Research Article  Open Access
Taher A. Nofal, "Approximate Solutions for Nonlinear Initial Value Problems Using the Modified Variational Iteration Method", Journal of Applied Mathematics, vol. 2012, Article ID 370843, 19 pages, 2012. https://doi.org/10.1155/2012/370843
Approximate Solutions for Nonlinear Initial Value Problems Using the Modified Variational Iteration Method
Abstract
We have used the modified variational iteration method (MVIM) to find the approximate solutions for some nonlinear initial value problems in the mathematical physics, via the BurgersFisher equation, the KuramotoSivashinsky equation, the coupled SchrodingerKdV equations, and the longshort wave resonance equations together with initial conditions. The results of these problems reveal that the modified variational iteration method is very powerful, effective, convenient, and quite accurate to systems of nonlinear equations. It is predicted that this method can be found widely applicable in engineering and physics.
1. Introduction
Nonlinear partial differential equations are known to describe a wide variety of phenomena not only in physics, where applications extend over magnetofluid dynamics, water surface gravity waves, electromagnetic radiation reactions, and ion acoustic waves in plasma, but also in biology, chemistry, and several other fields. It is one of the important tasks in the study of the nonlinear partial differential equations to seek exact and explicit solutions. In the past several decades both mathematicians and physicists have made many attempts in this direction. Various methods for obtaining exact solutions to nonlinear partial differential equations have been proposed. Among these methods are the Bäcklund transformation method [1, 2], the Hirota’s bilinear method [3], the inverse scattering transform method [4], extended tanh method [5–7], the AdomianPade approximation [8–10], the variational method [11–14], the variational iteration method [15, 16], the various LindstedtPoincare methods [17–20], the Adomian decomposition method [8, 21, 22], the expansion method [23, 24], the Expfunction method [25–27] and others [28–35]. Zayed et al. [36] investigated the travelling wave solutions for nonlinear initial value problems using the homotopy perturbation method. The modified variational iteration method is the couples of the variational iteration method with the homotopy pertirbation method. Recently Akbarzade and Langari [37] and Mei and Zhang [38] had used the modified variational iteration method for some nonlinear partial differential equations.
The main objective of the present paper is to use the modified variational iteration method (MVIM) for constructing the traveling wave solutions of the following nonlinear partial differential equations in mathematical physics:(i)the nonlinear BurgersFisher equation [39]: (ii)the nonlinear KuramotoSivashinsky equation [40]: (iii)the nonlinear coupled Schrodinger KdV equations [41]: (iv)the nonlinear longshort wave resonance equations [42]:
together with initial conditions, where , , , and are arbitrary constants while . It is interesting to point out that (1.1) includes the convection term and the dissipation term . Equation (1.2) describes the fluctuations of the position of a flame front, the motion of a fluid going down a vertical wall, or a spatially uniform oscillating chemical reaction in a homogeneous medium. Equation (1.3) describe various processes in dusty plasma such as Langmuri, dustacoustic wave and electromagnetic waves, while in (1.4) is the envelope of the short wave and is a complex function, and is the amplitude of the long wave which is a real function.
2. Basic Idea of He’s Homotopy Perturbation Method
We illustrate the following nonlinear differential equation [43–54]: with the boundary conditions: where is a general differential operator, is a boundary operator, is an analytic function, and is the boundary of the domain . Generally speaking, the operator can be divided into two parts and , where is linear but is nonlinear. Therefore, (2.1) can be rewritten in the following form: By the homotopy technique, we construct a homotopy which satisfies or where is an embedding parameter and is an initial approximation of (2.1) which satisfies the boundary conditions (2.2). Obviously, from (2.4) and (2.5), we have The changing process of from zero to unity is just that of from to . In topology, this is called the deformation but and are called the homotopies. According to the homotopy perturbation method, we can first use the embedding parameter “” as a small parameter and assume that (2.4) or (2.5) can be written as a power series in “” as follows: Letting in (2.7), the approximate solution of (2.3) takes the following form: The combination of the perturbation method and the homotopy method is called the homotopy perturbation method which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantage of the traditional perturbation techniques.
