A reliable method for the space-time fractional Burgers and time-fractional Cahn-Allen equations via the FRDTM
- Mahmoud S Rawashdeh^{1}Email author
DOI: 10.1186/s13662-017-1148-8
© The Author(s) 2017
Received: 9 February 2017
Accepted: 21 March 2017
Published: 31 March 2017
Abstract
We propose a new method called the fractional reduced differential transform method (FRDTM) to solve nonlinear fractional partial differential equations such as the space-time fractional Burgers equations and the time-fractional Cahn-Allen equation. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers and Cahn-Allen equations to show the nature of solutions as the fractional derivative parameter is changed. The results prove that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.
Keywords
fractional reduced differential transform method Caputo fractional derivative Burgers equations Cahn-Allen equationMSC
35J05 35J10 35A22 65M12 65M15 65M201 Introduction
The space-fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. They are also connected with applications in acoustic phenomena and have been used to model turbulence and certain steady-state viscous flows. Moreover, Burgers equations are used to model the formation and decay of nonplanar shock waves, where the variable x is a coordinate moving with the wave at the speed of sound and the dependent variable u represents the velocity fluctuations. The Burgers equations occur in various areas of applied sciences and physical applications, such as modeling of fluid mechanics and financial mathematics, and the equation has still interesting applications in physics and astrophysics.
The fractional differential equations (FDE) appear more and more frequently in different research areas and engineering applications. There are many physical applications in science and engineering that can be represented by models using fractional differential equations [1–10], which are quite useful for many physical problems. These equations are represented by fractional linear and nonlinear PDEs, and solving such fractional differential equations is very important [11–21].
Many approximation and numerical techniques have been used to solve fractional differential equations [12, 16, 22–25]. Lately, many new approaches to fractional differential equations have been proposed, a few of these methods are as follows: the fractional differential transform method (FDTM) [25–28], the fractional Adomian decomposition method (FADM) [2], the fractional variational iteration method (FVIM) [4], the fractional sub-equation method [23], the fractional natural decomposition method [17, 29] and the fractional homotopy perturbation method (FHPM) [22, 30]. Kurulay [26] found approximate and exact solutions of the space- and time-fractional Burgers equations. Bekir et al. [23] found exact solutions of the time-fractional Cahn-Allen equation. Khan et al. [22] used the generalized differential transform method (GDTM) and the homotopy perturbation method (HPM) to solve the time-fractional Burgers and coupled Burgers equations. Recently, Rawashdeh [16, 31] used the FRDTM to solve nonlinear fractional partial differential equations.
The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. Note that we call Eq. (1.1) the time-fractional Burgers and the space-fractional Burgers equation in the case \(0<\alpha\le1, \eta=0\) and \(0<\beta\le1, \alpha=1\), respectively.
2 Background of fractional calculus
Here are some definitions and facts that we shall use in our work. Some of these basic definitions are due to Liouville [3, 4, 33, 34].
Definition 2.1
A real function \(f(x)\), \(x>0\), is said to be in the space \(C_{\mu }\), \(\mu\in{\mathbb {R}}\), if there exists a real number \(q(>\mu )\) such that \(f(x)=x^{q} g(x)\), where \(g(x)\in C [0,\infty )\), and it is said to be in the space \(C_{\mu}^{m} \) if \(f^{(m)} \in C_{\mu}, m\in{\mathbb {N}}\).
Definition 2.2
Definition 2.3
Lemma 2.4
[6]
We use the Caputo fractional derivative because it allows traditional initial and boundary conditions to be included in the formulation of our work.
3 Analysis of the FRDTM
Definition 3.1
Basic operations of the FRDTM [ 39 ]
Functional form | Transformed form |
---|---|
u(x,t) | \(\frac{1}{\Gamma(k\alpha+1)} [\frac{\partial^{\alpha k } }{\partial t^{\alpha k} } u(x,t) ]_{ t=0} \) |
γu(x,t)±βv(x,t) | \(\gamma U_{k} (x)\pm \beta V_{k} (x)\), where γ and β are constants |
u(x,t).v(x,t) | \(\sum_{i=0}^{k}U_{i} (x)V_{k-i} (x) \) |
u(x,t).v(x,t).w(x,t) | \(\sum_{i=0}^{k}\sum_{j=0}^{i}U_{j} (x) V_{i-j} (x) W_{k-i} (x)\) |
\(\frac{\partial^{n\alpha} }{\partial t^{n\alpha } } u(x,t)\) | \(\frac{\Gamma (k\alpha+n\alpha +1 )}{\Gamma (k\alpha+1 )} U_{k+n} (x)\) |
\(\frac{\partial^{n} }{\partial x^{n} } u(x,t)\) | \(\frac{\partial^{n} }{\partial x^{n} } U_{k}(x)\) |
\(x^{m} t^{n} u(x,t)\) | \(x^{m} U_{k-n} (x )\) |
\(x^{m} t^{n} \) | \(x^{m} \delta (k\alpha-n )\), where \(\delta(k\alpha -n)=\left \{ \begin{array}{l@{\quad}l} 1, & \alpha k=n \\ 0, & \alpha k\ne n \end{array} \right \}\) |
Remark 3.2
In Table 1, Γ represents the gamma function, where \(\Gamma (z+1)=z\Gamma(z), z>0\).
3.1 Methodology
4 Worked examples
We shall employ the FRDTM to three different applications to illustrate the accuracy and efficiency of the method.
