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Journal of Applied Mathematics
Volume 2012, Article ID 103205, 13 pages
http://dx.doi.org/10.1155/2012/103205
Research Article

A Nonclassical Radau Collocation Method for Nonlinear Initial-Value Problems with Applications to Lane-Emden Type Equations

1School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
2Department of Mathematics, Islamic Azad University, Khorasgan Branch, Isfahan 71595, Iran
3Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
4Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 21 March 2012; Revised 16 September 2012; Accepted 16 September 2012

Academic Editor: Igor Andrianov

Copyright © 2012 Mohammad Maleki 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. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover, New York, NY, USA, 1967. View at Zentralblatt MATH
  2. C. M. Bender, K. A. Milton, S. S. Pinsky, and L. M. Simmons, Jr., “A new perturbative approach to nonlinear problems,” Journal of Mathematical Physics, vol. 30, no. 7, pp. 1447–1455, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. A.-M. Wazwaz, “A new method for solving singular initial value problems in the second-order ordinary differential equations,” Applied Mathematics and Computation, vol. 128, no. 1, pp. 45–57, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. 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
  5. S. 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
  6. M. S. H. Chowdhury and I. Hashim, “Solutions of Emden-Fowler equations by homotopy-perturbation method,” Nonlinear Analysis. Real World Applications, vol. 10, no. 1, pp. 104–115, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. 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
  8. M. I. Nouh, “Accelerated power series solution of polytropic and isothermal gas spheres,” New Astronomy, vol. 9, pp. 467–473, 2004. View at Google Scholar
  9. 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 Zentralblatt MATH
  10. J. I. Ramos, “Linearization techniques for singular initial-value problems of ordinary differential equations,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 525–542, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. Dehghan and F. Shakeri, “Approximate solution of a differential equation arising in astrophysics using the variational iteration method,” New Astronomy, vol. 13, pp. 53–59, 2008. View at Google Scholar
  12. A. Yıldırım and T. Öziş, “Solutions of singular IVPs of Lane-Emden type by the variational iteration method,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 6, pp. 2480–2484, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, ,NY, USA, 1962.
  14. 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 A, vol. 372, no. 37, pp. 5883–5886, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. K. Parand, A. R. Rezaei, and A. Taghavi, “Lagrangian method for solving Lane-Emden type equation arising in astrophysics on semi-infinite domains,” Acta Astronautica, vol. 67, p. 673, 2010. View at Google Scholar
  16. 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
  17. K. Parand and A. Pirkhedri, “Sinc-Collocation method for solving astrophysics equations,” New Astronomy, vol. 15, pp. 533–537, 2010. View at Google Scholar
  18. K. Parand, M. Shahini, and M. Dehghan, “Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type,” Journal of Computational Physics, vol. 228, no. 23, pp. 8830–8840, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. H. Adibi and A. M. Rismani, “On using a modified Legendre-spectral method for solving singular IVPs of Lane-Emden type,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 2126–2130, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. S. Karimi Vanani and A. Aminataei, “On the numerical solution of differential equations of Lane-Emden type,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2815–2820, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. G. P. Horedt, Polytropes Applications in Astrophysics and Related Fields, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004.
  22. G. Elnagar, M. A. Kazemi, and M. Razzaghi, “The pseudospectral Legendre method for discretizing optimal control problems,” IEEE Transactions on Automatic Control, vol. 40, no. 10, pp. 1793–1796, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. G. N. Elnagar and M. A. Kazemi, “Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems,” Computational Optimization and Applications, vol. 11, no. 2, pp. 195–217, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. D. Garg, M. Ptterson, W. W. Hager, A. V. Rao, and D. A. Benson, “A unified framework for the numerical solution of optimal control problems using pseudospectral methods,” Automatica, vol. 46, p. 1843, 2010. View at Google Scholar
  25. B. Shizgal, “A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems,” Journal of Computational Physics, vol. 41, no. 2, pp. 309–328, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. B. Shizgal and H. Chen, “The quadrature discretization method (QDM) in the solution of the Schrödinger equation with nonclassical basis functions,” Journal of Chemical Physics, vol. 104, 4137 pages, 1996. View at Google Scholar
  27. G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM Review, vol. 15, pp. 318–334, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. M. Maleki and M. Mashali-Firouzi, “A numerical solution of problems in calculus of variation using direct method and nonclassical parameterization,” Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1364–1373, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. J. Szabados, “Weighted Lagrange and Hermite-Fejér interpolation on the real line,” Journal of Inequalities and Applications, vol. 1, no. 2, pp. 99–123, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. S. B. Damelin, “The asymptotic distribution of general interpolation arrays for exponential weights,” Electronic Transactions on Numerical Analysis, vol. 13, pp. 12–21, 2002. View at Google Scholar · View at Zentralblatt MATH
  31. D. S. Lubinsky, “A survey of weighted polynomial approximation with exponential weights,” Surveys in Approximation Theory, vol. 3, pp. 1–105, 2007. View at Google Scholar · View at Zentralblatt MATH
  32. P. Vértesi, “On the Lebesgue function of weighted Lagrange interpolation—I. (Freud-type weights),” Constructive Approximation, vol. 15, no. 3, pp. 355–367, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. J. A. C. Weideman, “Spectral methods based on nonclassical orthogonal polynomials,” in Applications and Computation of Orthogonal Polynomials, vol. 131 of International Series of Numerical Mathematics, pp. 239–251, Birkhäuser, Basel, Switzerland, 1999. View at Google Scholar · View at Zentralblatt MATH
  34. H. Chen and B. D. Shizgal, “A spectral solution of the Sturm-Liouville equation: comparison of classical and nonclassical basis sets,” Journal of Computational and Applied Mathematics, vol. 136, no. 1-2, pp. 17–35, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. A. Alipanah, M. Razzaghi, and M. Dehghan, “Nonclassical pseudospectral method for the solution of brachistochrone problem,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1622–1628, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. G. Criscuolo, B. Della Vecchia, D. S. Lubinsky, and G. Mastroianni, “Functions of the second kind for Freud weights and series expansions of Hilbert transforms,” Journal of Mathematical Analysis and Applications, vol. 189, no. 1, pp. 256–296, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH