Research Article  Open Access
Qian Lijuan, Tian Lixin, Ma Kaiping, "Variational Iteration Method for Solving the Generalized DegasperisProcesi Equation", Abstract and Applied Analysis, vol. 2014, Article ID 629434, 6 pages, 2014. https://doi.org/10.1155/2014/629434
Variational Iteration Method for Solving the Generalized DegasperisProcesi Equation
Abstract
We introduce the variational iteration method for solving the generalized DegasperisProcesi equation. Firstly, according to the variational iteration, the Lagrange multiplier is found after making the correction functional. Furthermore, several approximations of which is converged to are obtained, and the exact solutions of DegasperisProcesi equation will be obtained by using the traditional variational iteration method with a suitable initial approximation . Finally, after giving the perturbation item, the approximate solution for original equation will be expressed specifically.
1. Introduction
The theory of soliton has extensive applications in physics, mechanics, and combustion science. In recent years, many researchers studied the soliton theory in the fields of shock wave [1, 2], light scattering, quantum mechanics, atmospheric physics, neural networks, explosion, and combustion [3]. There are many new methods for searching the soliton solution of nonlinear evolution equations such as hyperbolic tangent function method [4], the homogeneous balance methods [5], Jacobi elliptic function expansion method [3], and pseudospectral method [6].
The variational iteration method (VIM) was developed, in 1999, by He [7–13]. The VIM gives rapidly convergent successive approximations of the exact solution if such a solution exists; otherwise, a few approximations can be used for numerical purposes. The Adomian decomposition method suffers from the complicated computational work needed for the derivation of Adomian polynomials for nonlinear terms. The VIM has no specific requirements, such as linearization, small parameters for nonlinear operators. Therefore, the VIM can overcome the foregoing restrictions and limitations of perturbation techniques, so that it provides us with a possibility to analyze strongly nonlinear problems. On the other hand, the VIM is capable of greatly reducing the size of calculation while still maintaining high accuracy of the numerical solution [14]. Moreover, the power of the method gives it a wider applicability in handling a huge number of analytical and numerical applications. The VIM was successfully applied to study a variety of differential equations. It is based on Lagrange multiplier, and it has the merits of simplicity and easy execution. As a result, it has been proved by many authors to be a powerful mathematical tool for addressing various kinds of linear and nonlinear problem. For example, this method was used for solving nonlinear wave equations and the Laplace equation by Wazwaz [14]. The VIM for solving linear systems of ODEs with constant coefficients was studied by Khojasteh Salkuyeh [15]. Helmholtz equation was researched by Momani and Abuasad [16]. Geng [17] introduced the piecewise VIM for solving Riccati differential equation. Fractional vibration equation was researched by Das [18]. Furthermore, higher order boundary value problems were researched by Xu [19],Noor, and MohyudDin [20]. Noor et al. [21] applied a modified He’s variational iteration method for solving singular fourthorder parabolic partial differential equations. The proposed modification is made by introducing He’s polynomials in the correction functional. Ghorbani and SaberiNadjafi [22] modified the VIM by constructing an initial trial function without unknown parameters. Sevimlican [23] constructed approximate Green’s function for a vector equation for the electric field by using VIM.
In this paper, we are concerned with the variational iterations method for solving the generalized DegasperisProcesi equation. As a review, we will recall the VIM briefly in Section 2.
2. Variational Iteration Method
In this section, the basic concepts of variational iteration method (VIM) are introduced. Here, a description of method [7–15] is given to handle the general nonlinear problem. Consider the differential equation of the form where is a linear operator, is a nonlinear operator, and is the inhomogeneous term. According to He’s variational iteration method, we can construct a correction functional for (1) as follows: where is a general Lagrange multiplier, which can be identified optimally via variational theory [12, 24]. Here is considered as a restricted variation [14, 25] which means ; the subscript denotes the th approximations. The successive approximations , of the solution , can be obtained after using the obtained Lagrange multiplier and the zeroth approximation , which are selected from any function that satisfies the initial conditions. With determined, several approximations , follow. Consequently, the exaction solution may be obtained as In fact, the VIM depends on the suitable selection of the initial approximation . Moreover, we use a wellknown, powerful tool to prove the convergence of the sequence obtained via the VIM and its rate. It is the Banach’s fixed point theorem that follows.
