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Analytical study for time and timespace fractional Burgers’ equation
 KM Saad^{1, 2}Email author and
 Eman HF AlSharif^{1, 2}
https://doi.org/10.1186/s1366201713580
© The Author(s) 2017
 Received: 13 June 2017
 Accepted: 11 September 2017
 Published: 26 September 2017
Abstract
In this paper, the variational iteration method (VIM) is applied to solve the time and spacetime fractional Burgers’ equation for various initial conditions. VIM solutions are computed for the fractional Burgers’ equation to show the behavior of VIM solutions as the fractional derivative parameter is changed. The results obtained by VIM are compared with exact solutions and also with expansions of the exact solutions. VIM solutions are found to be in excellent agreement with these exact solutions.
Keywords
 variational iteration method
 spacetime fractional Burgers equation
 different initial values
1 Introduction
In recent years fractional calculus has been utilized to find solutions of equations governing the modeling of real materials in engineering and physics. Fractional differential equations model many phenomena in several fields such as fluid mechanics, chemistry [1, 2], biology [3], viscoelasticity [4], engineering, finance, and physics [5–7]. Therefore, the use of fractional calculus has gained the attention of scientists and engineers as, though counterintuitive, the fractional derivative does arise in physical problems. In this connection there is the work of [8–15]. Due to the difficulty of obtaining exact solutions of equations involving a fractional derivative, approximate and numerical techniques have tended to be used instead. Examples of such approximate and numerical methods are Taylor collocation method followed in [16]; Adomian’s decomposition method followed in [17, 18]; finite difference method followed in [19, 20]; homotopy analysis method and homotopy perturbation methods followed, respectively, in [21–23] and [24].
In addition, there are general papers giving an overview of the field of fractional differential equations [25–28]. Also, numerical methods have been used to find solutions of fractional differential equations [29–37].
In this paper we study approximate solutions using VIM for the time and spacetime fractional Burgers’ equation. To the best of our knowledge, this is the first study of the spacetime fractional Burgers’ equation by VIM. The work of [42, 43] studied only the time fractional Burgers’ equation and the space fractional Burgers’ equation, but did not study the spacetime fractional Burgers’ equation.
The present paper is organized as follows. The second and third sections are devoted to the basic ideas of fractional calculus and the standard VIM, respectively. The fourth and fifth sections are devoted to the application of VIM to evaluate solutions of the time and spacetime fractional Burgers’ equation, respectively. Conclusions are presented in section seven.
2 Fractional calculus
Definitions of fractional derivatives were given by Riemann, Liouville, Grunwald, Letnikov and Caputo [25–28], and they are based on generalized functions. The most commonly used definitions are those of Riemann [25] and Liouville and Caputo [27]. Here we give some basic definitions and properties of fractional calculus theory.
Definition 2.1
A real function \(f(t)\), \(t>0\), is said to be in the space \(C_{\mu}\), \(\mu\in R\), if there exists a real number \(p>\mu\) such that \(f(t)=t^{p} f_{1} (t)\), where \(f_{1}(t)\in C[0,\infty)\), and it is said to be in the space \(C_{\mu}^{m}\) iff \(f^{m} \in C_{m}\), \(m \in N \).

\(J^{\alpha} J^{\beta} f(t) = J^{\alpha+\beta} f(t)\),

\(J^{\alpha} J^{\beta} f(t) = J^{\beta} J^{\alpha} f(t)\),

\(J^{\alpha} t^{\gamma} = \frac{\Gamma(\gamma+1)}{\Gamma (\alpha+\gamma+1)} t^{\alpha+\gamma}\).
Definition 2.2
3 Basic ideas of VIM
4 The timefractional Burgers’ equation
5 The spacetime fractional Burgers’ equation
5.1 Tanh initial condition
5.2 Polynomial initial condition
6 Convergence analysis
In this section the existence of a unique solution is introduced in Theorem 6.2. Furthermore, the convergence of VIM solution (22) is proved in Theorem 6.3. Finally, the maximum absolute truncation error of VIM solution (22) is given in Theorem 6.4. In this section we prove theorems for the spacetime fractional Burgers’ equation. This theorem covers the timefractional Burgers’ equation on setting \(\beta=1\). We define \((C(I), \Vert \cdot \Vert )\) as a Banach space, the space of all continuous functions on \(I=R \times R^{+} \) with the norm \(\Vert v(x,t) \Vert =\max_{(x,t)\in I} \vert {v(x,t)} \vert\).
Lemma 6.1
Suppose that \(v(x,t)\) and their partial derivatives are continuous. Then the derivatives \(D_{t}^{\alpha}v(x,t)\), \(D_{x}^{\beta}v(x,t)\) and \(D_{x}^{2\beta }v(x,t)\) are bounded.
Proof
Theorem 6.2
Let \(F(v)=a v v_{x}^{\beta}\) satisfy the Lipschitz condition with the Lipschitz constant \(L_{4}\). Then problem (21) has the unique solution \(v(x,t)\) whenever \(0 < \gamma<1\).
Proof
Theorem 6.3
The sequence \(v_{n}(x,t)\) obtained from VIM iteration (22) converges to the exact solution of problem (21) for \(0<\sigma <1\) and \(0<\gamma_{1}<1\).
Proof
Theorem 6.4
The maximum absolute truncation error of the approximate solution \({v_{n}(x,t)}\) of the timespace fractional Burgers’ equation (21) can be estimated as \(\Vert E_{n}(x,t) \Vert \leq\frac{\sigma^{n}}{1\sigma} \Vert v_{1}(x,t) \Vert \).
Proof
7 Conclusions
Approximate solutions of the time and spacetime fractional Burgers’ equation have been evaluated using the VIM for different initial conditions by expanding the \(tanh\) initial condition in the basis functions \(e^{n x}\). The fractional derivative could then be easily calculated. An important point is that many authors avoid this initial condition as there was no direct method to calculate its fractional derivative. In Figure 1, the surface of VIM solutions with the exact solution of Burgers’ equation for \(\alpha=0.7\) was plotted. Also, the absolute error with \(\alpha=1\) for different values for b were displayed in Figure 2. The effect of changes of b was clear through the resulting decrease of the error. As the value of b decreases, the absolute error becomes very small. In Figure 3 the VIM solutions were plotted with different values of α and β. The VIM solutions approach the exact solution as the values of α and β approach 1, as shown in Figure 4. The results were shown to be in very good agreement with both exact solutions and previous approximate solutions. The calculations in this paper were performed using Mathematica 9.
Declarations
Acknowledgements
We thank Noel Frederick Smyth, JiHuan He and Saeid Abbasbandy for stimulating discussions during the preparation of this article.
Authors’ contributions
All authors drafted the manuscript, and they read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
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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