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Journal of Scientific Computing

, Volume 43, Issue 3, pp 388–401 | Cite as

Computer Assisted Proofs of Bifurcating Solutions for Nonlinear Heat Convection Problems

  • Mitsuhiro T. NakaoEmail author
  • Yoshitaka Watanabe
  • Nobito Yamamoto
  • Takaaki Nishida
  • Myoung-Nyoung Kim
Article

Abstract

In previous works (Nakao et al., Reliab. Comput., 9(5):359–372, 2003; Watanabe et al., J. Math. Fluid Mech., 6(1):1–20, 2004), the authors considered the numerical verification method of solutions for two-dimensional heat convection problems known as Rayleigh-Bénard problem. In the present paper, to make the arguments self-contained, we first summarize these results including the basic formulation of the problem with numerical examples. Next, we will give a method to verify the bifurcation point itself, which should be an important information to clarify the global bifurcation structure, and show a numerical example. Finally, an extension to the three dimensional case will be described.

Keywords

Navier-Stokes equation Nonlinear heat convection Bifurcation point Computer assisted proof 

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References

  1. 1.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, London (1961) zbMATHGoogle Scholar
  2. 2.
    Curry, J.H.: Bounded solutions of finite dimensional approximations to the Boussinesq equations. SIAM J. Math. Anal. 10, 71–79 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Getling, A.V.: Rayleigh-Bénard Convection: Structures and Dynamics. Advanced Series in Nonlinear Dynamics, vol. 11. World Scientific, Singapore (1998) zbMATHGoogle Scholar
  4. 4.
    Kawanago, T.: A symmetry-breaking bifurcation theorem and some related theorems applicable to maps having unbounded derivatives. Jpn. J. Ind. Appl. Math. 21, 57–74 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kim, M.-N., Nakao, M.T., Watanabe, Y., Nishida, T.: A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-Bénard problems. Numer. Math. 111, 389–406 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kearfott, R.B., Kreinovich, V. (eds.): Applications of Interval Computations. Kluwer Academic, Dordrecht (1996) zbMATHGoogle Scholar
  7. 7.
    Nagatou, K., Yamamoto, N., Nakao, M.T.: An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. Optim. 20, 543–565 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Nakao, M.T.: Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optim. 22(3&4), 321–356 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Nakao, M.T., Watanabe, Y., Yamamoto, N., Nishida, T.: Some computer assisted proofs for solutions of the heat convection problems. Reliab. Comput. 9(5), 359–372 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Nakao, M.T., Hashimoto, K., Watanabe, Y.: A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems. Computing 75, 1–14 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Nishida, T., Ikeda, T., Yoshihara, H.: Pattern formation of heat convection problems. In: Miyoshi, T., et al. (eds.) Proc. Internat. Symp. Math. Modeling and Numer. Simul. in Cont. Mech. (2000). Lecture Notes in Comput. Sci. Engin., vol. 19, pp. 209–218. Springer, Berlin (2002) Google Scholar
  12. 12.
    Watanabe, Y., Yamamoto, N., Nakao, M.T., Nishida, T.: A numerical verification of nontrivial solutions for the heat convection problem. J. Math. Fluid Mech. 6(1), 1–20 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Yamamoto, N.: A numerical verification method for solutions of boundary value problems with local uniqueness by Banach’s fixed point theorem. SIAM J. Numer. Anal. 35, 2004–2013 (1998) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Mitsuhiro T. Nakao
    • 1
    Email author
  • Yoshitaka Watanabe
    • 2
  • Nobito Yamamoto
    • 3
  • Takaaki Nishida
    • 4
  • Myoung-Nyoung Kim
    • 1
  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Research Institute for Information TechnologyKyushu UniversityFukuokaJapan
  3. 3.Department of Computer Science and Information MathematicsThe University of Electro-CommunicationsTokyoJapan
  4. 4.Faculty of Science and EngineeringWaseda UniversityTokyoJapan

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