Theorem 1 (Banach’s fixed point theorem). Assume that is a Banach space and is a nonlinear mapping, and suppose that for some constant . Then has a unique fixed point. Furthermore, the sequence with an arbitrary choice of converges to the fixed point of .
According to Theorem 1, for the nonlinear mapping a sufficient condition for the convergence of the variational iteration method is strict contraction of . Furthermore, the sequence (2) converges to the fixed point of which is also the solution of problem (1). Some modifications to prove the convergence speed and to lengthen the interval of convergence for VIM series solution are suggested in [17, 26–30].
3. The Variational Iteration of Generalized DegasperisProcesi Equation
Degasperis and Procesi consider the following family of thirdorder dispersive conservation laws [31], where , , , , , and are real constants. In this family, only three equations satisfy asymptotic integrability conditions [31]. That is, if , , , , , and , (8) is the KdV equation If , , , , , and , (8) is the CamassaHolm equation If , , , , , and , (8) is the DegasperisProcesi equation It should be mentioned that both CH and DP equations are derived as members of a oneparameter family of asymptotic shallow water approximations to the Euler equations. It shows that the two equations are physically relevant; otherwise, the DP equation would be of purely theoretical interest.
Variational iteration method for KdVBurgers and Lax’s seventhorder KdV equations has been studied by Soliman [32]. In this paper, we consider the generalized DegasperisProcesi equation that was proposed in [33]. is the generalized perturbation item. We suppose is a sufficiently smooth function of the variable.
Step 1. Make the independent variable transformation: Here, is an arbitrary complex number. is wave number; is wave velocity. Substituting (13) into (12), we have Here, is the derivative of with respect to ; that is, . .
Step 2. From [34], we find the special solution, when is identical to :
Remark 2. Notice that , , , , and , where . These solutions contain the other four types of forms named , , , and .
Step 3. Make the correction functional
Here, , , , and are considered as a restricted variation [35]. That is,
Step 4. Under the above condition, make the correct functional stationary with respect to ; noticing that , we have
For arbitrary , from the above relation, we obtain the EulerLanguage equation:
Solve (19), we derive
Substituted (20) into (16), we have the integration form:
From the above solution procedure, we can see that the approximate solutions converge to its exact solution. That is, , is the approximate solution with arbitrary degree of accurate solitary wave of DegasperisProcesi equation.
Step 5. Calculation of the approximate solution.
According to the integration form (21),we can calculate the approximate solution. Firstly, let (15) be the zeroorder approximate solution:
Substitute (15) into (21). We obtain the oneorder approximate solution
in which .
Then, substitute (23) into (21). We can obtain the secondorder approximate solution :
Using the same method, we can get the higher order approximate solution.
4. The Optical Soliton Perturbation Solution and the Numerical Example
Specially, we set Then (12) change to We obtain the zeroorder and the oneorder approximate solution of (20)
We set we will obtain the following numerical example. See Tables 1 and 2.


5. Conclusion
By the analysis of structure on the left side of (8) and the properties of about the variable and the analytical variational iteration formula, we can prove that the sequence of functions of decided by (21) is uniform convergence. So the limit function of is the solution of the equation. Moreover, the zeroorder approximate solution is the soliton of (8) in which ; it should be specially pointed out that the more accurate the identification of the multiplier is, the faster the approximations converge to their exact solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work is supported by the Youth Foundation of National Natural Science Foundation of China (Grant no. 71101072).