3. Variational Iteration Method
Consider the following nonhomogeneous, nonlinear partial differential equation: where is a linear differential operator with respect to time, is a nonlinear operator and is a given function.
According to the variational iteration method, we can construct correct functionals as follows: which is variational iteration algorithm I, and is a general Lagrange multipliers. The variational iteration method can be identified optimally via variational theory [6, 7]. The second term on the righthand side in (3.2) is called the corrections, the subscript denotes the th order approximation, and is restricted variations. We can assume that the aforementioned correctional functionals are stationary (i.e., ), and then the Lagrange multipliers can be identified.
Now we can start with the given initial approximation and by the previous iteration formulas we can obtain the approximate solutions. He [55] has used the fractional iteration method to obtain the approximate solutions for nonlinear fractional differential equations.
4. The Modified Variational Iteration Method
To convey the basic idea of the variational homotopy perturbation method [2, 3], we consider the following general differential equation: where is a linear differential operator, is a nonlinear operator and is an inhomogeneous term. According to the variational iteration method [4–13], we can construct a correct functional as follows: where is a Lagrange multipliers, which can be identified optimally via variational theory [6, 7]. The subscripts denote the th approximation, and is considered as a restricted variation. That is, is called a correct functional. Now, we apply the homotopy perturbation method to (4.2): which is the variational iteration algarithm II and is formulated by the modified variational iteration method. The embedding parameter can be considered as an expanding parameter [14–19].
The homotopy perturbation method uses the homotopy parameter as an expanding parameter [14–19] to obtain If , then (4.4) becomes the approximate solution of the following form:
A comparison of like powers of gives solutions of various orders.
The application of the Adomain polynomial is too complex so that we consider the variational iteration method and He’s polynomial to calculate the approximate solutions (see, e.g., [56–60]).
5. Applications
In this section, we construct the approximate solutions for some nonlinear evolution equations in the mathematical physics, namely, the BurgersFisher equation (1.1), the KuramotoSivashinsky equation (1.2), the coupled SchrodingerKdV equations (1.3), and the longshort wave resonance equations (1.4) together with initial conditions by using the the modified variational iteration method. Applications of this method to similar equations can be found in [61–67].
5.1. Approximate Solution of BurgersFisher Equation with Initial Conditions Using Modified Variational Iteration Method
In this subsection, we use the MVIM to find the solution of an initial value problem consisting of the nonlinear BurgersFisher equation (1.1) and the following initial condition [39]: This initial condition follows by setting in the following exact solution of (1.1): This exact solution has been derived by Wazwaz [39] using the tanhcoth method. To this end, we construct the modified variational iteration method for the nonlinear Burgers Fisher equation (1.1) which satisfies Comparing the different coefficient of like power of , we have and so on. Consequently after some reduction with help of Maple or Mathematica, we get: In this manner the other components can be obtained.
Substituting from (5.5) into (4.5), we obtain the following approximate solution of the initial value problem (1.1) and (5.1): Note that if we expand the exact solution (5.2) in Taylor series near , we obtain the approximate solution (5.6). To demonstrate the convergence of the variational homotopy perturbation method, the results of the numerical example are presented and only few terms are required to obtain accurate solutions. The accuracy of the modified variational iteration method for the nonlinear BurgersFisher equation is controllable and absolute errors are very small with the present choice of and . These results are listed in Table 1. Both the exact solution (5.2) and the approximate solution (5.6) obtained for the first three approximations are plotted in Figure 1. There are no visible differences in diagrams. It is also evident that when more terms for the modified variational iteration method are computed, the numerical results get much more closer to the corresponding exact solution with the initial condition (5.1).

