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Variational iteration method for solving twelfth-order boundary-value problems using He’s polynomials

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In this paper, we apply the variational iteration method using He’s polynomials (VIMHP) for solving the twelfth-order boundary-value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The suggested iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.

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References

  1. S. Abbasbandy, “Numerical solutions of nonlinear Klein – Gordon equation by variational iteration method,” Int. J. Numer. Mech. Eng., 70, 876–881 (2007).

    Article  MathSciNet  Google Scholar 

  2. S. Abbasbandy, “A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials,” J. Comput. Appl. Math., 207, 59–63 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  3. M. A. Abdou and A. A. Soliman, “New applications of variational iteration method,” Phys. D., 211, No. 1–2, 1–8 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  4. R. P. Agarwal, Boundary-Value Problems for Higher-Order Differential Equations, World Scientific, Singapore (1986).

    MATH  Google Scholar 

  5. A. Boutayeb and E. H. Twizell, “Finite-difference methods for the solution of special eighth-order boundary-value problems,” Int. J. Comput. Math., 48, 63–75 (1993).

    Article  MATH  Google Scholar 

  6. R. E. D. Bishop, S. M. Cannon, and S. Miao, “On coupled bending and torsional vibration of uniform beams,” J. Sound. Vib., 131, 457–464 (1989).

    Article  Google Scholar 

  7. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, New York (1981).

    Google Scholar 

  8. K. Djidjeli, E. H. Twizell, and A. Boutayeb, “Numerical methods for special nonlinear boundary-value problems of order 2m,” J. Comput. Appl. Math., 47, 35–45 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Ghorbani and J. S. Nadjafi, “He’s homotopy perturbation method for calculating Adomian’s polynomials,” Int. J. Nonlin. Sci. Numer. Simul., 8, No. 2, 229–332 (2007).

    Google Scholar 

  10. A. Ghorbani, “Beyond Adomian’s polynomials: He polynomials,” Chaos Solitons Fractals, in press (2007).

  11. J. H. He, “Homotopy perturbation method for solving boundary-value problems,” Phys. Lett. A, 350, 87–88 (2006).

    Article  MathSciNet  Google Scholar 

  12. J. H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Appl. Math. Comput., 156, 527–539 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  13. J. H. He, “Homotopy perturbation method for bifurcation of nonlinear problems,” Int. J. Nonlin. Sci. Numer. Simul., 6, No. 2, 207–208 (2005).

    Google Scholar 

  14. J. H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Appl. Math. Comput., 151, 287–292 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  15. J. H. He, “A coupling method of homotopy technique and perturbation technique for nonlinear problems,” Int. J. Nonlin. Mech., 35, No. 1, 115–123 (2000).

    Article  Google Scholar 

  16. J. H. He, “Variational iteration method: Some recent results and new interpretations,” J. Comput. Appl. Math., 207, 3–17 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  17. J. H. He, “Variational iteration method: A kind of non-linear analytical technique, some examples,” Int. J. Nonlin. Mech., 34, No. 4, 699–708 (1999).

    Article  MATH  Google Scholar 

  18. J. H. He, “Variational iteration method for autonomous ordinary differential systems,” Appl. Math. Comput., 114, No. 2–3, 115–123 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  19. J. H. He and X. H. Wu, “Construction of solitary solution and compaction-like solution by variational iteration method,” Chaos Solitons Fractals, 29, No. 1, 108–113 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  20. J. H. He, “The variational iteration method for eighth-order initial boundary-value problems,” Phys. Scr., 76, No. 6, 680–682 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  21. J. H. He, “Some asymptotic methods for strongly nonlinear equation,” Int. J. Mod. Phys., 20, No. (20)10, 1144–1199 (2006).

    Google Scholar 

  22. M. Inokuti, H. Sekine, and T. Mura, “General use of the Lagrange multiplier in nonlinear mathematical physics,” in: S. Nemat-Naseer (Ed.), Variational Method in the Mechanics of Solids, Pergamon Press, New York (1978), pp. 156–162.

    Google Scholar 

  23. S. Momani and Z. Odibat, “Application of He’s variational iteration method to Helmholtz equations,” Chaos Solitons Fractals, 27, No. 5, 1119–1123 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  24. S. T. Mohyud-Din, “A reliable algorithm for blasius equation,” in: Proceedings of International Conference on Mathematical Sciences, Selangor, Malaysia (2007), pp. 616–626.

  25. S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving fourth-order boundary-value problems,” Math. Prob. Eng., 1–15, Article ID 98602, doi:10.1155/2007/98602 (2007).

