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ISRN Mathematical Physics
Volume 2013 (2013), Article ID 650208, 9 pages
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.
- H. B. Keller, Numerical Solution of Two Point Boundary Value problems, SIAM, Philadelphia, Pa, USA, 1976.
- 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.
- R. M. M. Mattheij and G. W. M. Staarink, “On optimal shooting intervals,” Mathematics of Computation, vol. 42, no. 165, pp. 25–40, 1984.
- J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, vol. 12 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 1993.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 3rd edition, 1978.
- J. M. Ortega, “The Newton-Kantorovich Theorem,” The American Mathematical Monthly, vol. 75, pp. 658–660, 1968.
- R. A. Tapia, “Classroom notes: the Kantorovich theorem for Newton's method,” The American Mathematical Monthly, vol. 78, no. 4, pp. 389–392, 1971.
- L. B. Rall, “A note on the convergence of Newton's method,” SIAM Journal on Numerical Analysis, vol. 11, pp. 34–36, 1974.
- 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.
- S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, UK, 1961.
- P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 2nd edition, 2004.