`Journal of Applied MathematicsVolume 2012, Article ID 370843, 19 pageshttp://dx.doi.org/10.1155/2012/370843`
Research Article

## Approximate Solutions for Nonlinear Initial Value Problems Using the Modified Variational Iteration Method

1Mathematics Department, Faculty of Science, El-Minia University, El-Minia 61519, Egypt
2Mathematics Department, Faculty of Science, Taif University, Taif 21974, Saudi Arabia

Received 19 February 2012; Revised 27 March 2012; Accepted 28 March 2012

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.

1. S.-F. Deng, “Bäcklund transformation and soliton solutions for KP equation,” Chaos, Solitons and Fractals, vol. 25, no. 2, pp. 475–480, 2005.
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.
3. O. Pashaev and G. Tanoğlu, “Vector shock soliton and the Hirota bilinear method,” Chaos, Solitons & Fractals, vol. 26, no. 1, pp. 95–105, 2005.
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.
5. L. De-Sheng, G. Feng, and Z. Hong-Qing, “Solving the $\left(2+1\right)$-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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
45. J. H. He, “Asymptotology by homotopy perturbation method,” Applied Mathematics and Computation, vol. 156, no. 3, pp. 591–596, 2004.
46. J. H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters. A, vol. 350, no. 1-2, pp. 87–88, 2006.
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.
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.
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.
50. J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999.
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.
53. J. H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.