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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 461837, 7 pages

http://dx.doi.org/10.1155/2013/461837

## Revised Variational Iteration Method for Solving Systems of Nonlinear Fractional-Order Differential Equations

^{1}Department of Mathematics, Faculty of Science, Istanbul University, Vezneciler, 34134 Istanbul, Turkey^{2}International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mmabatho 2735, South Africa^{3}Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran^{4}Department 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.

#### Abstract

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.

#### 1. Introduction

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 [18]. 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 [19], 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 [28] have explored the Adomian decomposition method to obtain solution of a system of linear and nonlinear fractional differential equations. Further in [29], 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 [30].

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 [9] 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 [26], He showed that the variational iteration method is also valid for fractional differential equations. In this section, following the discussion presented in [26], 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 [32].

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 [33].

#### 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.

Figures 5(a), 5(b), and 5(c), show approximate solutions , , and of (19) using VIM and RVIM for the special case and the numerical solutions, respectively.

#### 6. Conclusion

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.

#### References

- D. Baleanu and S. I. Muslih, “Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives,”
*Physica Scripta*, vol. 72, no. 2-3, pp. 119–121, 2005. View at Zentralblatt MATH · View at MathSciNet · View at Scopus - D. Bǎleanu, O. G. Mustafa, and R. P. Agarwal, “On the solution set for a class of sequential fractional differential equations,”
*Journal of Physics A*, vol. 43, no. 38, Article ID 385209, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - K. Diethelm,
*The Analysis of Fractional Differential Equations*, vol. 2004 of*Lecture Notes in Mathematics*, Springer, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - R. Hilfer,
*Applications of Fractional Calculus in Physics*, Academic Press, Orlando, Fla, USA, 1999. - A. A. Kilbas, H. H. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. - R. L. Magin,
*Fractional Calculus in Bioengineering*, Begell House, Connecticut, Hartford, USA, 2006. - R. L. Magin, O. Abdullah, D. Baleanu, and X. J. Zhou, “Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation,”
*Journal of Magnetic Resonance*, vol. 190, no. 2, pp. 255–270, 2008. View at Publisher · View at Google Scholar · View at Scopus - F. Mainardi,
*Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models*, Imperial College Press, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, Academic Press, San Diego, Calif, USA, 1999. - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives—Theory and Applications*, Gordon and Breach, Linghorne, Pa, USA, 1993. - 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 · View at Google Scholar · View at Scopus - V. E. Tarasov, “Fractional vector calculus and fractional Maxwell's equations,”
*Annals of Physics*, vol. 323, no. 11, pp. 2756–2778, 2008. View at Publisher · View at Google Scholar · View at Scopus - O. P. Agrawal and D. Baleanu, “A hamiltonian formulation and a direct numerical scheme for fractional optimal control problems,”
*Journal of Vibration and Control*, vol. 13, no. 9-10, pp. 1269–1281, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - R. S. Barbosa, J. A. T. Machado, and I. M. Ferreira, “Tuning of PID controllers based on bode's ideal transfer function,”
*Nonlinear Dynamics*, vol. 38, no. 1–4, pp. 305–321, 2004. View at Publisher · View at Google Scholar · View at Scopus - G. M. Mophou, “Optimal control of fractional diffusion equation,”
*Computers and Mathematics with Applications*, vol. 61, no. 1, pp. 68–78, 2011. View at Publisher · View at Google Scholar · View at Scopus - J. A. Tenreiro Machado, “Optimal tuning of fractional controllers using genetic algorithms,”
*Nonlinear Dynamics*, vol. 62, no. 1-2, pp. 447–452, 2010. View at Publisher · View at Google Scholar · View at Scopus - H.-F. Raynaud and A. Zergaïnoh, “State-space representation for fractional order controllers,”
*Automatica*, vol. 36, no. 7, pp. 1017–1021, 2000. View at Publisher · View at Google Scholar · View at Scopus - T. M. Atanackovic and B. Stankovic, “On a system of differential equations with fractional derivatives arising in rod theory,”
*Journal of Physics A*, vol. 37, no. 4, pp. 1241–1250, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. H. He, “A new approach to nonlinear partial differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 2, pp. 230–235, 1997. - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. K. Golmankhaneh, T. Khatuni, N. A. Porghoveh, and D. Baleanu, “Comparison of iterative methods by solving nonlinear Sturm-Liouville, Burgers and Navier-Stokes equations,”
*Central European Journal of Physics*, vol. 10, no. 4, pp. 966–976, 2012. - 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 Zentralblatt MATH · View at MathSciNet · View at Scopus - G.-C. Wu and D. Baleanu, “Variational iteration method for the Burgers' flow with fractional derivatives—new Lagrange multipliers,”
*Applied Mathematical Modelling*, vol. 37, no. 9, pp. 6183–6190, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Y.-J. Yang, D. Baleanu, and X.-J. Yang, “A local fractional variational iteration method for Laplace equation within local fractional operators,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 202650, 6 pages, 2013. View at MathSciNet - 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. - 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 Scopus - 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 · View at Google Scholar · View at Scopus - V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 301, no. 2, pp. 508–518, 2005. View at Publisher · View at Google Scholar · View at Scopus - H. Jafari and V. Daftardar-Gejji, “Revised Adomian decomposition method for solving a system of nonlinear equations,”
*Applied Mathematics and Computation*, vol. 175, no. 1, pp. 1–7, 2006. View at Publisher · View at Google Scholar · View at Scopus - H. Jafari and V. Daftardar-Gejji, “Revised Adomian decomposition method for solving systems of ordinary and fractional differential equations,”
*Applied Mathematics and Computation*, vol. 181, no. 1, pp. 598–608, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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*, pp. 156–162, Pergamon Press, 1978. - M. Tatari and M. Dehghan, “Improvement of He's variational iteration method for solving systems of differential equations,”
*Computers and Mathematics with Applications*, vol. 58, no. 11-12, pp. 2160–2166, 2009. View at Publisher · View at Google Scholar · View at Scopus