References
 M. J. McPhaden and D. Zhang, “Slowdown of the meridional overturning circulation in the upper Pacific Ocean,” Nature, vol. 415, no. 6872, pp. 603–608, 2002. View at: Publisher Site  Google Scholar
 D. Gu and S. G. H. Philander, “Interdecadal climate fluctuations that depend on exchanges between the tropics and extratropics,” Science, vol. 275, no. 5301, pp. 805–807, 1997. View at: Publisher Site  Google Scholar
 S. K. Liu, Z. T. Fu, and S. D. Liu, “The envelope periodic nonlinear wave equations with Jacobi Elliptic function solutions,” Acta Physica Sinica, vol. 51, no. 1, pp. 10–14, 2002 (Chinese). View at: Google Scholar  Zentralblatt MATH  MathSciNet
 W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” American Journal of Physics, vol. 60, no. 7, pp. 650–654, 1992. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 E. Fan and H. Zhang, “A note on the homogeneous balance method,” Physics Letters A, vol. 246, no. 5, pp. 403–406, 1998. View at: Google Scholar
 P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelength,” Physical Review Letters, vol. 70, no. 5, pp. 564–567, 1993. View at: Publisher Site  Google Scholar
 J. H. He, “Approximate solution of non linear differential equations with convolution product nonlinearities,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 12, pp. 69–73, 1998. View at: Publisher Site  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. 12, pp. 57–68, 1998. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J.H. He, “Variational iteration method for autonomous ordinary differential systems,” Applied Mathematics and Computation, vol. 114, no. 23, pp. 115–123, 2000. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J.H. He, “Variational principles for some nonlinear partial differential equations with variable coefficients,” Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 847–851, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 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  MathSciNet
 J.H. He and X.H. Wu, “Variational iteration method: new development and applications,” Computers & Mathematics with Applications, vol. 54, no. 78, pp. 881–894, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J.H. He, “Variational iteration method—some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A.M. Wazwaz, “The variational iteration method: a reliable analytic tool for solving linear and nonlinear wave equations,” Computers & Mathematics with Applications, vol. 54, no. 78, pp. 926–932, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 D. Khojasteh Salkuyeh, “Convergence of the variational iteration method for solving linear systems of ODEs with constant coefficients,” Computers & Mathematics with Applications, vol. 56, no. 8, pp. 2027–2033, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. Momani and S. Abuasad, “Application of He's variational iteration method to Helmholtz equation,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1119–1123, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. Geng, “A modified variational iteration method for solving Riccati differential equations,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 1868–1872, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. Das, “Solution of fractional vibration equation by the variational iteration method and modified decomposition method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 4, pp. 361–366, 2008. View at: Google Scholar
 L. Xu, “The variational iteration method for fourth order boundary value problems,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1386–1394, 2009. View at: Publisher Site  Google Scholar
 M. A. Noor and S. T. MohyudDin, “Variational iteration method for solving higherorder nonlinear boundary value problems using He's polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 141–156, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. A. Noor, K. I. Noor, and S. T. MohyudDin, “Modified variational iteration technique for solving singular fourthorder parabolic partial differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e630–e640, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. Ghorbani and J. SaberiNadjafi, “An effective modification of He's variational iteration method,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 2828–2833, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. Sevimlican, “Constructing approximate Green's function for a vector equation for the electric field using the variational iteration method,” Applied Mathematics Letters, vol. 23, no. 5, pp. 533–536, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. Inokuti, H. Sekine, and T. Mura, “General use of the Lagrange multiplier in nonlinear mathematical physics,” in Variational Method in the Mechanics of Solids, S. NematNasser, Ed., pp. 159–162, Pergamon Press, NewYork, NY, USA, 1978. View at: Google Scholar
 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
 A.M. Wazwaz, “A comparison between the variational iteration method and Adomian decomposition method,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 129–136, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 T. A. Abassy, M. A. ElTawil, and H. El Zoheiry, “Solving nonlinear partial differential equations using the modified variational iteration Padé technique,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 73–91, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. M. Odibat, “Reliable approaches of variational iteration method for nonlinear operators,” Mathematical and Computer Modelling, vol. 48, no. 12, pp. 222–231, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. Tatari and M. Dehghan, “Improvement of He's variational iteration method for solving systems of differential equations,” Computers & Mathematics with Applications, vol. 58, no. 1112, pp. 2160–2166, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. M. Odibat, “A study on the convergence of variational iteration method,” Mathematical and Computer Modelling, vol. 51, no. 910, pp. 1181–1192, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. Degasperis and M. Procesi, Asymoptotic Integrability, Symmetry and Perturbation Theory, World Scientific, River Edge, NJ, USA, 1999.
 A. A. Soliman, “A numerical simulation and explicit solutions of KdVBurgers' and Lax's seventhorder KdV equations,” Chaos, Solitons and Fractals, vol. 29, no. 2, pp. 294–302, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. Q. Yu and L. X. Tian, “A new traveling wave solutions for the DegasperisProcesi equation,” Journal of Jiangsu University, vol. 26, no. 3, pp. 231–234, 2005 (Chinese). View at: Google Scholar  MathSciNet
 J. H. He, Approximate Nonlinear Analytical Methods in Engineering and Sciences, Henan Science and Technology Press, Zhengzhou, China, 2002, (Chinese).
Copyright
Copyright © 2014 Qian Lijuan 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.