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Inclusions and Existence Proofs for Solutions of a Nonlinear Boundary Value Problem by Spectral Numerical Methods

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Topics in Numerical Analysis

Part of the book series: Computing Supplementa ((COMPUTING,volume 15))

Abstract

For a specific elliptic boundary value problem with quadratic nonlinearity, we give a partial positive answer to an old conjecture concerning the number of solutions. This result is obtained via an existence and enclosure method. For computing the highly accurate solutions needed for this method, a spectral two-grid procedure (combined with a numerical Mountain-Pass algorithm and a Newton iteration) is proposed. Furthermore, Emden’s equation is shown to admit completely spurious approximate solutions which nevertheless have “small” defects—a powerful argument for rigorous enclosure methods.

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References

  1. Ambrosetti, A., Prodi, G.: On the Inversion of some Differentiable Mappings with Singularities between Banach Spaces. Ann. Math. Pura Appl. 93, 231–247 (1973).

    MathSciNet  Google Scholar 

  2. Breuer, B., Plum, M.: Lsungseinschließungen bei einem nichtlinearen Randwertproblem mittels eines Fourierreihenansatzes. Internal Report, Preprint Nr. 98/9, IWRMM Karlsruhe, 1998.

    Google Scholar 

  3. Chatelin, F.: Spectral approximations of linear operators. New York: Academic Press 1983.

    Google Scholar 

  4. Chen, Y., McKenna, P. J.: Traveling waves in a nonlinearly suspended beam: theoretical results and numerical observations. J. Differ. Eq. 136, 325–355 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  5. Choi, Y. S., McKenna, P. J.: A mountain pass method for the numerical solution of semilinear elliptic problems. Nonlinear Anal. Theory Meth. Appl. 20, 411–431 (1993).

    Article  MathSciNet  Google Scholar 

  6. Dancer, E. N.: A counterexample to the Lazer-McKenna conjecture. Nonlinear Anal. T.M.A. 13, 19–22 (1982).

    Article  MathSciNet  Google Scholar 

  7. Gidas, B., Ni, W. M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  8. Hackbusch, W.: Multi-grid methods and applications. Springer Series in Computational Mathematics, Vol. 4. Berlin Heidelberg New York: Springer 1985.

    Google Scholar 

  9. Hofer, H.: Variational and topological methods in partially ordered Hilbert spaces. Math. Ann. 261, 493–514 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  10. Kazdan, J. L., Warner, F.: Remarks on quasilinear elliptic equations. Comm. Pure Appl. Math. 28, 567–597 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  11. Knüppel, O.: Programmer’s Runtime Optimized Fast Interval Library. Bericht 93.4, Technische Universität Hamburg-Harburg, 1993.

    Google Scholar 

  12. Lazer, A. C, McKenna, P. J.: On the number of solutions of a nonlinear Dirichlet problem. J. Math. Anal. Appl. 84, 282–294 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  13. Lions, P.-L.: On positive solutions of semilinear elliptic equations in unbounded domains. Nonlinear diffusion equations and their equilibrium states, II, Berkeley, CA, 85–122 (1986). Math. Sci. Res. Inst. Publ., 13. Berlin Heidelberg New York Tokyo: Springer 1988.

    Google Scholar 

  14. Nakao, M. T.: Solving nonlinear elliptic problems with result verification using an H-l type residual iteration. Computing [Suppl] 9, 161–173 (1993).

    Google Scholar 

  15. Plum, M.: Explicit Hi-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems. J. Math. Anal. Appl. 165, 36–61 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  16. Plum, M.: Enclosures for weak solutions of nonlinear elliptic boundary value problems, pp. 505–521. Singapore: World Scientific 1994.

    Google Scholar 

  17. Plum, M.: Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems. Computing 49, 25–44 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  18. Plum, M.: Guaranteed numerical bounds for eigenvalues. In: Spectral theory and computational methods of sturm-liouville problems (Hinton, D., Schaefer, W., eds.), Lecture Notes in Pure and Applied Mathematics, vol. 191. New York, Basel: Marcel Dekker 1997.

    Google Scholar 

  19. Solimini, S.: Some remarks on the number of solutions of some nonlinear elliptic equations. Anal. Nonlin., I.H.P. 2, 143–156 (1985).

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematisches Institut, I Universität Karlsruhe, Englerstrasse 2, D-76128, Karlsruhe, Germany

    B. Breuer & M. Plum

  2. Department of Mathematics U9, University of Connecticut, 196 Auditorium Road, Storrs, CT, 06269-3009, USA

    P. J. McKenna

Authors
  1. B. Breuer
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  2. M. Plum
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  3. P. J. McKenna
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Editor information

Editors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Karlsruhe, Karlsruhe, Germany

    Goetz Alefeld

  2. Department of Mathematical Sciences, Shimane University, Matsue, Japan

    Xiaojun Chen

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© 2001 Springer-Verlag Wien

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Breuer, B., Plum, M., McKenna, P.J. (2001). Inclusions and Existence Proofs for Solutions of a Nonlinear Boundary Value Problem by Spectral Numerical Methods. In: Alefeld, G., Chen, X. (eds) Topics in Numerical Analysis. Computing Supplementa, vol 15. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6217-0_6

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  • DOI: https://doi.org/10.1007/978-3-7091-6217-0_6

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-83673-6

  • Online ISBN: 978-3-7091-6217-0

  • eBook Packages: Springer Book Archive

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