##
**Approximate solutions for nonlinear initial value problems using the modified variational iteration method.**
*(English)*
Zbl 1251.65148

Summary: We use the modified variational iteration method (MVIM) to find the approximate solutions for some nonlinear initial value problems in the mathematical physics, via the Burgers-Fisher equation, the Kuramoto-Sivashinsky equation, the coupled Schrödinger-KdV equations, and the long-short 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.

### MSC:

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35Q53 | KdV equations (Korteweg-de Vries equations) |

PDF
BibTeX
XML
Cite

\textit{T. A. Nofal}, J. Appl. Math. 2012, Article ID 370843, 19 p. (2012; Zbl 1251.65148)

Full Text:
DOI

### References:

[1] | S.-F. Deng, “Bäcklund transformation and soliton solutions for KP equation,” Chaos, Solitons and Fractals, vol. 25, no. 2, pp. 475-480, 2005. · Zbl 1070.35059 |

[2] | G. Tsigaridas, A. Fragos, I. Polyzos et al., “Evolution of near-soliton initial conditions in non-linear wave equations through their Bäcklund transforms,” Chaos, Solitons and Fractals, vol. 23, no. 5, pp. 1841-1854, 2005. · Zbl 1069.35066 |

[3] | O. Pashaev and G. Tano\uglu, “Vector shock soliton and the Hirota bilinear method,” Chaos, Solitons & Fractals, vol. 26, no. 1, pp. 95-105, 2005. · Zbl 1070.35056 |

[4] | 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. · Zbl 1030.37047 |

[5] | L. De-Sheng, G. Feng, and Z. Hong-Qing, “Solving the (2+1)-dimensional higher order Broer-Kaup system via a transformation and tanh-function method,” Chaos, Solitons and Fractals, vol. 20, no. 5, pp. 1021-1025, 2004. · Zbl 1049.35157 |

[6] | E. M. E. Zayed, H. A. Zedan, and K. A. Gepreel, “Group analysis and modified extended tanh-function 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. · Zbl 1401.35014 |

[7] | H. A. Abdusalam, “On an improved complex tanh-function method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 99-106, 2005. · Zbl 1401.35012 |

[8] | T. A. Abassy, M. A. El-Tawil, 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. · Zbl 1401.65122 |

[9] | S. M. El-Sayed, “The decomposition method for studying the Klein-Gordon equation,” Chaos, Solitons and Fractals, vol. 18, no. 5, pp. 1025-1030, 2003. · Zbl 1068.35069 |

[10] | D. Kaya and S. M. El-Sayed, “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. · Zbl 1030.35139 |

[11] | H. M. Liu, “Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method,” Chaos, Solitons and Fractals, vol. 23, no. 2, pp. 573-576, 2005. · Zbl 1135.76597 |

[12] | 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. · Zbl 06942051 |

[13] | 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. · Zbl 1135.35303 |

[14] | 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. · Zbl 1107.37021 |

[15] | J. H. He, “Variational iteration method - A kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699-708, 1999. · Zbl 1342.34005 |

[16] | G.E. Draganescu and V. Capalnasan, “Nonlinear relaxation phenomena in polycrys-talline solids,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 4, pp. 219-226, 2003. |

[17] | J. H. He, “Modified Lindstedt-Poincaré methods for some strongly non-linear oscillations. I. Expansion of a constant,” International Journal of Non-Linear Mechanics, vol. 37, no. 2, pp. 309-314, 2002. · Zbl 1116.34320 |

[18] | J. H. He, “Modified Lindstedt-Poincaré methods for some strongly non-linear oscillations. II. A new transformation,” International Journal of Non-Linear Mechanics, vol. 37, no. 2, pp. 315-320, 2002. · Zbl 1116.34321 |

[19] | J. H. He, “Modified Lindsted-Poincare 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. · Zbl 1072.34507 |

[20] | H. M. Liu, “Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method,” Chaos, Solitons and Fractals, vol. 23, no. 2, pp. 573-576, 2005. · Zbl 1135.76597 |

[21] | 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. · Zbl 0671.34053 |

[22] | A. M. Wazwaz, “A reliable technique for solving the wave equation in an infinite one-dimensional medium,” Applied Mathematics and Computation, vol. 92, no. 1, pp. 1-7, 1998. · Zbl 0942.65107 |

[23] | D. Wang and H.-Q. Zhang, “Further improved F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equation,” Chaos, Solitons and Fractals, vol. 25, no. 3, pp. 601-610, 2005. · Zbl 1083.35122 |

[24] | 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. · Zbl 1092.37054 |

[25] | X. H. Wu and J. H. He, “Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 966-986, 2007. · Zbl 1143.35360 |

[26] | J. H. He and X. H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700-708, 2006. · Zbl 1141.35448 |

[27] | J. H. He and M. A. Abdou, “New periodic solutions for nonlinear evolution equations using Exp-function method,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1421-1429, 2007. · Zbl 1152.35441 |

[28] | J. H. He, Gongcheng Yu Kexue Zhong de jinshi feixianxing feixi fangfa, Henan Science and Technology Press, Zhengzhou, China, 2002. |