4.1 The time-fractional Burgers equation
Remark 4.1
4.2 The space-fractional Burgers equation
4.3 The time-fractional Cahn-Allen equation
Remark 4.3
5 Tables of numerical calculations
The results obtained by the FRDTM for different values of α for Example 4.1
x | t | α = 0.25 | α = 0.5 | α = 0.75 | α = 1 | |
---|---|---|---|---|---|---|
Numerical | Numerical | Numerical | Numerical | Exact | ||
−10 | 2 | 1 | 1 | 1 | 1 | 1 |
4 | 1 | 1 | 1 | 1 | 1 | |
6 | 1 | 1 | 1 | 1 | 1 | |
8 | 1 | 1 | 1 | 1 | 1 | |
−5 | 2 | 1.00014 | 1.00014 | 1 | 1 | 1 |
4 | 1.02059 | 1.00261 | 1.00001 | 1 | 1 | |
6 | 1.37155 | 1.02059 | 1.00014 | 1.00002 | 1 | |
8 | 3.86303 | 1.02059 | 1.00112 | 1.00014 | 1 | |
5 | 2 | 0.200786 | 0.20001 | 0.200001 | 0.2 | 0.2 |
4 | 0.280682 | 0.200786 | 0.200059 | 0.20001 | 0.20004 | |
6 | 1.48828 | 0.211558 | 0.200786 | 0.200123 | 0.204848 | |
8 | 9.53999 | 0.280682 | 0.205256 | 0.200786 | 0.540446 | |
10 | 2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |
4 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | |
6 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | |
8 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |
The results obtained by the FRDTM for different values of β for Example 4.2
x | t | β = 0.25 | β = 0.5 | β = 0.75 | β = 1 | |
---|---|---|---|---|---|---|
Numerical | Numerical | Numerical | Numerical | Exact | ||
0.1 | 2 | −0.0764845 | −0.0764173 | −0.0765568 | −0.076893 | −0.073117 |
3 | −0.0224361 | −0.0224164 | −0.0224575 | −0.0225564 | −0.0214477 | |
4 | −0.00609528 | −0.00608992 | −0.00610107 | −0.00612793 | −0.00582667 | |
6 | 0.00309842 | 0.00309573 | 0.00310145 | 0.00311518 | 0.00296201 | |
0.3 | 2 | −0.243822 | −0.24332 | −0.244843 | −0.248451 | −0.21341 |
3 | −0.0721999 | −0.0720561 | −0.0725197 | −0.0736135 | −0.0631537 | |
4 | −0.0195606 | −0.019521 | −0.0196462 | −0.019942 | −0.0171018 | |
6 | 0.0102373 | 0.0102193 | 0.0102888 | 0.0104509 | 0.008961 | |
0.5 | 2 | −0.419375 | −0.418177 | −0.4228 | −0.433774 | −0.336983 |
3 | −0.126549 | −0.126227 | −0.127726 | −0.131255 | −0.101393 | |
4 | −0.034099 | −0.0340076 | −0.0344084 | −0.0353558 | −0.0272609 | |
6 | 0.0188912 | 0.0188622 | 0.0191169 | 0.0197044 | 0.0151832 | |
0.6 | 2 | −0.505904 | −0.504266 | −0.511001 | −0.527077 | −0.389893 |
3 | −0.154611 | −0.154193 | −0.156467 | −0.161845 | −0.118574 | |
4 | −0.0414992 | −0.041377 | −0.0419815 | −0.0434176 | −0.0317049 | |
6 | 0.0238959 | 0.0238714 | 0.0242898 | 0.0252544 | 0.0184231 |
The results obtained by the FRDTM for different values of α for Example 4.3
x | t | α = 0.25 | α = 0.5 | α = 0.75 | α = 1 | |
---|---|---|---|---|---|---|
Numerical | Numerical | Numerical | Numerical | Exact | ||
0.1 | 0.002 | 0.768025 | 0.515858 | 0.487054 | 0.483079 | 0.483079 |
0.003 | 0.79101 | 0.523384 | 0.488734 | 0.483453 | 0.483453 | |
0.004 | 0.807412 | 0.529721 | 0.490276 | 0.483828 | 0.483828 | |
0.006 | 0.830055 | 0.540329 | 0.493102 | 0.484577 | 0.484577 | |
0.3 | 0.002 | 0.7419 | 0.480518 | 0.451845 | 0.447907 | 0.447907 |
0.003 | 0.766709 | 0.488048 | 0.453511 | 0.448278 | 0.448278 | |
0.004 | 0.784528 | 0.494401 | 0.455041 | 0.448649 | 0.448649 | |
0.006 | 0.809302 | 0.505062 | 0.457846 | 0.449391 | 0.449391 | |
0.5 | 0.002 | 0.713958 | 0.445372 | 0.417112 | 0.413248 | 0.413248 |
0.003 | 0.740586 | 0.452832 | 0.418748 | 0.413612 | 0.413612 | |
0.004 | 0.759881 | 0.459137 | 0.420251 | 0.413976 | 0.413976 | |
0.006 | 0.787034 | 0.469743 | 0.423008 | 0.414704 | 0.414704 | |
0.6 | 0.002 | 0.699334 | 0.427979 | 0.400028 | 0.396214 | 0.396214 |
0.003 | 0.726851 | 0.435377 | 0.401642 | 0.396573 | 0.396573 | |
0.004 | 0.746891 | 0.441635 | 0.403127 | 0.396932 | 0.396932 | |
0.006 | 0.775296 | 0.452175 | 0.40585 | 0.397651 | 0.397651 |
6 Conclusion
In this paper, we successfully implemented the FRDTM to find approximate solutions of the space-time fractional Burgers equations and the time-fractional Cahn-Allen equation for different values of α and β and the results we obtained in Examples 4.1, 4.2 and 4.3 were in excellent agreement with the exact solutions. The FRDTM introduces a significant improvement in the field over the existing methods.
Declarations
Acknowledgements
The author would like to thank the editor and the anonymous referees for their comments and suggestions on this paper.
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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