About this Journal Submit a Manuscript Table of Contents
ISRN Mathematical Physics
Volume 2013 (2013), Article ID 650208, 9 pages
http://dx.doi.org/10.1155/2013/650208
Research Article

A Nonlinear Shooting Method and Its Application to Nonlinear Rayleigh-Bénard Convection

Department of Mathematics, Guru Nanak Dev University, Amritsar 143005, India

Received 5 June 2013; Accepted 3 July 2013

Academic Editors: D. Dürr and A. Qadir

Copyright © 2013 Jitender Singh. 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. H. B. Keller, Numerical Solution of Two Point Boundary Value problems, SIAM, Philadelphia, Pa, USA, 1976. View at Zentralblatt MATH · View at MathSciNet
  2. A. Granas, R. B. Guenther, and J. W. Lee, “The shooting method for the numerical solution of a class of nonlinear boundary value problems,” SIAM Journal on Numerical Analysis, vol. 16, no. 5, pp. 828–836, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R. M. M. Mattheij and G. W. M. Staarink, “On optimal shooting intervals,” Mathematics of Computation, vol. 42, no. 165, pp. 25–40, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, vol. 12 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 1993. View at MathSciNet
  5. M. E. Kramer and R. M. M. Mattheij, “Application of global methods in parallel shooting,” SIAM Journal on Numerical Analysis, vol. 30, no. 6, pp. 1723–1739, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A.-M. Wazwaz, “Approximate solutions to boundary value problems of higher order by the modified decomposition method,” Computers & Mathematics with Applications, vol. 40, no. 6-7, pp. 679–691, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. N. Ha, “A nonlinear shooting method for two-point boundary value problems,” Computers & Mathematics with Applications, vol. 42, no. 10-11, pp. 1411–1420, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. M. Wazwaz, “A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems,” Computers & Mathematics with Applications, vol. 41, no. 10-11, pp. 1237–1244, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. B. S. Attili and M. I. Syam, “Efficient shooting method for solving two point boundary value problems,” Chaos, Solitons and Fractals, vol. 35, no. 5, pp. 895–903, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. C.-S. Liu, “Cone of non-linear dynamical system and group preserving schemes,” International Journal of Non-Linear Mechanics, vol. 36, no. 7, pp. 1047–1068, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. C. S. Liu, “The Lie-group shooting method for boundary-layer problms with suction/injection/reverse flow conditions for power-law fluids,” International Journal of Non-Linear Mechanics, vol. 46, pp. 1001–1008, 2011.
  12. G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 3rd edition, 1978. View at MathSciNet
  13. J. M. Ortega, “The Newton-Kantorovich Theorem,” The American Mathematical Monthly, vol. 75, pp. 658–660, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. R. A. Tapia, “Classroom notes: the Kantorovich theorem for Newton's method,” The American Mathematical Monthly, vol. 78, no. 4, pp. 389–392, 1971. View at Publisher · View at Google Scholar · View at MathSciNet
  15. L. B. Rall, “A note on the convergence of Newton's method,” SIAM Journal on Numerical Analysis, vol. 11, pp. 34–36, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. W. B. Gragg and R. A. Tapia, “Optimal error bounds for the Newton-Kantorovich theorem,” SIAM Journal on Numerical Analysis, vol. 11, pp. 10–13, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, UK, 1961. View at MathSciNet
  18. P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 2nd edition, 2004. View at Publisher · View at Google Scholar · View at MathSciNet