About this Journal Submit a Manuscript Table of Contents
ISRN Astronomy and Astrophysics
Volume 2011 (2011), Article ID 351747, 7 pages
http://dx.doi.org/10.5402/2011/351747
Research Article

Solution of the Lane-Emden Equation Using the Bernstein Operational Matrix of Integration

1Department of Mathematics, PDPM Indian Institute of Information Technology, Design and Manufacturing Jabalpur, Airport Road, Madhya Pardish, Jabalpur 482005, India
2Department of Mathematics (DIPMAT), University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

Received 28 November 2011; Accepted 22 December 2011

Academic Editors: M. Ding and J. Robertson

Copyright © 2011 Narayan Kumar 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. N. Riazi and M. R. Bordbar, “Generalized Lane-Emden equation and the structure of galactic dark matter,” International Journal of Theoretical Physics, vol. 45, no. 3, pp. 495–510, 2006. View at Publisher · View at Google Scholar
  2. H. J. de Vega and N. G. Sanchez, “Model-independent analysis of dark matter points to a particle mass at the keV scale,” Monthly Notices of the Royal Astronomical Society, vol. 404, no. 2, pp. 885–894, 2010. View at Publisher · View at Google Scholar
  3. G. E. Marsh, “Dark matter and charged exotic dust,” In press, http://arxiv.org/abs/1107.0315.
  4. J. H. Lane, “On theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its internal heat and depending on the laws of gases known to terrestrial experiment,” The American Journal of Science and Arts, vol. 50, pp. 57–74, 1870.
  5. R. Emden, Gaskugeln: Anwendungen der Mechanischen Wärmetheorie auf Kosmologische und Meteorologische Probleme, Teubner, Berlin, Germany, 1907.
  6. H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, NY, USA, 1962.
  7. S. Chandrasekhar, Introduction to Study of Stellar Structure, Dover, New York, NY, USA, 1967.
  8. N. T. Shawagfeh, “Nonperturbative approximate solution for Lane-Emden equation,” Journal of Mathematical Physics, vol. 34, no. 9, pp. 4364–4369, 1993.
  9. A. M. Wazwaz, “A new algorithm for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 287–310, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. V. B. Mandelzweig and F. Tabakin, “Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs,” Computer Physics Communications, vol. 141, no. 2, pp. 268–281, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Krivec and V. B. Mandelzweig, “Numerical investigation of quasilinearization method in quantum mechanics,” Computer Physics Communications, vol. 138, no. 1, pp. 69–79, 2001. View at Publisher · View at Google Scholar
  12. R. Krivec and V. B. Mandelzweig, “Quasilinearization approach to computations with singular potentials,” Computer Physics Communications, vol. 179, no. 12, pp. 865–867, 2008. View at Publisher · View at Google Scholar
  13. J. I. Ramos, “Linearization methods in classical and quantum mechanics,” Computer Physics Communications, vol. 153, no. 2, pp. 199–208, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  14. C. M. Khalique and P. Ntsime, “Exact solutions of the Lane-Emden-type equation,” New Astronomy, vol. 13, no. 7, pp. 476–480, 2008. View at Publisher · View at Google Scholar
  15. S. J. Liao, “A new analytic algorithm of Lane-Emden type equations,” Applied Mathematics and Computation, vol. 142, no. 1, pp. 1–16, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. A. Van Gorder and K. Vajravelu, “Analytic and numerical solutions to the Lane-Emden equation,” Physics Letters, Section A, vol. 372, no. 39, pp. 6060–6065, 2008. View at Publisher · View at Google Scholar
  17. A. Yildirim and T. Öziş, “Solutions of singular IVPs of Lane-Emden type by the variational iteration method,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 6, pp. 2480–2484, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J. H. He, “Variational approach to the Lane-Emden equation,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 539–541, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. A. Yousefi, “Legendre wavelets method for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1417–1422, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. H. R. Marzban, H. R. Tabrizidooz, and M. Razzaghi, “Hybrid functions for nonlinear initial-value problems with applications to Lane-Emden type equations,” Physics Letters, Section A, vol. 372, no. 37, pp. 5883–5886, 2008. View at Publisher · View at Google Scholar
  21. M. Dehghan and F. Shakeri, “Approximate solution of a differential equation arising in astrophysics using the variational iteration method,” New Astronomy, vol. 13, no. 1, pp. 53–59, 2008. View at Publisher · View at Google Scholar
  22. H. Adibi and A. M. Rismani, “On using a modified Legendre-spectral method for solving singular IVPs of Lane-Emden type,” Computers and Mathematics with Applications, vol. 60, no. 7, pp. 2126–2130, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. O. P. Singh, R. K. Pandey, and V. K. Singh, “An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified Homotopy analysis method,” Computer Physics Communications, vol. 180, no. 7, pp. 1116–1124, 2009. View at Publisher · View at Google Scholar
  24. C. Yang and J. Hou, “A Numerical Method for Lane-Emden Equations Using Chebyshev Polynomials and the Collocation Method,” in Proceedings of the IEEE International Conference on Computational and Information Sciences, pp. 97–100, 2010.
  25. K. Parand, M. Dehghan, A. R. Rezaei, and S. M. Ghaderi, “An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method,” Computer Physics Communications, vol. 181, no. 6, pp. 1096–1108, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. K. Parand, A. R. Rezaei, and A. Taghavi, “Lagrangian method for solving LaneEmden type equation arising in astrophysics on semi-infinite domains,” Acta Astronautica, vol. 67, no. 7-8, pp. 673–680, 2010. View at Publisher · View at Google Scholar
  27. K. Parand, S. Abbasbandy, S. Kazem, and A. R. Rezaei, “An improved numerical method for a class of astrophysics problems based on radial basis functions,” Physica Scripta, vol. 83, no. 1, article 015011, 2011. View at Publisher · View at Google Scholar
  28. A. H. Bhrawy and A. S. Alofi, “A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 62–70, 2012. View at Publisher · View at Google Scholar
  29. S. Iqbal and A. Javed, “Application of optimal homotopy asymptotic method for the analytic solution of singular Lane-Emden type equation,” Applied Mathematics and Computation, vol. 217, no. 19, pp. 7753–7761, 2011. View at Publisher · View at Google Scholar
  30. R. A. Van Gorder, “Exact first integrals for a Lane-Emden equation of the second kind modeling a thermal explosion in a rectangular slab,” Celestial Mechanics and Dynamical Astronomy, vol. 109, pp. 137–145, 2011.
  31. R. A. Van Gorder, “An elegant perturbation solution for the Lane-Emden equation of the second kind,” New Astronomy, vol. 16, no. 2, pp. 65–67, 2011. View at Publisher · View at Google Scholar
  32. S. Bernstein, “Démonstration du théorème de Weierstrass fondée sur le calcul des probabilities,” Communications of the Kharkov Mathematical Society, vol. 13, pp. 1–2, 1912.
  33. A. Kilicman and Z. A. A. Al Zhour, “Kronecker operational matrices for fractional calculus and some applications,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 250–265, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet