Analytical study of time-fractional Navier-Stokes equation by using transform methods
- Kangle Wang^{1}Email author and
- Sanyang Liu^{1}
https://doi.org/10.1186/s13662-016-0783-9
© Wang and Liu 2016
Received: 12 December 2015
Accepted: 8 February 2016
Published: 29 February 2016
Abstract
In this paper, we establish a modified reduced differential transform method and a new iterative Elzaki transform method, which are successfully applied to obtain the analytical solutions of the time-fractional Navier-Stokes equations. The obtained results show that the proposed techniques are simple, efficient, and easy to implement for fractional differential equations.
Keywords
Elzaki transform fractional Navier-Stokes equation reduced differential transform method1 Introduction
In recent years, the fractional differential equations have been used in various fields such as colored noise, electromagnetic waves, boundary layer effects in ducts, viscoelastic mechanics, diffusion processes, and so on [1–5]. However, most fractional differential equations are very difficult to exactly solve, so numerical and approximation techniques have to be used. Recently, many powerful methods have been used to approximate linear and nonlinear fractional differential equations. These methods include the Adomain decomposition method (ADM) [6, 7], the homotopy perturbation method (HPM) [8–12], the variational iteration method (VIM) [13, 14], and so on.
In this paper, we consider the unsteady flow of a viscous fluid in a tube, the velocity field is a function of only one space coordinate, the time is a dependent variable. This kind of time-fractional Navier-Stokes equation has been studied by Momani and Odibat [15], Kumar et al. [16, 17], and Khan [18] by using the Adomian decomposition method (ADM), the homotopy perturbation transform method (HPTM), the modified Laplace decomposition method (MLDM), the variational iteration method (VIM), and the homotopy perturbation method (HPM), respectively.
In 2006, Daftardar-Gejji and Jafari [19] were first to propose the Gejji-Jafari iteration method for solving a linear and nonlinear fractional differential equation. The Gejji-Jafari iteration method is easy to implement and obtains a highly accurate result. The reduced differential transform method (RDTM) was first proposed by Keskin and Oturanc [20, 21]. The RDTM was also applied by many researchers to handle nonlinear equations arising in science and engineering. In recent years, Kumar et al. [22–28] used various methods to study the solutions of linear and nonlinear fractional differential equation combined with a Laplace transform.
Based on the Gejji-Jafari iteration method and RDTM, we established the new iterative Elzaki transform method (NIETM) and the modified reduced differential transform method (MRDTM) with the help of the Elzaki transform [29, 30] and we successfully applied this to time-fractional Navier-Stokes equations. The results show that our proposed methods are efficient and easy to implement with less computation for fractional differential equations.
2 Basic definitions
In this section, we set up notation and review some basic definitions from fractional calculus and Elzaki transforms.
Definition 2.1
A real function \(f(x)\), \(x>0\), is said to be in the space \(C_{\mu}\), \(\mu\in{R}\) if there exists a real number p (\(p>\mu\)), such that \(f(x)=x^{p}f_{1}(x)\), where \(f_{1}(x)\in{C[0,\infty)}\), and it is said to be in the space \(C_{\mu}^{m}\) if \(f^{(m)}\in{C_{\mu}}\), \(m\in{N}\).
Definition 2.2
Definition 2.3
Definition 2.4
Lemma 2.1
3 New iterative Elzaki transform method (NIETM)
Similarly, for the proof of the convergence of the NIETM, see [19].
4 Modified reduced differential transform method (MRDTM)
In this section, the basic definition of the modified reduced differential transform method is introduced as in [31, 32].
Definition 4.1
Definition 4.2
According to (4.1) and (4.2), the following theorems can be obtained.
Theorem 4.1
If \(w(x,t)=u(x,t)\pm{v(x,t)}\), then \(MRDT[w(x,t)]=U_{k}(x)\pm{V_{k}(x)}\).
Theorem 4.2
If \(w(x,t)=\lambda{u(x,t)}\), then \(MRDT[w(x,t)]=\lambda{U_{k}(x)}\).
Theorem 4.3
Theorem 4.4
If \(w(x,t)=u(x,t)v(x,t)\), then \(MRDT[w(x,t)]=\sum^{k}_{r=0}U_{r}(x)V_{k-r}(x)\).
Theorem 4.5
If \(w(x,t)=\frac{\partial^{r}}{\partial{t^{r}}}u(x,t)\), then \(MRDT[w(x,t)]=\frac{(k+r)!}{k!}\frac{\partial^{r}}{\partial{t^{r}}}U_{k+r}(x)\).
Theorem 4.6
If \(w(x,t)=\frac{\partial^{N\alpha}}{\partial{t^{N\alpha}}}u(x,t)\), then \(MRDT[w(x,t)]=\frac{\Gamma(k\alpha+N\alpha+1)}{\Gamma(k\alpha +1)}U_{k+N}(x)\).
Theorem 4.7
5 Illustrative examples
Example 1
5.1 Applying the NIETM
Remark 5.1
The result is the same as ADM, HPTM, HPM, and VIM by Momani and Odibat [15], Kumar et al. [16, 17], and Khan [18].
Remark 5.2
5.2 Applying the MRDTM
Example 2
5.3 Applying the NIETM
5.4 Applying the MRDTM
Remark 5.5
We apply the NIETM and MRDTM to solve the time-fractional Navier-Stokes equations, and we get complete agreement with HPM, HPTM, ADM, and VIM. By comparing, NIETM and MRDTM are more easy to understand and implement than other methods with less computation.
Remark 5.6
6 Conclusion
In this paper, we apply the modified reduced differential transform method and new iterative Elzaki transform method for solving the time-fractional Navier-Stokes equation. The numerical results show that the MRDTM and NIETM are very powerful and efficient techniques for fractional differential equations.
Declarations
Acknowledgements
The authors are very grateful to the editor and referees for their insightful comments and suggestions. This work was supported by the National Natural Science Foundation of China (No.: 61373174).
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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