} | 6.89082786 × 10^{−4} | 1.71691547 × 10^{−6} | 1.71691547 × 10^{−6} |
11 | 5.71482080 × 10^{−4} | 5.71482080 × 10^{−4} | 1.98386814 × 10^{−8} | 1.98386814 × 10^{−8} |
Example 4: Absolute errors \(\pmb{\vert u_{7}-u_{6}\vert}\) and \(\pmb{\vert v_{7}-v_{6}\vert}\) with \(\pmb{\alpha=\beta=0.5}\) , in the final time
x | TQF | FDM | ||
---|---|---|---|---|
\(\vert u_{7}-u_{6}\vert\) | \(\vert v_{7}-v_{6}\vert\) | \(\vert u_{7}-u_{6}\vert\) | \(\vert v_{7}-v_{6}\vert\) | |
0.1 | 1.920638996 × 10^{−2} | 1.920638996 × 10^{−2} | 1.775781437 × 10^{−2} | 1.775781437 × 10^{−2} |
0.2 | 3.057241885 × 10^{−3} | 3.057241885 × 10^{−3} | 2.819522500 × 10^{−3} | 2.819522500 × 10^{−3} |
0.3 | 3.102165273 × 10^{−3} | 3.102165273 × 10^{−3} | 2.850429678 × 10^{−3} | 2.850429678 × 10^{−3} |
0.4 | 2.564731408 × 10^{−3} | 2.564731408 × 10^{−3} | 2.345949628 × 10^{−3} | 2.345949628 × 10^{−3} |
0.5 | 1.882240640 × 10^{−3} | 1.882240640 × 10^{−3} | 1.712524225 × 10^{−3} | 1.712524225 × 10^{−3} |
0.6 | 1.281492551 × 10^{−3} | 1.281492551 × 10^{−3} | 1.158901227 × 10^{−3} | 1.158901227 × 10^{−3} |
0.7 | 8.355109022 × 10^{−4} | 8.355109022 × 10^{−4} | 7.506877573 × 10^{−4} | 7.506877573 × 10^{−4} |
0.8 | 5.381590409 × 10^{−4} | 5.381590409 × 10^{−4} | 4.805394710 × 10^{−4} | 4.805394710 × 10^{−4} |
0.9 | 3.545769785 × 10^{−4} | 3.545769785 × 10^{−4} | 3.151826781 × 10^{−4} | 3.151826781 × 10^{−4} |
Example 5
Also, we can compare our results by the variational iteration method (VIM) [52] for different α and β. We report the results obtained by the proposed method and VIM [52] for \(u(x,t)\) at the final time \(T=1\) while \(\alpha=\beta=0.5\) in Figure 3 (right).
6 Conclusion
In this paper we presented a numerical method for solving the time-fractional Burgers equation by utilizing the exponential Chebyshev functions and TQF and FDM as well. Numerical results illustrate the validity and efficiency of the method and comparison for the maximum absolute errors between spectral collection method with TQF and FDM. This technique can be used to solve fractional time partial differential equations.
Declarations
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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