Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2014, Article ID 649471, 6 pages
http://dx.doi.org/10.1155/2014/649471
Research Article

The Solution of Fourth Order Boundary Value Problem Arising out of the Beam-Column Theory Using Adomian Decomposition Method

Department of Civil Engineering, Technology Faculty, Firat University, 23119 Elazig, Turkey

Received 23 November 2013; Accepted 2 March 2014; Published 30 March 2014

Academic Editor: Sergii V. Kavun

Copyright © 2014 Omer Kelesoglu. 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. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, Mass, USA, 1994. View at MathSciNet
  2. G. Adomian, “A review of the decomposition method and some recent results for nonlinear equations,” Mathematical and Computer Modelling, vol. 13, no. 7, pp. 17–43, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. G. Adomian and R. Rach, “Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations,” Computers & Mathematics with Applications, vol. 19, no. 12, pp. 9–12, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Inc, “On numerical solutions of partial differential equations by the decomposition method,” Kragujevac Journal of Mathematics, vol. 26, pp. 153–164, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Inc, “Decomposition method for solving parabolic equations in finite domains,” Journal of Zhejiang University SCIENCE A, vol. 6, no. 10, pp. 1058–1064, 2005. View at Publisher · View at Google Scholar · View at Scopus
  6. M. Inc, Y. Cherruault, and K. Abbaoui, “A computational approach to the wave equations: an application of the decomposition method,” Kybernetes, vol. 33, no. 1, pp. 80–97, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. Z. M. Odibat, “A new modification of the homotopy perturbation method for linear and nonlinear operators,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 746–753, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. R. S. Alizadeh, G. G. Domairry, and S. Karimpour, “An approximation of the analytical solution of the linear and nonlinear integro-differential equations by homotopy perturbation method,” Acta Applicandae Mathematicae, vol. 104, no. 3, pp. 355–366, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y.-G. Wang, H.-F. Song, and D. Li, “Solving two-point boundary value problems using combined homotopy perturbation method and Green's function method,” Applied Mathematics and Computation, vol. 212, no. 2, pp. 366–376, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. Biazar and H. Ghazvini, “Convergence of the homotopy perturbation method for partial differential equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 2633–2640, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. C.-S. Liu, “The essence of the homotopy analysis method,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1299–1303, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. E. M. Abulwafa, M. A. Abdou, and A. A. Mahmoud, “Nonlinear fluid flows in pipe-like domain problem using variational-iteration method,” Chaos, Solitons & Fractals, vol. 32, no. 4, pp. 1384–1397, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. N. H. Sweilam and M. M. Khader, “Variational iteration method for one dimensional nonlinear thermoelasticity,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 145–149, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. L. Xu, “Variational iteration method for solving integral equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1071–1078, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J.-H. He, A.-M. Wazwaz, and L. Xu, “The variational iteration method: reliable, efficient, and promising,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 879–880, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. L. Xu, J.-H. He, and A.-M. Wazwaz, “Preface variational iteration method—reality, potential, and challenges,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 1–2, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  18. S. B. Coşkun and M. T. Atay, “Analysis of convective straight and radial fins with temperature-dependent thermal conductivity using variational iteration method with comparison with respect to finite element analysis,” Mathematical Problems in Engineering, vol. 2007, Article ID 42072, 15 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. T. Atay and S. B. Coşkun, “Effects of nonlinearity on the variational iteration solutions of nonlinear two-point boundary value problems with comparison with respect to finite element analysis,” Mathematical Problems in Engineering, vol. 2008, Article ID 857296, 10 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Inc and Y. Uǧurlu, “Numerical simulation of the regularized long wave equation by He's homotopy perturbation method,” Physics Letters A: General, Atomic and Solid State Physics, vol. 369, no. 3, pp. 173–179, 2007. View at Publisher · View at Google Scholar · View at Scopus
  21. A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Homotopy analysis method for singular IVPs of Emden-Fowler type,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1121–1131, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. Abbasbandy, E. Babolian, and M. Ashtiani, “Numerical solution of the generalized Zakharov equation by homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 12, pp. 4114–4121, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. M. Rashidi and S. Dinarvand, “Purely analytic approximate solutions for steady three-dimensional problem of condensation film on inclined rotating disk by homotopy analysis method,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2346–2356, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. L. Song and H. Zhang, “Solving the fractional BBM-Burgers equation using the homotopy analysis method,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 1616–1622, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. S. Liao, “On the relationship between the homotopy analysis method and Euler transform,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 6, pp. 1421–1431, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, Calif, USA, 1986. View at MathSciNet
  27. M. Inc and M. Işık, “Adomian decomposition method for three-dimensional parabolic equation with non-classic boundary conditions,” Journal of Analysis, vol. 11, pp. 43–51, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. F. Abdelwahid, “A mathematical model of Adomian polynomials,” Applied Mathematics and Computation, vol. 141, no. 2-3, pp. 447–453, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. E. Babolian and S. Javadi, “New method for calculating Adomian polynomials,” Applied Mathematics and Computation, vol. 153, no. 1, pp. 253–259, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Y. Cherruault, M. Inc, and K. Abbaoui, “On the solution of the non-linear Korteweg-de Vries equation by the decomposition method,” Kybernetes, vol. 31, no. 5, pp. 766–772, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  31. A. M. A. El-Sayed and M. Gaber, “The Adomian decomposition method for solving partial differential equations of fractal order in finite domains,” Physics Letters A, vol. 359, no. 3, pp. 175–182, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. E. Momoniat, T. A. Selway, and K. Jina, “Analysis of Adomian decomposition applied to a third-order ordinary differential equation from thin film flow,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 10, pp. 2315–2324, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. L. L. Thompson and P. M. Pinsky, “A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation,” International Journal for Numerical Methods in Engineering, vol. 38, no. 3, pp. 371–397, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. J. Dolbow and T. Belytschko, “Numerical integration of the Galerkin weak form in meshfree methods,” Computational Mechanics, vol. 23, no. 3, pp. 219–230, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. S. N. Atluri and T. Zhu, “A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics,” Computational Mechanics, vol. 22, no. 2, pp. 117–127, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. A. M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, Rotterdam, The Netherlands, 2002.
  37. Y. Cherruault, “Convergence of Adomian's method,” Kybernetes, vol. 18, no. 2, pp. 31–38, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. K. Abbaoui and Y. Cherruault, “New ideas for proving convergence of decomposition methods,” Computers & Mathematics with Applications, vol. 29, no. 7, pp. 103–108, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. S. P. Timoshenko and J.M. Gere, Theory of Elastic Stability, McGraw-Hill, New York, NY, USA, 2nd edition, 1961. View at MathSciNet
  40. S. P. Timoshenko and J.M. Gere, Theory of Elastic Stability, McGraw-Hill, New York, NY, USA, 1985. View at MathSciNet
  41. W. F. Chen and T. Atsuta, Theory of Beam-Columns, vol. 1 of In-plane Behavior and Design, McGraw-Hill, New York, NY, USA, 1976.
  42. W. F. Chen and T. Atsuta, Theory of Beam-Columns, vol. 2 of Space Behavior and Design, McGraw-Hill, New York, NY, USA, 1977.
  43. O. A. Taiwo and O. M. Ogunlaran, “A non-polynomial spline method for solving linear fourth-order boundary-value problems,” International Journal of Physical Sciences, vol. 6, no. 13, pp. 3246–3254, 2011. View at Google Scholar · View at Scopus