    MathSciNet  Google Scholar 

  26. M. A. Noor and S. T. Mohyud-Din, “An efficient algorithm for solving fifth-order boundary-value problems,” Math. Comput. Model., 45, 954–964 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  27. M. A. Noor and S. T. Mohyud-Din, “Homotopy perturbation method for solving sixth-order boundary-value problems,” Comput. Math. Appl., 55, No. 12, 2953–2972 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  28. M. A. Noor and S. T. Mohyud-Din, “Homotopy perturbation method for nonlinear higher-order boundary-value problems,” Int. J. Nonlin. Sci. Numer. Simul., 9, No. 4, 395–408 (2008).

    Google Scholar 

  29. M. A. Noor and S. T. Mohyud-Din, “Approximate solutions of Flierl Petviashivili equation and its variants,” Int. J. Math. Comput. Sci., 2, No. 4, 345–360 (2007).

    MATH  MathSciNet  Google Scholar 

  30. M. A. Noor and S. T. Mohyud-Din, “Variational homotopy perturbation method for solving higher-dimensional initial boundary-value problems,” Math. Prob. Eng., in press (2008).

  31. M. A. Noor and S. T. Mohyud-Din, “Modified variational iteration method for heat and wave-like equations,” Acta Appicandae Mathematicae, DOI: 10.1007/s10440–008–9255-x (2008).

    Google Scholar 

  32. M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for solving higher-order nonlinear boundary-value problems using He’s polynomials,” Int. J. Nonlin. Sci. Numer. Simul., 9, No. 2, 141–157 (2008).

    Google Scholar 

  33. M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for unsteady flow of gas through a porous medium using He’s polynomials and Pade approximants,” Comput. Math. Appl. (2008).

  34. M. A. Noor and S. T. Mohyud-Din, “Modified variational iteration method for solving fourth-order boundary-value problems,” J. Appl. Math. Comput., Article ID: 90, DOI: 10.1007/s12190–008–0090-z (2008).

    MathSciNet  Google Scholar 

  35. M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for solving fifth-order boundary-value problems using He’s polynomials,” Math. Prob. Eng., Article ID 954794, doi: 10:1155/2008/954794 (2008).

    Google Scholar 

  36. M. A Noor and S. T. Mohyud-Din, “An efficient method for fourth-order boundary-value problems,” Comput. Math. Appl., 54, 1101–1111 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  37. M. A. Noor and S. T. Mohyud-Din, “Variational iteration technique for solving higher-order boundary-value problems,” Appl. Math. Comput., 189, 1929–1942 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  38. M. A. Noor and S. T. Mohyud-Din, “Solution of singular and nonsingular initial and boundary-value problems by modified variational iteration method,” Math. Prob. Eng., in press (2008).

  39. M. A. Noor and S. T. Mohyud-Din, “Solution of twelfth-order boundary-value problems by variational iteration technique,” J. Appl. Math. Comput, DOI: 10.1007/s12190–008–0081–0 (2008).

    MathSciNet  Google Scholar 

  40. M. A. Noor and S. T. Mohyud-Din, “Variational iteration decomposition method for solving eighth-order boundary-value problems,” Differ. Equat. Nonlin. Mech., doi: 10.1155/2007/19529 (2007).

    Google Scholar 

  41. J. I. Ramos, “On the variational iteration method and other iterative techniques for nonlinear differential equations,” Appl. Math. Comput., 199, 39–69 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  42. S. S. Siddiqi and E. H. Twizell, “Spline solution of linear eighth-order boundary-value problems,” Comput. Math. Appl. Mech. Eng., 131, 309–325 (1996).

    Article  MATH  Google Scholar 

  43. A. M. Wazwaz, “Approximate solutions to boundary-value problems of higher order by the modified decomposition method,” Comput. Math. Appl., 40, 679–691 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  44. A. M. Wazwaz, “The modified decomposition method for solving linear and nonlinear boundary-value problems of tenth-order and twelfth-order,” Int. J. Nonlin. Sci. Numer. Simul., 1, 17–24 (2000).

    MATH  MathSciNet  Google Scholar 

  45. L. Xu, “The variational iteration method for fourth-order boundary-value problems,” Chaos Solitons Fractals, in press (2007).

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Authors and Affiliations

  1. Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan

    Muhammad Aslam Noor & Syed Tauseef Mohyud-Din

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  1. Muhammad Aslam Noor
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  2. Syed Tauseef Mohyud-Din
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Correspondence to Syed Tauseef Mohyud-Din.

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Noor, M.A., Mohyud-Din, S.T. Variational iteration method for solving twelfth-order boundary-value problems using He’s polynomials. Comput Math Model 21, 239–251 (2010). https://doi.org/10.1007/s10598-010-9068-4

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