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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 461837, 7 pages
Revised Variational Iteration Method for Solving Systems of Nonlinear Fractional-Order Differential Equations
1Department of Mathematics, Faculty of Science, Istanbul University, Vezneciler, 34134 Istanbul, Turkey
2International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mmabatho 2735, South Africa
3Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran
4Department of Mathematics and Computer Science, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey
Received 9 July 2013; Accepted 22 July 2013
Academic Editor: Juan J. Trujillo
Copyright © 2013 C. Ünlü 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.
A modification of the variational iteration method (VIM) for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE) obtained using the traditional variational iteration method give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. Illustrative examples are presented to show the validity of this modification.
Recently, fractional-order calculus has been studied as an alternative calculus in mathematics. Numerous problems in physics, chemistry, biology, and engineering can be modeled with fractional derivatives [1–12]. On the other hand, in control society, fractional-order dynamic systems and controls have gained an increasing attention [13–17], and also motion of an elastic column fixed at one end loaded at the other can be formulated in terms of a system of fractional differential equations . Since most fractional differential equations do not have exact analytic solutions, approximate and numerical techniques, therefore, are used extensively.
The variational iteration method is relatively new approaches to provide approximate solutions to linear and nonlinear problems. The variational iteration method, which is proposed by He , was successfully applied to find the solutions of several classes of variational problems. Some research works in this field are [20–25]. Recently, the application of the method is extended for fractional differential equations [26, 27].
Daftardar-Gejji and Jafari  have explored the Adomian decomposition method to obtain solution of a system of linear and nonlinear fractional differential equations. Further in , they have suggested a modification (termed as “revised ADM”) of this method and applied revised method for solving systems of linear/nonlinear ordinary and fractional differential equations .
The objective of this paper is the use of revised variational iteration method (RVIM) for solving systems of nonlinear fractional-order differential equations. We demonstrate that the approximate solution thus obtained converges faster relative to the approximate solutions by standard variational iteration method. Several illustrative examples have been presented.
2. Definitions and Preliminaries
In this section, we give some definitions and properties of the fractional calculus  which are used further in this paper.
Definition 1. A real function is said to be in the space if there exists a real number p , such that , where .
Definition 2. A function is said to be in the space .
Definition 3. The left-sided Riemann-Liouville fractional integral of order , of a function is defined as
Definition 4. The left-sided Caputo fractional derivative of , , , is defined as
3. The VIM for FDE
The principles of the variational iteration method and its applicability for various kinds of differential equations are given in [20, 31]. In , He showed that the variational iteration method is also valid for fractional differential equations. In this section, following the discussion presented in , we extend the application of the variational iteration method to solve the fractional differential equation as follows: where is an operator with respect to and is a known function. According to the variational iteration method, we can construct the correction functional for (3) as follows: where is the general Lagrange multiplier, which can be identified optimally via variational theory .
To identify approximately Lagrange multiplier, some approximations must be made. The correction functional equation (4) can be approximately expressed as follows: Here we apply restricted variations to the term , and in this case, we can easily determine the multiplier. Making the aforementioned functional stationary, noticing that , yields the Lagrange multiplier , and substituting into the functional equation (4), we obtain the following iteration formula: The initial approximation can be freely chosen if it satisfies the initial conditions of the problem. Finally, we approximate the solution by the th term .
4. The System of FDE and Revised VIM
Let us consider the following system of fractional differential equations: where are operators with respect to and are known functions. In this case, the correction functionals are obtained as follows: Here we construct the following iteration formula instead of the iteration formula obtained with the standard variational iteration method equation (9): In fact, the updated values are used for finding . We called it the revised variational iteration method (RVIM). This technique can accelerate the convergence of iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. The effect of this correction is clear in because the updated values are used to compute it. Tatari and Dehghan have employed this technique for systems of ordinary differential equations .
5. Test Examples
In this section, we illustrate the applicability of revised variational iteration method to systems of nonlinear fractional-order differential equations.
Example 1. Consider the following system of nonlinear fractional-order Van der pol: with the initial condition The standard VIM for (11) leads to the following iteration formula: The use of the revised VIM for (11) results in the following formula: Starting with the initial approximations and , we can easily obtain the results using (13) and (14).
In Figures 1 and 2, the results from VIM and RVIM are shown. Figures 1(a) and 1(b) show comparison between the approximate solutions , of (11) obtained using VIM and RVIM for the special case and the numerical solutions for the special case , respectively. Figures 2(a) and 2(b), show approximate solutions , of (11) using VIM and RVIM for the special case and the numerical solutions, respectively.
Example 2. Consider the system of nonlinear fractional-order Brusselator: with the initial conditions In this example, the use of the variational iteration method leads to Using the RVIM, we obtain We consider the initial approximations and . Results are shown in Figure 3. Figures 3(a) and 3(b) show comparison between the approximate solutions , of (15) obtained using VIM and RVIM for the special case and the numerical solutions for the special case , respectively.
Example 3. Consider the system of nonlinear fractional-order Genesio-Tesi where , , , and are system parameters. With the initial conditions, In view of the variational iteration method, we set The revised variational iteration method would lead to Beginning with the initial approximations and , we can easily obtain the results. Results are shown in Figures 4 and 5. Figures 4(a), 4(b), and 4(c) show comparison between the approximate solutions , and of (19) obtained using VIM and RVIM for the special case and the numerical solutions for the special case , respectively.
The variational iteration method is an efficient method for solving various kinds of problems. In this paper, we have suggested a modification of this method which is called “revised variational iteration method.” We employ the revised VIM for solving a systems of nonlinear fractional-order differential equations. The revised method yields a series solution which converges faster than the series obtained by standard VIM. Illustrative examples presented clear support for this claim.
Mathematica has been used for computation and graphs presented in this paper.
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