About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 960289, 16 pages
http://dx.doi.org/10.1155/2012/960289
Research Article

On Spectral Homotopy Analysis Method for Solving Linear Volterra and Fredholm Integrodifferential Equations

Department of Mathematics, Universiti Putra Malaysia (UPM), Selangor, 43400 Serdang, Malaysia

Received 12 June 2012; Revised 19 September 2012; Accepted 4 October 2012

Academic Editor: Lishan Liu

Copyright © 2012 Z. Pashazadeh Atabakan 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. L. K. Forbes, S. Crozier, and D. M. Doddrell, “Calculating current densities and fields produced by shielded magnetic resonance imaging probes,” SIAM Journal on Applied Mathematics, vol. 57, no. 2, pp. 401–425, 1997. View at Publisher · View at Google Scholar
  2. K. Parand, S. Abbasbandy, S. Kazem, and J. A. Rad, “A novel application of radial basis functions for solving a model of first-order integro-ordinary differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4250–4258, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. P. Darania and A. Ebadian, “A method for the numerical solution of the integro-differential equations,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 657–668, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. Karamete and M. Sezer, “A Taylor collocation method for the solution of linear integro-differential equations,” International Journal of Computer Mathematics, vol. 79, no. 9, pp. 987–1000, 2002. View at Publisher · View at Google Scholar
  5. S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems [Ph.D. thesis], Shanghai Jiao Tong University, 1992.
  6. A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, London, UK, 1992.
  7. 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
  8. G. Adomian and R. Rach, “Noise terms in decomposition solution series,” Computers & Mathematics with Applications, vol. 24, no. 11, pp. 61–64, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. G. Adomian and R. Rach, “Analytic solution of nonlinear boundary value problems in several dimensions by decomposition,” Journal of Mathematical Analysis and Applications, vol. 174, no. 1, pp. 118–137, 1993. View at Publisher · View at Google Scholar
  10. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60, Kluwer Academic, Boston, Mass, USA, 1994.
  11. P. K. Bera and J. Datta, “Linear delta expansion technique for the solution of anharmonic oscillations,” Pramana Journal of Physics, vol. 68, no. 1, pp. 117–122, 2007. View at Publisher · View at Google Scholar · View at Scopus
  12. 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
  13. J. H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004. View at Publisher · View at Google Scholar · View at Scopus
  14. G. Domairry, A. Mohsenzadeh, and M. Famouri, “The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery-Hamel flow,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 1, pp. 85–95, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Z. Ziabakhsh and G. Domairry, “Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1284–1294, 2009. View at Publisher · View at Google Scholar · View at Scopus
  16. 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 Zentralblatt MATH
  17. T. Hayat, S. Noreen, and M. Sajid, “Heat transfer analysis of the steady flow of a fourth grade fluid,” International Journal of Thermal Sciences, vol. 47, no. 5, pp. 591–599, 2008. View at Publisher · View at Google Scholar · View at Scopus
  18. M. Jalaal, D. D. Ganji, and G. Ahmadi, “Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media,” Advanced Powder Technology, vol. 21, no. 3, pp. 298–304, 2010. View at Publisher · View at Google Scholar · View at Scopus
  19. D. D. Ganji, G. A. Afrouzi, and R. A. Talarposhti, “Application of variational iteration method and homotopy-perturbation method for nonlinear heat diffusion and heat transfer equations,” Physics Letters, Section A, vol. 368, no. 6, pp. 450–457, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. S. Abbasbandy, “Modified homotopy perturbation method for nonlinear equations and comparison with Adomian decomposition method,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 431–438, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. S. S. Motsa, P. Sibanda, and S. Shateyi, “A new spectral-homotopy analysis method for solving a nonlinear second order BVP,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2293–2302, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. S. S. Motsa, P. Sibanda, F. G. Awad, and S. Shateyi, “A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem,” Computers & Fluids, vol. 39, no. 7, pp. 1219–1225, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. S. S. Motsa and P. Sibanda, “A new algorithm for solving singular IVPs of Lane-Emden type,” in Proceedings of the 4th International Conference on Applied Mathematics, Simulation, Modelling (ASM '10), pp. 176–180, Corfu Island, Greece, July 2010. View at Scopus
  24. W. S. Don and A. Solomonoff, “Accuracy and speed in computing the Chebyshev collocation derivative,” SIAM Journal on Scientific Computing, vol. 16, no. 6, pp. 1253–1268, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, UK, 1985. View at Publisher · View at Google Scholar
  26. K. Maleknejad, S. Sohrabi, and Y. Rostami, “Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 123–128, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. S. Yalçinbaş, M. Sezer, and H. H. Sorkun, “Legendre polynomial solutions of high-order linear Fredholm integro-differential equations,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 334–349, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. H. M. Jaradat, F. Awawdeh, and O. Alsayyed, “Series solution to the high-order integro-differential equations,” Fascicola Matematica, vol. 16, pp. 247–257, 2009. View at Zentralblatt MATH