[29] | J. H. He, “Determination of limit cycles for strongly nonlinear oscillators,” Physical Review Letters, vol. 90, no. 17, Article ID 174301, 3 pages, 2003. |

[30] | J. Shen and W. Xu, “Bifurcations of smooth and non-smooth travelling wave solutions of the Degasperis-Procesi equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 5, no. 4, pp. 397-402, 2004. · Zbl 1401.35276 |

[31] | S. Ma and Q. Lu, “Dynamical bifurcation for a predator-prey metapopulation model with delay,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 1, pp. 13-17, 2005. · Zbl 1401.92032 |

[32] | Y. Zhang and J. Xu, “Classification and computation of non-resonant double Hopf bifurcations and solutions in delayed van der Pol-Duffing system,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 1, pp. 63-68, 2005. · Zbl 1401.37058 |

[33] | Z. Zhang and Q. Bi, “Bifurcations of a generalized Camassa-Holm equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 1, pp. 81-86, 2005. · Zbl 1401.37057 |

[34] | 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. · Zbl 1401.74196 |

[35] | E. Fan, “Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled MKdV equation,” Physics Letters. A, vol. 282, no. 1-2, pp. 18-22, 2001. · Zbl 0984.37092 |

[36] | E. M. E. Zayed, T. A. Nofal, and K. A. Gepreel, “The travelling wave solutions for non-linear initial-value problems using the homotopy perturbation method,” Applicable Analysis, vol. 88, no. 4, pp. 617-634, 2009. · Zbl 1167.35493 |

[37] | 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. · Zbl 1314.34076 |

[38] | 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. 7-8, pp. 1092-1100, 2007. · Zbl 1267.65102 |

[39] | 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. · Zbl 1132.65098 |

[40] | A. M. Wazwaz, “New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawahara equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1642-1650, 2006. · Zbl 1107.65094 |

[41] | 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. · Zbl 1198.81109 |

[42] | Y. Shang, “The extended hyperbolic function method and exact solutions of the long-short wave resonance equations,” Chaos, Solitons and Fractals, vol. 36, no. 3, pp. 762-771, 2008. · Zbl 1153.35374 |

[43] | J. H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287-292, 2004. · Zbl 1039.65052 |

[44] | J. H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 527-539, 2004. · Zbl 1062.65074 |

[45] | J. H. He, “Asymptotology by homotopy perturbation method,” Applied Mathematics and Computation, vol. 156, no. 3, pp. 591-596, 2004. · Zbl 1061.65040 |

[46] | J. H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters. A, vol. 350, no. 1-2, pp. 87-88, 2006. · Zbl 1195.65207 |

[47] | J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695-700, 2005. · Zbl 1072.35502 |

[48] | 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. · Zbl 1401.65085 |

[49] | J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695-700, 2005. · Zbl 1072.35502 |

[50] | J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999. · Zbl 0956.70017 |

[51] | J. H. He, “New interpretation of homotopy method,” International Journal of Modern Physics B, vol. 20, no. 18, pp. 2561-2568, 2006. |

[52] | J. H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141-1199, 2006. · Zbl 1102.34039 |

[53] | J. H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73-79, 2003. · Zbl 1030.34013 |

[54] | J. H. He, “A Note on the homotopy perturbation method,” Thermal Science, vol. 14, no. 2, pp. 565-568, 2010. |

[55] | J. H. He, “A short remark on fractional variational iteration method,” Physics Letters. A, vol. 375, no. 38, pp. 3362-3364, 2011. · Zbl 1252.49027 |

[56] | 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. · Zbl 1241.35216 |

[57] | S. T. Mohyud-Din 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. |

[58] | A. Yıldırım, “Applying He’s variational iteration method for solving differential-difference equation,” Mathematical Problems in Engineering, vol. 2008, Article ID 869614, 7 pages, 2008. · Zbl 1155.65384 |

[59] | S. T. Mohyud-Din 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. · Zbl 1188.65093 |

[60] | S. T. Mohyud-Din, A. Yildirim, S. A. Sezer, and M. Usman, “Modified variational iteration method for free-convective boundary-layer equation using Padé approximation,” Mathematical Problems in Engineering, vol. 2010, Article ID 318298, 11 pages, 2010. · Zbl 1191.76080 |

[61] | 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. · Zbl 1387.65110 |

[62] | M. Dehghan, A. Hamidi, and M. Shakourifar, “The solution of coupled Burgers’ equations using Adomian-Pade technique,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1034-1047, 2007. · Zbl 1122.65388 |

[63] | 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. · Zbl 1197.65106 |

[64] | 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. · Zbl 1120.65108 |

[65] | 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. · Zbl 06942263 |

[66] | 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. · Zbl 1343.65127 |

[67] | E. M. E. Zayed, T. A. Nofal, and K. A. Gepreel, “The homotopy perturbation method for solving nonlinear burgers and new coupled modified korteweg-de vries equations,” Zeitschrift fur Naturforschung, Section A, vol. 63, no. 10-11, pp. 627-633, 2008. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.