Computability and Incomputability of Differential Equations

  • Guido Gherardi


In the following discussion we are going to deal with the problem of computability of differential equations, and we will outline some of the most important results achieved in this area, mainly due to K. Weihrauch and N. Zhong. In particular, a large part of the paper will concern the debate about the computability of the wave equation.


Wave Equation Cauchy Problem Turing Machine Computable Function Solution Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Blum, F. Cucker, M. Schub and S. Smale: Complexity and Real Computation (Springer-Verlag, Berlin Heidelberg New York 1998)Google Scholar
  2. 2.
    V. Brattka: Effective Borel measurability and reducibility of functions. Mathematical Logic Quarterly 51 (2005) pp 19–44CrossRefGoogle Scholar
  3. 3.
    V. Brattka and G. Gherardi: Borel Complexity of Topological Operations on Computable Metric Spaces. To appear in Journal of Logic and Computation Google Scholar
  4. 4.
    G. Chaitin: Meta Math! The Quest for Omega (Vintage Books 2006)Google Scholar
  5. 5.
    D. Cenzer and J. B. Remmel: Index sets for computable differential equations. Mathematical Logic Quarterly 50 (2004) pp 329–344CrossRefGoogle Scholar
  6. 6.
    R. Feynman: The Character of Physical Law (The MIT Press, Boston 1965)Google Scholar
  7. 7.
    W. Gay, B. Y. Zhang and N. Zhong: Computability of solutions of the Korteweg-de Vries equation. Mathematical Logic Quarterly 47 (2001) pp 93–110CrossRefGoogle Scholar
  8. 8.
    G. Gherardi: Effective Borel Degrees of some Topological Functions. Mathematical Logic Quarterly 52, 6 (2006) pp 625–642CrossRefGoogle Scholar
  9. 9.
    G. Gherardi: Paradigmi di computazione per i numeri reali. Bollettino dell’Unione Matematica Italiana To appearGoogle Scholar
  10. 10.
    R. Guénon: Les principes du calcul infinitésimal (Gallimard, Paris 1946)Google Scholar
  11. 11.
    M. B. Pour-El and J. I. Richards: A computable ordinary differential equation which possesses no computable solution. Annals Math. Logic 17 (1979) pp 61–90CrossRefGoogle Scholar
  12. 12.
    M. B. Pour-El and J. I. Richards: Computability in Analysis and Physics (Springer-Verlag, Berlin Heidelberg New York 1989)Google Scholar
  13. 13.
    S. G. Simpson: Subsystems of Second Order Arithmetic (Springer-Verlag, Berlin Heidelberg New York 1999)Google Scholar
  14. 14.
    K. Weihrauch: Computable Analysis (Springer-Verlag, Berlin Heidelberg New York 2000)Google Scholar
  15. 15.
    K. Weihrauch and N. Zhong: The wave propagator is Turing computable. Lecture Notes in Computer Science 1644 (1999) pp 697–706CrossRefGoogle Scholar
  16. 16.
    K. Weihrauch and N. Zhong: Is the linear Schrödinger propagator Turing computable? Lecture Notes in Computer Science 2064 (2001) pp 369–377CrossRefGoogle Scholar
  17. 17.
    K. Weihrauch and N. Zhong: Is wave propagation computable or can wave computers beat the Turing machine? Proceedings of the London Mathematical Society 85 (2002) pp 312–332CrossRefGoogle Scholar
  18. 18.
    K. Weihrauch and N. Zhong: Computing the solution of the Korteweg-de Vries equation with arbitrary precision on Turing machines. Theoretical Computer Science 332 (2005) pp 337–366CrossRefGoogle Scholar
  19. 19.
    K. Weihrauch and N. Zhong: Computable analysis of the abstract Cauchy problem in a Banach space and its applications I. Mathematical Logic Quarterly 53 (2007) pp 511–531CrossRefGoogle Scholar
  20. 20.
    N. Zhong: Computable analysis of a boundary-value problem for the Kortewegde Vires equation. Theory of Computing Systems 41 (2007) pp 155–175CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2008

Authors and Affiliations

  • Guido Gherardi
    • 1
  1. 1.Dipartimento di FilosofiaUniversità di BolognaBolognaItaly

Personalised recommendations