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5.2. Approximate Solution of the Nonlinear KuramotoSivashinsky Equation with Initial Conditions Using MVIM
In this subsection, we use the MVIM to find the solution of an initialvalue problem consisting of the nonlinear KuramotoSivashinsky equation (1.2) with the following initial condition [40]: where , and are constants. This initial condition follows by setting in the following exact solution of (1.2): This exact solution has been derived by Wazwaz [40] using the tanh method and the extended tanh method. Let us now apply the MVIM to the initial value problem (1.2) and (5.7). To this end, we construct an MVIM for the nonlinear KuramotoSivashinsky equation (1.2) which satisfies Comparing the different coefficient of like power of , we have and so on. Consequently after some reduction with help of Maple or Mathematica, we get and so on. Substituting from (5.11)(5.12) into (4.5), we obtain the approximate solution of the initial value problem (1.2): which is in agreement with the exact solution (5.8) using Taylor series expansion near The comparison between the exact solution (5.8) and the approximate solution (5.12) is shown in Table 2 and Figure 2. It seems that the errors are very small if .

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5.3. Approximate Solutions for the NonlinearCoupled SchrodingerKdV Equations with Initial Conditions Using MVIM
In this subsection, we find the solutions and satisfying the nonlinear coupled SchrodingerKdV equations (1.3) with the following initial conditions [41]: where are arbitrary constants and . These initial conditions follow by setting in the following exact solutions of (1.3): These exact solutions have been derived by Zhang [41] using a direct algebraic approach. Let us now apply the MVIM to the initial value problem (1.3) and (5.13): Comparing the different coefficient of like power of , we have and so on. Consequently after some reduction with help of Maple or Mathematica, we get In this manner the other components can be obtained. Substituting (5.18) into (4.5), we obtain the approximate solutions of the initial value problem (1.3) and (5.13): which are in the closed form of the exact solutions (5.14) and (5.15) using Taylor series expansion near .
The comparison between the exact solutions (5.14), (5.15) and the approximate solutions (5.19), (5.20) respectively, are shown in Table 3 and Figures 3 and 4. It seems that the errors are very small if , .

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5.4. Approximate Solution of the Nonlinear Long and Short Wave Resonance Equations with Initial Conditions Using MVIM
In this subsection, we find the solutions and satisfying the nonlinear long–short wave resonance equations (1.4) with the following initial conditions [42]: where and are arbitrary constants. These initial conditions follow by setting in the following exact solutions of (1.4): where is constant. These exact solutions have been derived by Shang [42] using the extended hyperbolic function method, which describes the resonance interaction between the long wave and the short wave. Let us now apply the MVIM to the initial value problem (1.4) and (5.21): Comparing the different coefficient of like power of , we have and so on. On substituting (5.21) into (5.25), we deduce that
In this manner the other components can be obtained. Consequently, we obtain the following approximate solutions of the initial value problem (1.4) and (5.21): which are in the closed forms of the exact solutions (5.22) and (5.23) using Taylor series expansion near .
The comparison between the exact solutions (5.22), (5.23) and the approximate solutions (5.27), (5.28) respectively is shown in Table 4 and Figures 5 and 6. It seems that the errors are very small if, .

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6. Conclusions
In this paper, the modified variational iteration method was applied for finding the approximate solutions for some nonlinear evolution equations in mathematical physics via the nonlinear BurgersFisher equation, nonlinear KuramotoSivashinsky equation, nonlinear coupled Schrodinger KdV equations, and nonlinear longshort wave resonance equations with wellknown initial conditions. It seems to us that the modified variational iteration method presents a rapid convergence solutions. It can be concluded that this method is very powerful and efficient technique in finding approximate solutions for wide classes of nonlinear problems.
