Skip to main content
Springer Nature Link
Log in

Fuzzy Numerical Schemes for Hyperbolic Differential Equations

  • Conference paper
KI 2009: Advances in Artificial Intelligence (KI 2009)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5803))

Included in the following conference series:

Abstract

The numerical solution of hyperbolic partial differential equations (PDEs) is an important topic in natural sciences and engineering. One of the main difficulties in the task stems from the need to employ several basic types of approximations that are blended in a nonlinear way. In this paper we show that fuzzy logic can be used to construct novel nonlinear blending functions. After introducing the set-up, we show by numerical experiments that the fuzzy-based schemes outperform methods based on conventional blending functions. To the knowledge of the authors, this paper represents the first work where fuzzy logic is applied for the construction of simulation schemes for PDEs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Chiang, C.K., Chung, H.Y., Lin, J.J.: A Self-Learning Fuzzy Logic Controller Using Genetic Algorithms with Reinforcements. Transaction on Fuzzy Systems 5 (1997)

    Google Scholar 

  2. Chin, T.C., Qi, X.M.: Genetic algorithms for learning the rule base of fuzzy logic controller. Fuzzy Sets and Systems, 1–7 (1998)

    Google Scholar 

  3. Clarke, F., Ekeland, I.: Nonlinear oscillations and boundary-value problems for Hamiltonian systems. Arch. Rat. Mech. Anal. 78, 315–333 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Clarke, F., Ekeland, I.: Solutions périodiques, du période donnée, des équations hamiltoniennes. Note CRAS Paris 287, 1013–1015 (1978)

    Google Scholar 

  5. Evans, L.C.: Partial Differential Equations. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  6. Goodridge, S.G., Kay, M.G., Luo, R.C.: Multi-layered fuzzy behavior fusion for reactive control of an autonomous mobile robot. In: Procs. of the 6th IEEE Int. Conf. on Fuzzy Systems, Barcelona, SP, pp. 579–584 (1996)

    Google Scholar 

  7. Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic, Theory and Applications (1995)

    Google Scholar 

  8. Harris, J.: Fuzzy Logic Applications in Engineering Science. Springer, Heidelberg (2005)

    Google Scholar 

  9. Kosko, B.: Fuzzy Thinking: The New Science of Fuzzy Logic, New York (1993)

    Google Scholar 

  10. Lee, C.C.: Fuzzy logic in control systems: Fuzzy logic controller, part I. IEEE, Trans. Fuzzy Syst., 2, 4–15 (1994)

    Article  Google Scholar 

  11. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems (Cambridge Texts in Applied Mathematics). Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  12. Lim, M.H., Rahardja, S., Gwee, B.H.: A GA paradigm for learning fuzzy rules School of EEE, Nanyang Technological University (1996)

    Google Scholar 

  13. Michalek, R., Tarantello, G.: Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems. J. Diff. Eq. 72, 28–55 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rabinowitz, P.: On subharmonic solutions of a Hamiltonian system. Comm. Pure Appl. Math. 33, 609–633 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tarantello, G.: Subharmonic solutions for Hamiltonian systems via a ℤ p pseudoindex theory. Annali di Matematica Pura (to appear)

    Google Scholar 

  16. Terano, T., Asai, K., Sugeno, M.: Fuzzy Systems Theory and Its Applications. Academic, New York (1992)

    MATH  Google Scholar 

  17. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd edn. Springer, Heidelberg (1999)

    Book  MATH  Google Scholar 

  18. Von Altrock, C.: Recent Successful Fuzzy Logic Applications in Industrial Automation IEEE. In: Fifth International Conference on Fuzzy Systems (1996)

    Google Scholar 

  19. Xu, D., Keller, J.M., Popescu, M.: Applications of Fuzzy Logic in Bioinformatics (Advances in Bioinformatics and Computational Biology) (2008)

    Google Scholar 

  20. Zadeh, L.A.: Fuzzy Sets. Information and Control (1965)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Saarland University, Germany

    Michael Breuss

  2. German Research Centre for Artificial Intelligence (DFKI GmbH), Bremen, Germany

    Dominik Dietrich

Authors
  1. Michael Breuss
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dominik Dietrich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. GET Lab, University of Paderborn, Pohlweg 47-49, 33098, Paderborn, Germany

    Bärbel Mertsching , Marcus Hund  & Zaheer Aziz ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Breuss, M., Dietrich, D. (2009). Fuzzy Numerical Schemes for Hyperbolic Differential Equations. In: Mertsching, B., Hund, M., Aziz, Z. (eds) KI 2009: Advances in Artificial Intelligence. KI 2009. Lecture Notes in Computer Science(), vol 5803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04617-9_53

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04617-9_53

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04616-2

  • Online ISBN: 978-3-642-04617-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics