Research Article | Open Access

# Variational Iteration Method for a Fractional-Order Brusselator System

**Academic Editor:**Alexander Domoshnitsky

#### Abstract

This paper presents approximate analytical solutions for the fractional-order Brusselator system using the variational iteration method. The fractional derivatives are described in the Caputo sense. This method is based on the incorporation of the correction functional for the equation. Two examples are solved as illustrations, using symbolic computation. The numerical results show that the introduced approach is a promising tool for solving system of linear and nonlinear fractional differential equations.

#### 1. Introduction

In recent years, fractional differential equations (FDEs) have been the focus of many studies due to their appearance in various fields such as physics, chemistry, and engineering [1–3]. On the other hand, much attention has been paid to the solutions of fractional differential equations. Since most fractional differential equations do not have exact analytic solutions and approximate and numerical techniques, therefore, they are used extensively. Recently, the Adomian decomposition method, homotopy perturbation method, homotopy analysis method, and differential transform method have been used for solving a wide range of problems [4–10].

Another powerful analytical method, called the variational iteration method (VIM), was first introduced in [11]. This technique has successfully been applied to many situations: for example, see [12–17]. Reference [18] was the first where the variational iteration method was applied to fractional differential equations. Odibat and Momani [19] implemented the variational iteration method to solve partial differential equations of fractional order.

In this paper, we introduce a new application of the variational iteration method to provide approximate solutions of the fractional-order Brusselator system in the following form: subject to the initial conditions with , , , and are constants.

is used to represent the Caputo-type fractional derivative of order .

The Riemann-Liouville definition of the fractional integration [2] is given by

For our purpose in this paper, we adopt Caputo’s fractional derivative [2]:

where is a positive integer and is the Gamma function. In particular, , and we have

The fractional-order Brusselator system has been considered by several authors recently [20–22]. Gafiychuk and Datsko investigated its stability [20]. Wang and Li proved by numerical method that the solutions of the fractional-order Brusselator system have a limit cycle [22]. We used the variational iteration method to investigate the approximate solutions of the fractional-order Brusselator system.

#### 2. Variational Iteration Method

The principles of the variational iteration method and its applicability for various kinds of differential equations are given in [23, 24]. In [18], it was shown that the variational iteration method is also valid for fractional differential equations. In this section, following the discussion presented in [18], we extend the application of the variational iteration method to solve the fractional Brusselator equation:

According to the variational iteration method, we can construct the correction functional for (6) as

where are the general Lagrange multiplier, which can be identified optimally via variational theory [25, 26].

To identify approximately Lagrange multiplier, some approximations must be made. The correction functional equation (7) can be approximately expressed as follows:

where and are considered as restricted variations, in which . To find the optimal and , we proceed as follows:

The stationary conditions can be obtained as follows:

We substitute , into the functional equation (11) to obtain the following iteration formula:

The initial approximations and can be freely chosen if they satisfy the initial conditions of the problem. Finally, we approximate the solutions and by the th terms and .

#### 3. Illustrative Examples

For purposes of illustration of (VIM) for solving Brusselator equation, we present two examples.

*Example 1. *Consider the following fractional-order Brusselator system:
with the initial conditions:

According to the variational iteration method and (11), the iteration formula for (12) is given by

By using the above variational iteration formula, if we start with the initial approximations and , we can obtain directly the other components as
and so on; in the same way the rest of the components of the iteration formula can be obtained. Figures 1(a) and 1(b) show comparison between the approximate solutions , of (12) obtained using VIM for the special case and the numerical solutions for the special case , respectively. Figures 2(a) and 2(b) show the approximate solutions , of (12) using VIM for the special case and the numerical solutions, respectively.

**(a)**

**(b)**

**(a)**

**(b)**

*Example 2. *Consider the following fractional-order Brusselator system:
with the initial conditions:

The correction functional for (16) turns out to be

By the above variational iteration formula and beginning with the initial approximations and , we can obtain directly the other components as
and so on; in the same manner the remaining set of the components of the iteration formula can be obtained. Figures 3(a) and 3(b) show comparison between the approximate solutions , of (16) obtained using VIM for the special case and the numerical solutions for the special case , respectively. Figures 4(a) and 4(b) show the approximate solutions , of (16) using VIM for the special case and the numerical solutions, respectively.

**(a)**

**(b)**

**(a)**

**(b)**

#### 4. Conclusions

The variational iteration method is a powerful method which is able to handle linear/nonlinear fractional differential equations. The method has been applied to fractional-order Brusselator system in order to find its approximate solutions. The results show that the applied method is suitable and inexpensive for obtaining the approximate solutions.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### References

- R. Hilfer,
*Applications of Fractional Calculus in Physics*, Academic Press, Orlando, Fla, USA, 1999. - I. Podlubny,
*Fractional Differential Equations*, Academic Press, San Diego, Calif, USA, 1999. - A. M. Spasic and M. P. Lazarevic, “Electroviscoelasticity of liquid/liquid interfaces: fractional-order model,”
*Journal of Colloid and Interface Science*, vol. 282, no. 1, pp. 223–230, 2005. View at: Publisher Site | Google Scholar - V. Erturk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 13, no. 8, pp. 1642–1654, 2008. View at: Publisher Site | Google Scholar - H. Jafari and V. Daftardar-Gejji, “Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition,”
*Applied Mathematics and Computation*, vol. 180, no. 2, pp. 488–497, 2006. View at: Publisher Site | Google Scholar - H. Jafari and S. Momani, “Solving fractional diffusion and wave equations by modified homotopy perturbation method,”
*Physics Letters A*, vol. 370, no. 5-6, pp. 388–396, 2007. View at: Publisher Site | Google Scholar - H. Jafari and S. Seifi, “Solving a system of nonlinear fractional partial differential equations using homotopy analysis method,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 1962–1969, 2009. View at: Publisher Site | Google Scholar - H. Jafari and S. Seifi, “Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 2006–2012, 2009. View at: Publisher Site | Google Scholar - H. Jafari, N. Kadkhoda, H. Tajadodi, and S. A. Hosseini Matikolai, “Homotopy perturbation pade technique for solving fractional Riccati differential equations,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 11, pp. 271–275, 2010. View at: Google Scholar - H. Jafari and H. Tajadodi, “He's variational iteration method for solving fractional Riccati differential equation,”
*International Journal of Differential Equations*, vol. 2010, Article ID 764738, 8 pages, 2010. View at: Publisher Site | Google Scholar - J. H. He, “A new approach to nonlinear partial differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 2, no. 4, pp. 230–235, 1997. View at: Publisher Site | Google Scholar - M. A. Abdou and A. A. Soliman, “New applications of variational iteration method,”
*Physica D*, vol. 211, no. 1-2, pp. 1–8, 2005. View at: Publisher Site | Google Scholar - J. H. He, “Variational iteration method for autonomous ordinary differential systems,”
*Applied Mathematics and Computation*, vol. 114, no. 2-3, pp. 115–123, 2000. View at: Google Scholar - J. H. He, “Variational principles for some nonlinear partial differential equations with variable coefficients,”
*Chaos, Solitons and Fractals*, vol. 19, no. 4, pp. 847–851, 2004. View at: Publisher Site | Google Scholar - H. Jafari, Ch. Chun, and C. M. Khalique, “The variational iterationmethod for finding exact solution of nonlinear gas dynamics equations,”
*Zeitschrift fur Naturforschung*, vol. 66, no. 3-4, pp. 161–164, 2011. View at: Google Scholar - Y. Khan, N. Faraz, A. Yildirim, and Q. Wu, “Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science,”
*Computers and Mathematics with Applications*, vol. 62, no. 5, pp. 2273–2278, 2011. View at: Publisher Site | Google Scholar - X. J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,”
*Thermal Science*, vol. 17, no. 2, pp. 625–628, 2013. View at: Google Scholar - J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 167, no. 1-2, pp. 57–68, 1998. View at: Google Scholar - Z. Odibat and S. Momani, “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics,”
*Computers and Mathematics with Applications*, vol. 58, no. 11-12, pp. 2199–2208, 2009. View at: Publisher Site | Google Scholar - V. Gafiychuk and B. Datsko, “Stability analysis and limit cycle in fractional system with Brusselator nonlinearities,”
*Physics Letters A*, vol. 372, no. 29, pp. 4902–4904, 2008. View at: Publisher Site | Google Scholar - S. Kumar, Y. Khan, and A. Yildirim, “A mathematical modeling arising in the chemical systems and its approximate numerical solution,”
*Asia-Pacific Journal of Chemical Engineering*, vol. 7, no. 6, pp. 835–840, 2012. View at: Publisher Site | Google Scholar - Y. Wang and C. Li, “Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle?”
*Physics Letters A*, vol. 363, no. 5-6, pp. 414–419, 2007. View at: Publisher Site | Google Scholar - J. H. He, “Variational iteration method for delay differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 2, no. 4, pp. 235–236, 1997. View at: Publisher Site | Google Scholar - S. Momani and S. Abuasad, “Application of He's variational iteration method to Helmholtz equation,”
*Chaos, Solitons and Fractals*, vol. 27, no. 5, pp. 1119–1123, 2006. View at: Publisher Site | Google Scholar - M. Inokuti, H. Sekine, and T. Mura, “General use of the Lagrange multiplier in non-linear mathematical physics,” in
*Variational Method in the Mechanics of Solids*, S. Nemat-Nasser, Ed., pp. 156–162, Pergamon Press, Oxford, UK, 1978. View at: Google Scholar - H. Jafari and A. Alipoor, “A new method for calculating general lagrange multiplier in the variational iteration method,”
*Numerical Methods for Partial Differential Equations*, vol. 27, no. 4, pp. 996–1001, 2011. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2014 H. Jafari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.