References
 S.F. Deng, “Bäcklund transformation and soliton solutions for KP equation,” Chaos, Solitons and Fractals, vol. 25, no. 2, pp. 475–480, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 G. Tsigaridas, A. Fragos, I. Polyzos et al., “Evolution of nearsoliton initial conditions in nonlinear wave equations through their Bäcklund transforms,” Chaos, Solitons and Fractals, vol. 23, no. 5, pp. 1841–1854, 2005. View at: Publisher Site  Google Scholar
 O. Pashaev and G. Tanoğlu, “Vector shock soliton and the Hirota bilinear method,” Chaos, Solitons & Fractals, vol. 26, no. 1, pp. 95–105, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 V. O. Vakhnenko, E. J. Parkes, and A. J. Morrison, “A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation,” Chaos, Solitons and Fractals, vol. 17, no. 4, pp. 683–692, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 L. DeSheng, G. Feng, and Z. HongQing, “Solving the $(2+1)$dimensional higher order BroerKaup system via a transformation and tanhfunction method,” Chaos, Solitons and Fractals, vol. 20, no. 5, pp. 1021–1025, 2004. View at: Publisher Site  Google Scholar
 E. M. E. Zayed, H. A. Zedan, and K. A. Gepreel, “Group analysis and modified extended tanhfunction to find the invariant solutions and soliton solutions for nonlinear Euler equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 5, no. 3, pp. 221–234, 2004. View at: Google Scholar
 H. A. Abdusalam, “On an improved complex tanhfunction method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 99–106, 2005. View at: Google Scholar
 T. A. Abassy, M. A. ElTawil, and H. K. Saleh, “The solution of KdV and mKdV equations using adomian padé approximation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 5, no. 4, pp. 327–340, 2004. View at: Google Scholar
 S. M. ElSayed, “The decomposition method for studying the KleinGordon equation,” Chaos, Solitons and Fractals, vol. 18, no. 5, pp. 1025–1030, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 D. Kaya and S. M. ElSayed, “An application of the decomposition method for the generalized KdV and RLW equations,” Chaos, Solitons and Fractals, vol. 17, no. 5, pp. 869–877, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 H. M. Liu, “Generalized variational principles for ion acoustic plasma waves by He's semiinverse method,” Chaos, Solitons and Fractals, vol. 23, no. 2, pp. 573–576, 2005. View at: Publisher Site  Google Scholar
 H. M. Liu, “Variational Approach to Nonlinear Electrochemical System,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 5, no. 1, pp. 95–96, 2004. 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  Zentralblatt MATH
 A. M. Mesón and F. Vericat, “Variational analysis for the multifractal spectra of local entropies and Lyapunov exponents,” Chaos, Solitons and Fractals, vol. 19, no. 5, pp. 1031–1038, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He, “Variational iteration method  A kind of nonlinear analytical technique: some examples,” International Journal of NonLinear Mechanics, vol. 34, no. 4, pp. 699–708, 1999. View at: Google Scholar
 G.E. Draganescu and V. Capalnasan, “Nonlinear relaxation phenomena in polycrystalline solids,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 4, pp. 219–226, 2003. View at: Google Scholar
 J. H. He, “Modified LindstedtPoincaré methods for some strongly nonlinear oscillations. I. Expansion of a constant,” International Journal of NonLinear Mechanics, vol. 37, no. 2, pp. 309–314, 2002. View at: Publisher Site  Google Scholar
 J. H. He, “Modified LindstedtPoincaré methods for some strongly nonlinear oscillations. II. A new transformation,” International Journal of NonLinear Mechanics, vol. 37, no. 2, pp. 315–320, 2002. View at: Publisher Site  Google Scholar
 J. H. He, “Modified LindstedPoincare methods for some strongly nonlinear oscillations part III : double series expansion,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 2, no. 4, pp. 317–320, 2001. View at: Google Scholar
 H. M. Liu, “Generalized variational principles for ion acoustic plasma waves by He's semiinverse method,” Chaos, Solitons and Fractals, vol. 23, no. 2, pp. 573–576, 2005. View at: Publisher Site  Google Scholar
 G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. M. Wazwaz, “A reliable technique for solving the wave equation in an infinite onedimensional medium,” Applied Mathematics and Computation, vol. 92, no. 1, pp. 1–7, 1998. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 D. Wang and H.Q. Zhang, “Further improved $F$expansion method and new exact solutions of KonopelchenkoDubrovsky equation,” Chaos, Solitons and Fractals, vol. 25, no. 3, pp. 601–610, 2005. View at: Publisher Site  Google Scholar
 M. Wang and X. Li, “Applications of $F$expansion to periodic wave solutions for a new Hamiltonian amplitude equation,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1257–1268, 2005. View at: Publisher Site  Google Scholar
 X. H. Wu and J. H. He, “Solitary solutions, periodic solutions and compactonlike solutions using the Expfunction method,” Computers & Mathematics with Applications, vol. 54, no. 78, pp. 966–986, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He and X. H. Wu, “Expfunction method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He and M. A. Abdou, “New periodic solutions for nonlinear evolution equations using Expfunction method,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1421–1429, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He, Gongcheng Yu Kexue Zhong de jinshi feixianxing feixi fangfa, Henan Science and Technology Press, Zhengzhou, China, 2002.
 J. H. He, “Determination of limit cycles for strongly nonlinear oscillators,” Physical Review Letters, vol. 90, no. 17, Article ID 174301, 3 pages, 2003. View at: Google Scholar
 J. Shen and W. Xu, “Bifurcations of smooth and nonsmooth travelling wave solutions of the DegasperisProcesi equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 5, no. 4, pp. 397–402, 2004. View at: Google Scholar
 S. Ma and Q. Lu, “Dynamical bifurcation for a predatorprey metapopulation model with delay,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 1, pp. 13–17, 2005. View at: Google Scholar
 Y. Zhang and J. Xu, “Classification and computation of nonresonant double Hopf bifurcations and solutions in delayed van der PolDuffing system,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 1, pp. 63–68, 2005. View at: Google Scholar
 Z. Zhang and Q. Bi, “Bifurcations of a generalized CamassaHolm equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 1, pp. 81–86, 2005. View at: Google Scholar
 Y. Zheng and Y. Fu, “Effect of damage on bifurcation and chaos of viscoelastic plates,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 1, pp. 87–92, 2005. View at: Google Scholar
 E. Fan, “Soliton solutions for a generalized HirotaSatsuma coupled KdV equation and a coupled MKdV equation,” Physics Letters. A, vol. 282, no. 12, pp. 18–22, 2001. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 E. M. E. Zayed, T. A. Nofal, and K. A. Gepreel, “The travelling wave solutions for nonlinear initialvalue problems using the homotopy perturbation method,” Applicable Analysis, vol. 88, no. 4, pp. 617–634, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Akbarzade and J. Langari, “Determination of natural frequencies by coupled method of homotopy perturbation and variational method for strongly nonlinear oscillators,” Journal of Mathematical Physics, vol. 52, no. 2, Article ID 023518, 10 pages, 2011. View at: Publisher Site  Google Scholar
 S. L. Mei and S. W. Zhang, “Coupling technique of variational iteration and homotopy perturbation methods for nonlinear matrix differential equations,” Computers & Mathematics with Applications, vol. 54, no. 78, pp. 1092–1100, 2007. View at: Publisher Site  Google Scholar
 A. M. Wazwaz, “Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 754–761, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. M. Wazwaz, “New solitary wave solutions to the KuramotoSivashinsky and the Kawahara equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1642–1650, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 H. Zhang, “A complex ansatz method applied to nonlinear equations of Schrödinger type,” Chaos, Solitons and Fractals, vol. 41, no. 1, pp. 183–189, 2009. View at: Publisher Site  Google Scholar
 Y. Shang, “The extended hyperbolic function method and exact solutions of the longshort wave resonance equations,” Chaos, Solitons and Fractals, vol. 36, no. 3, pp. 762–771, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004. View at: Publisher Site  Google Scholar
