Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 671527, 7 pages
http://dx.doi.org/10.1155/2015/671527
Research Article

A Simple Modification of Homotopy Perturbation Method for the Solution of Blasius Equation in Semi-Infinite Domains

1Young Researchers and Elite Club, Islamic Azad University, Ilkhchi Branch, Ilkhchi, Iran
2Department of Civil Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia
3Department of Civil Engineering, University of Tabriz, Tabriz, Iran

Received 8 July 2015; Revised 28 August 2015; Accepted 9 September 2015

Academic Editor: Gerhard-Wilhelm Weber

Copyright © 2015 M. Aghakhani et al. 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.

Linked References

  1. S. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Methoc [i.e Method], Kluwer Academic Publishers, Dordrecht, The Netherlands, 2013.
  3. 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 · View at Google Scholar · View at Scopus
  4. M.-J. Jang, C.-L. Chen, and Y.-C. Liy, “On solving the initial-value problems using the differential transformation method,” Applied Mathematics and Computation, vol. 115, no. 2-3, pp. 145–160, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. N. Herisanu, V. Marinca, and Gh. Madescu, “An analytical approach to non-linear dynamical model of a permanent magnet synchronous generator,” Wind Energy, vol. 18, pp. 1657–1670, 2015. View at Publisher · View at Google Scholar · View at Scopus
  7. H. Vazquez-Leal, “Generalized homotopy method for solving nonlinear differential equations,” Computational and Applied Mathematics, vol. 33, no. 1, pp. 275–288, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. M. N. Alam, M. A. Akbar, and S. T. Mohyud-Din, “A novel (G′/G)-expansion method and its application to the Boussinesq equation,” Chinese Physics B, vol. 23, no. 2, Article ID 020203, 2014. View at Publisher · View at Google Scholar · View at Scopus
  9. M. A. Jafari and A. Aminataei, “Improved homotopy perturbation method,” International Mathematical Forum, vol. 5, no. 29–32, pp. 1567–1579, 2010. View at Google Scholar · View at MathSciNet
  10. E. Yusufoğlu, “An improvement to homotopy perturbation method for solving system of linear equations,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2231–2235, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. S. H. Hosseinnia, A. Ranjbar, and S. Momani, “Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part,” Computers and Mathematics with Applications, vol. 56, no. 12, pp. 3138–3149, 2008. View at Publisher · View at Google Scholar · View at Scopus
  12. D. Kumar, J. Singh, and S. Kumar, “Numerical computation of fractional multi-dimensional diffusion equations by using a modified homotopy perturbation method,” Journal of the Association of Arab Universities for Basic and Applied Sciences, vol. 17, pp. 20–26, 2015. View at Publisher · View at Google Scholar · View at Scopus
  13. C. Dong, Z. Chen, and W. Jiang, “A modified homotopy perturbation method for solving the nonlinear mixed Volterra-Fredholm integral equation,” Journal of Computational and Applied Mathematics, vol. 239, pp. 359–366, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. H. Blasius, “Grenzschichten in Flussigkeiten mit kleiner Reibung,” Zeitschrift für Angewandte Mathematik und Physik, vol. 56, pp. 1–37, 1908. View at Google Scholar
  15. P. Cheng and W. J. Minkowycz, “Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike,” Journal of Geophysical Research, vol. 82, no. 14, pp. 2040–2044, 1977. View at Publisher · View at Google Scholar · View at Scopus
  16. O. E. Potter, “Laminar boundary layers at the interface of co-current parallel streams,” The Quarterly Journal of Mechanics and Applied Mathematics, vol. 10, pp. 302–311, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. J. A. Ackroyd, “On the laminar compressible boundary layer with stationary origin on a moving flat wall,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 63, no. 3, pp. 871–888, 1967. View at Publisher · View at Google Scholar
  18. J.-H. He, “A simple perturbation approach to Blasius equation,” Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 217–222, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. S. Abbasbandy, “A numerical solution of Blasius equation by Adomian's decomposition method and comparison with homotopy perturbation method,” Chaos, Solitons & Fractals, vol. 31, no. 1, pp. 257–260, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. L.-T. Yu and C.-K. Chen, “The solution of the blasius equation by the differential transformation method,” Mathematical and Computer Modelling, vol. 28, no. 1, pp. 101–111, 1998. View at Publisher · View at Google Scholar
  21. J. H. He, “Approximate analytical solution of Blasius' equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 4, no. 1, pp. 75–78, 1999. View at Publisher · View at Google Scholar · View at Scopus
  22. B. I. Yun, “Constructing uniform approximate analytical solutions for the Blasius problem,” Abstract and Applied Analysis, vol. 2014, Article ID 495734, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. I. Ahmad and M. Bilal, “Numerical solution of blasius equation through neural networks algorithm,” American Journal of Computational Mathematics, vol. 4, no. 3, pp. 223–232, 2014. View at Publisher · View at Google Scholar
  24. V. Marinca and N. Herişanu, “The optimal homotopy asymptotic method for solving Blasius equation,” Applied Mathematics and Computation, vol. 231, pp. 134–139, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. A. Ebaid and N. Al-Armani, “A new approach for a class of the blasius problem via a transformation and adomian's method,” Abstract and Applied Analysis, vol. 2013, Article ID 753049, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  26. O. Costin, T. E. Kim, and S. Tanveer, “A quasi-solution approach to nonlinear problems—the case of the Blasius similarity solution,” Fluid Dynamics Research, vol. 46, no. 3, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. 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 · View at Google Scholar · View at MathSciNet · View at Scopus
  28. L. Howarth, “On the solution of the laminar boundary layer equations,” Proceedings of the Royal Society A, vol. 164, no. 919, pp. 547–579, 1938. View at Publisher · View at Google Scholar
  29. J. H. He, “Approximate analytical solution of Blasius' equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 3, no. 4, pp. 260–263, 1998. View at Publisher · View at Google Scholar · View at Scopus
  30. G. A. Baker and P. R. Graves-Morris, Essentials of Padé Approximants, Academic Press, New York, NY, USA, 1975.
  31. J. P. Boyd, “Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain,” Computers in Physics, vol. 11, no. 3, article 299, 1997. View at Publisher · View at Google Scholar