Abstract and Applied Analysis

The Optimal Homotopy Asymptotic Method for the Solution of Higher-Order Boundary Value Problems in Finite Domains

Javed Ali, S. Islam, Hamid Khan, and Syed Inayat Ali Shah

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Abstract

We solve some higher-order boundary value problems by the optimal homotopy asymptotic method (OHAM). The proposed method is capable to handle a wide variety of linear and nonlinear problems effectively. The numerical results given by OHAM are compared with the exact solutions and the solutions obtained by Adomian decomposition (ADM), variational iteration (VIM), homotopy perturbation (HPM), and variational iteration decomposition method (VIDM). The results show that the proposed method is more effective and reliable.

Article information

Source
Abstr. Appl. Anal. Volume 2012, Special Issue (2012), Article ID401217, 14 pages.

Dates
First available: 15 February 2012

Permanent link to this document
http://projecteuclid.org/euclid.aaa/1329337692

Digital Object Identifier
doi:10.1155/2012/401217

Mathematical Reviews number (MathSciNet)
MR2872301

Zentralblatt MATH identifier
1252.65126

Citation

Ali, Javed; Islam, S.; Khan, Hamid; Ali Shah, Syed Inayat. The Optimal Homotopy Asymptotic Method for the Solution of Higher-Order Boundary Value Problems in Finite Domains. Abstract and Applied Analysis 2012, Special Issue (2012), 1--14. doi:10.1155/2012/401217. http://projecteuclid.org/euclid.aaa/1329337692.


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References

  • S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, UK, 1961.
  • R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Teaneck, NJ, USA, 1986.
  • G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
  • G. Adomian, “A review of the decomposition method and some recent results for nonlinear equations,” Computers & Mathematics with Applications, vol. 21, no. 5, pp. 101–127, 1991.
  • 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.
  • J.-H. He, “Variational approach to the sixth-order boundary value problems,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 537–538, 2003.
  • J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999.
  • J.-H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters. A, vol. 350, no. 1-2, pp. 87–88, 2006.
  • H. E. Ji-Huan, “A Note on the homotopy perturbation method,” Thermal Science, vol. 14, no. 2, pp. 565–568, 2010.
  • S. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004.
  • S. Liao, “Notes on the homotopy analysis method: some definitions and theorems,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 983–997, 2009.
  • J. K. Zhou, Deferential Transformation and its Application for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.
  • V. Marinca, N. Herişanu, and I. Nemeş, “Optimal homotopy asymptotic method with application to thin film flow,” Central European Journal of Physics, vol. 6, no. 3, pp. 648–653, 2008.
  • N. Herisanu, V. Marinca, T. Dordea, and G. Madescu, “A new analytical approach to nonlinear vibration of an electric machine,” Proceedings of the Romanian Academy, vol. 9, no. 3, 2008.
  • V. Marinca, N. Herişanu, C. Bota, and B. Marinca, “An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate,” Applied Mathematics Letters, vol. 22, no. 2, pp. 245–251, 2009.
  • V. Marinca and N. Herişanu, “Application of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, vol. 35, no. 6, pp. 710–715, 2008.
  • V. Marinca and N. Herişanu, “Optimal homotopy perturbation method for strongly nonlinear differential equations,” Nonlinear Science Letters A, vol. 1, pp. 273–280, 2010.
  • 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.
  • A.-M. Wazwaz, “The numerical solution of sixth-order boundary value problems by the modified decomposition method,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 311–325, 2001.
  • M. Aslam 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.
  • M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for fifth-order boundary value problems using He's polynomials,” Mathematical Problems in Engineering, vol. 2008, Article ID 954794, 12 pages, 2008.
  • M. Aslam 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.
  • 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, vol. 2, no. 2, pp. 163–172, 2007.
  • M. A. Noor and S. T. Mohyud-Din, “Homotopy perturbation method for solving sixth-order boundary value problems,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2953–2972, 2008.
  • M. A. Noor, S. T. Mohyud-Din, and M. Tahir, “Variational iteration decomposition method for solving eighth-order boundary value problems,” Differential Equations and Nonlinear Mechanics, vol. 2007, Article ID 19529, 16 pages, 2007.
  • S. T. Mohyud-Din, A. Yildirim, and M. M. Hosseini, “An iterative algorithm for fifth-order boundary value problems,” World Applied Sciences Journal, vol. 8, no. 5, pp. 531–535, 2010.
  • S. T. Mohyud-Din and A. Yildirim, “Solutions of tenth and ninth-order boundary value problems by modified variational iteration method,” Applications and Applied Mathematics, vol. 5, no. 1, pp. 11–25, 2010.
  • K. N. S. Kasi Viswanadham and P. Murali Krishna, “Quintic B-Spline Galerkin method for fifth order boundary value problems,” ARPN Journal of Engineering and Applied Sciences, vol. 5, no. 2, 2010.
  • Siraj-ul-Islam and M. Azam Khan, “A numerical method based on polynomial sextic spline functions for the solution of special fifth-order boundary-value problems,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 356–361, 2006.
  • Siraj-ul-Islam, I. A. Tirmizi, Fazal-i-Haq, and M. A. Khan, “Non-polynomial splines approach to the solution of sixth-order boundary-value problems,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 270–284, 2008.
  • J. Ali, Siraj-ul-Islam, S. Islam, and G. Zaman, “The solution of multipoint boundary value problems by the Optimal Homotopy Asymptotic Method,” Computers and Mathematics with Applications, vol. 59, no. 6, pp. 2000–2006, 2010.
  • J. Ali, S. Islam, M. Tariq Rahim, and G. Zaman, “The solution of special twelfth order boundary value problems by the optimal homotopy asymptotic method,” World Applied Sciences Journal, vol. 11, no. 3, pp. 371–378, 2010.
  • C. H. Che Hussin and A. Kiliçman, “On the solutions of nonlinear higher-order boundary value problems by using differential transformation method and Adomian decomposition method,” Mathematical Problems in Engineering, vol. 2011, Article ID 724927, 19 pages, 2011.