 J. H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 527–539, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He, “Asymptotology by homotopy perturbation method,” Applied Mathematics and Computation, vol. 156, no. 3, pp. 591–596, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters. A, vol. 350, no. 12, pp. 87–88, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005. View at: Publisher Site  Google Scholar
 J. H. He, “Homotopy perturbation method for bifurcation of nonlinear problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 207–208, 2005. View at: Google Scholar
 J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005. View at: Publisher Site  Google Scholar
 J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 34, pp. 257–262, 1999. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He, “New interpretation of homotopy method,” International Journal of Modern Physics B, vol. 20, no. 18, pp. 2561–2568, 2006. View at: Publisher Site  Google Scholar
 J. H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. H. He, “A Note on the homotopy perturbation method,” Thermal Science, vol. 14, no. 2, pp. 565–568, 2010. View at: Google Scholar
 J. H. He, “A short remark on fractional variational iteration method,” Physics Letters. A, vol. 375, no. 38, pp. 3362–3364, 2011. View at: Publisher Site  Google Scholar
 S. Guo and L. Mei, “The fractional variational iteration method using He's polynomials,” Physics Letters. A, vol. 375, no. 3, pp. 309–313, 2011. View at: Publisher Site  Google Scholar
 S. T. MohyudDin and A. Yildirim, “Variational iteration method for delay differential equations using he's polynomials,” Zeitschrift fur Naturforschung, Section A, vol. 65, no. 12, pp. 1045–1048, 2010. View at: Google Scholar
 A. Yıldırım, “Applying He's variational iteration method for solving differentialdifference equation,” Mathematical Problems in Engineering, vol. 2008, Article ID 869614, 7 pages, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 S. T. MohyudDin and A. Yildirim, “Solving nonlinear boundary value problems using He's polynomials and Padé approximants,” Mathematical Problems in Engineering, vol. 2009, Article ID 690547, 17 pages, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 S. T. MohyudDin, A. Yildirim, S. A. Sezer, and M. Usman, “Modified variational iteration method for freeconvective boundarylayer equation using Padé approximation,” Mathematical Problems in Engineering, vol. 2010, Article ID 318298, 11 pages, 2010. View at: Publisher Site  Google Scholar
 M. Basto, V. Semiao, and F. L. Calheiros, “Numerical study of modified Adomian's method applied to Burgers equation,” Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 927–949, 2007. View at: Publisher Site  Google Scholar
 M. Dehghan, A. Hamidi, and M. Shakourifar, “The solution of coupled Burgers' equations using AdomianPade technique,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1034–1047, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. Biazar, M. Eslami, and H. Ghazvini, “Homotopy perturbation method for systems of partial differential equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 413–418, 2007. View at: Google Scholar
 A. Sadighi and D. D. Ganji, “Solution of the generalized nonlinear boussinesq equation using homotopy perturbation and variational iteration methods,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 435–444, 2007. View at: Google Scholar
 H. Tari, D. D. Ganji, and M. Rostamian, “Approximate solutions of K (2,2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 203–210, 2007. View at: Google Scholar
 E. M. E. Zayed, T. A. Nofal, and K. A. Gepreel, “Homotopy perturbation and Adomain decomposition methods for solving nonlinear Boussinesq equations,” Communications on Applied Nonlinear Analysis, vol. 15, no. 3, pp. 57–70, 2008. View at: Google Scholar
 E. M. E. Zayed, T. A. Nofal, and K. A. Gepreel, “The homotopy perturbation method for solving nonlinear burgers and new coupled modified kortewegde vries equations,” Zeitschrift fur Naturforschung, Section A, vol. 63, no. 1011, pp. 627–633, 2008. View at: Google Scholar
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Copyright © 2012 Taher A. Nofal. 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.