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Noor, Muhammad Aslam; Mohyud-Din, Syed Tauseef

Variational iteration method for fifth-order boundary value problems using He’s polynomials. (English) Zbl 1151.65334

Math. Probl. Eng. 2008, Article ID 954794, 12 p. (2008).
Summary: We apply the variational iteration method using J.-H. He’s polynomials (VIMHP) [Phys. Scr. 76, No. 6, 680–682 (2007; Zbl 1134.34307)]’s for solving the fifth-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods and is mainly due to A. Ghorbani [“Beyond Adomian’s polynomials: He’s polynomials”, to appear in Chaos Solitons Fractals]. 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 discritization, 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.

Cited in 35 Documents

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations

Citations:

Zbl 1134.34307
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\textit{M. A. Noor} and \textit{S. T. Mohyud-Din}, Math. Probl. Eng. 2008, Article ID 954794, 12 p. (2008; Zbl 1151.65334)
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References:

[1] H. N. \cCaglar, S. H. \cCaglar, and E. H. Twizell, “The numerical solution of fifth-order boundary value problems with sixth-degree B-spline functions,” Applied Mathematics Letters, vol. 12, no. 5, pp. 25-30, 1999. · Zbl 0941.65073
[2] A. R. Davies, A. Karageorghis, and T. N. Phillips, “Spectral Galerkin methods for the primary two-point boundary value problem in modelling viscoelastic flows,” International Journal for Numerical Methods in Engineering, vol. 26, no. 3, pp. 647-662, 1988. · Zbl 0635.73091
[3] D. J. Fyfe, “Linear dependence relations connecting equal interval Nth degree splines and their derivatives,” Journal of the Institute of Mathematics and Its Applications, vol. 7, no. 3, pp. 398-406, 1971. · Zbl 0219.65010
[4] M. A. Noor and S. T. Mohyud-Din, “An efficient algorithm for solving fifth-order boundary value problems,” Mathematical and Computer Modelling, vol. 45, no. 7-8, pp. 954-964, 2007. · Zbl 1133.65052
[5] M. A. Noor and S. T. Mohyud-Din, “Variational iteration technique for solving higher order boundary value problems,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1929-1942, 2007. · Zbl 1122.65374
[6] M. A. Noor and S. T. Mohyud-Din, “Modified decomposition method for solving linear and nonlinear fifth-order boundary value problems,” International Journal of Applied Mathematics and Computer Science. In press. · Zbl 1133.65052
[7] A.-M. Wazwaz, “The numerical solution of fifth-order boundary value problems by the decomposition method,” Journal of Computational and Applied Mathematics, vol. 136, no. 1-2, pp. 259-270, 2001. · Zbl 0986.65072
[8] 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
[9] 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
[10] 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
[11] 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
[12] J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for nonlinear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37-43, 2000. · Zbl 1068.74618
[13] 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. · Zbl 1119.65049
[14] J.-H. He, “Variational iteration method-a kind of nonlinear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699-708, 1999. · Zbl 1342.34005
[15] J.-H. He, “Variational iteration method for autonomous ordinary differential systems,” Applied Mathematics and Computation, vol. 114, no. 2-3, pp. 115-123, 2000. · Zbl 1027.34009
[16] J. H. He and X. H. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 108-113, 2006. · Zbl 1147.35338
[17] 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
[18] J.-H. He, “The variational iteration method for eighth-order initial-boundary value problems,” Physica Scripta, vol. 76, no. 6, pp. 680-682, 2007. · Zbl 1134.34307
[19] 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. Nemat-Naseer, Ed., pp. 156-162, Pergamon Press, New York, NY, USA, 1978.
[20] S. Abbasbandy, “Numerical solution of nonlinear Klein-Gordon equations by variational iteration method,” International Journal for Numerical Methods in Engineering, vol. 70, no. 7, pp. 876-881, 2007. · Zbl 1194.65120
[21] S. Abbasbandy, “A new application of He/s variational iteration method for quadratic Riccati differential equation by using Adomian/s polynomials,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 59-63, 2007. · Zbl 1120.65083
[22] S. Abbasbandy, “Numerical method for nonlinear wave and diffusion equations by the variational iteration method,” International Journal for Numerical Methods in Engineering, vol. 73, no. 12, pp. 1836-1843, 2008. · Zbl 1159.76372
[23] A. Ghorbani and J. Saberi-Nadjafi, “He/s homotopy perturbation method for calculating adomian polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 229-232, 2007. · Zbl 1401.65056
[24] A. Ghorbani, “Beyond Adomian/s polynomials: He polynomials,” Chaos, Solitons & Fractals. In press. · Zbl 1197.65061
[25] 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. · Zbl 1086.65113
[26] S. T. Mohyud-Din, “A reliable algorithm for Blasius equation,” in Proceedings of the International Conference of Mathematical Sciences (ICMS /07), pp. 616-626, Bangi-Putrajaya, Malaysia, November 2007.
[27] S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving fourth-order boundary value problems,” Mathematical Problems in Engineering, vol. 2007, Article ID 98602, 15 pages pages, 2007. · Zbl 1144.65311
[28] M. A. Noor and S. T. Mohyud-Din, “Homotopy perturbation method for solving sixth-order boundary value problems,” Computers and Mathematics with Applications. In press. · Zbl 1142.65386
[29] M. A. Noor and S. T. Mohyud-Din, “Homotopy method for solving eighth order boundary value problems,” Journal of Mathematical Analysis and Approximation Theory, vol. 1, no. 2, pp. 161-169, 2006. · Zbl 1204.65086
[30] M. A. Noor and S. T. Mohyud-Din, “Approximate solutions of Flieral Petviashivili equation and its variants,” International Journal of Mathematics and Computer Science, vol. 2, no. 4, pp. 345-360, 2007. · Zbl 1136.65073
[31] M. A. Noor and S. T. Mohyud-Din, “A reliable approach for solving linear and nonlinear sixth-order boundary value problems,” International Journal of Computational and Applied Mathematics. In press. · Zbl 1142.65386
[32] M. A. Noor and S. T. Mohyud-Din, “An efficient method for fourth-order boundary value problems,” Computers and Mathematics with Applications, vol. 54, no. 7-8, pp. 1101-1111, 2007. · Zbl 1141.65375
[33] M. A. Noor and S. T. Mohyud-Din, “Variational iteration technique for solving tenth-order boundary value problems,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1929-1942, 2007. · Zbl 1122.65374
[34] M. A. Noor and S. T. Mohyud-Din, “Variational iteration decomposition method for solving eighth-order boundary value problems,” Differential Equations and Nonlinear Mechanics. In press. · Zbl 1143.49023
[35] J. I. Ramos, “On the variational iteration method and other iterative techniques for nonlinear differential equations,” Applied Mathematics and Computation. In press. · Zbl 1142.65082
[36] L. Xu, “The variational iteration method for fourth-order boundary value problems,” Chaos, Solitons & Fractals. In press. · Zbl 1197.65114
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