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Modified variational iteration method for solving Helmholtz equations

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In this paper, we apply the modified variational iteration method (MVIM) for solving the Helmholtz equations. The proposed modification is made by introducing He's polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed 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 the Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.

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References

  1. S. M. El-Sayed and D. Kaya, “Comparing numerical methods for Helmholtz equation model problem,” Appl. Math. Comp., 150, 763–773 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  2. C. F. Gerald and P. O. Wheatly, Applied Numerical Analysis, Addison-Wesley, California (1994).

    MATH  Google Scholar 

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

    Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. 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 

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

    Google Scholar 

  11. 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 

  12. J. H. He and X. Wu, “Variational iteration method: New developments and applications,” Comput. Math. Appl., 54, 881–894 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  13. 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 

  14. 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 

  15. 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 

  16. 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 

  17. 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 

  18. 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 

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

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

  21. 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 

  22. 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 

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

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

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

  27. 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. Num. Simul., 9, No. 2, 141–157 (2008).

    Google Scholar 

  28. 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).

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

  30. 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. (2008), Article ID 954794, doi: 10: 1155/2008/954794.

  31. 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 

  32. 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 

  33. 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. (2008), in press.

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

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

  36. M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for solving initial value problems of Bratu type,” Appl. Math. (2008).

  37. M. A. Noor and S. T. Mohyud-Din, “An iterative method for solving Helmholtz equations,” Arab J. Math. Math. Sci., 9–15 (2007).

  38. 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 

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

    M. A. Noor & S. T. Mohyud-Din

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  1. M. A. Noor
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  2. S. T. Mohyud-Din
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Correspondence to S. T. Mohyud-Din.

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Noor, M.A., Mohyud-Din, S.T. Modified variational iteration method for solving Helmholtz equations. Comput Math Model 20, 40–50 (2009). https://doi.org/10.1007/s10598-009-9018-1

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