Variational Iteration Decomposition Method for Solving Eighth-Order Boundary Value Problems

Noor, Muhammad Aslam; Mohyud-Din, Syed Tauseef

doi:https://doi.org/10.1155/2007/19529

International Journal of Differential Equations

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Research Article | Open Access

Volume 2007 | Article ID 019529 | https://doi.org/10.1155/2007/19529

Variational Iteration Decomposition Method for Solving Eighth-Order Boundary Value Problems

Muhammad Aslam Noor¹and Syed Tauseef Mohyud-Din¹

Academic Editor: Yong Zhou

Received18 Sept 2007

Revised10 Nov 2007

Accepted11 Dec 2007

Published13 Apr 2008

Abstract

We implement a relatively new analytical technique, the variational iteration decomposition method (VIDM), for solving the eighth-order boundary value problems. The proposed method is an elegant combination of variational iteration method and decomposition method. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Numerical work is given to check the efficiency of the method. Comparisons are made to confirm the reliability and accuracy of the technique. The technique can be used as an alternative for solving nonlinear boundary value problems.

References

S. ChandraSekhar, in Hydrodynamic and Hydro Magnetic Stability, Dover, New York, NY, USA, 1981.
K. Djidjeli, E. H. Twizell, and A. Boutayeb, “Numerical methods for special nonlinear boundary value problems of order $2 m$ ,” Journal of Computational and Applied Mathematics, vol. 47, no. 1, pp. 35–45, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.
View at: Google Scholar | MathSciNet
S. S. Siddiqi and E. H. Twizell, “Spline solution of linear eighth-order boundary value problems,” Computer Methods in Applied Mechanics and Engineering, vol. 131, no. 3-4, pp. 309–325, 1996.
View at: Publisher Site | Google Scholar
A.-M. Wazwaz, “The numerical solution of special eight-order boundary value problems by the modified decomposition method,” Neural, Parallel & Scientific Computations, vol. 8, no. 2, pp. 133–146, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Boutayeb and E. H. Twizell, “Finite-difference methods for the solution of special eighth-order boundary-value problem,” International Journal of Computer Mathematics, vol. 48, no. 1, pp. 63–75, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
R. E. D. Bishop, S. M. Cannon, and S. Miao, “On coupled bending and torsional vibration of uniform beams,” Journal of Sound and Vibration, vol. 131, no. 1, pp. 457–464, 1989.
View at: Publisher Site | Google Scholar
R. P. Agarwal, in Boundary Value Problems for Higher Order Differential Equations, p. xii+307, World Scientific, Teaneck, NJ, USA, 1986.
View at: Zentralblatt MATH | MathSciNet
J. H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87–88, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
J. H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 527–539, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.
View at: Google Scholar
J. H. He, “The homotopy perturbation method nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.
View at: Publisher Site | Google Scholar
J. H. He, “Variational iteration method for autonomous ordinary differential systems,” Applied Mathematics and Computation, vol. 114, no. 2-3, pp. 115–123, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.
View at: Publisher Site | Google Scholar | MathSciNet
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, pp. 156–162, Pergamon Press, New York, NY, USA, 1978.
View at: Google Scholar
M. A. Noor and S. T. Mohyd-Din, “Variational iteration method for solving sixth order boundary value problems,” preprint, 2007.
View at: Google Scholar
M. A. Noor and S. T. Mohyd-Din, “An efficient method for fourth order boundary value problems,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1101–1111, 2007.
View at: Publisher Site | Google Scholar
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.
View at: Google Scholar | MathSciNet
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.
View at: Publisher Site | Google Scholar | MathSciNet
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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A.-M. Wazwaz, “Approximate solutions to boundary value problems of higher order by the modified decomposition method,” Computers & Mathematics with Applications, vol. 40, no. 6-7, pp. 679–691, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A.-M. Wazwaz, “A reliable modification of Adomian decomposition method,” Applied Mathematics and Computation, vol. 102, no. 1, pp. 77–86, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A.-M. Wazwaz, “A new algorithm for calculating adomian polynomials for nonlinear operators,” Applied Mathematics and Computation, vol. 111, no. 1, pp. 33–51, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Abbasbandy, “Numerical solution of non-linear Klein-Gordon equations by variational iteration method,” International Journal for Numerical Methods in Engineering, vol. 70, no. 7, pp. 876–881, 2007.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2007 Muhammad Aslam Noor and Syed Tauseef Mohyud-Din. 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.

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