Beyond the First Main Theorem – When Is the Solution of a Linear Cauchy Problem Computable?

  • Klaus Weihrauch
  • Ning Zhong
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3959)

Abstract

We study computabilty of the abstract linear Cauchy problem

du(t)/dt = Au(t), t > 0, u(0) = x ∈ X

where A is a linear operator on a Banach space X. We give necessary and sufficient conditions for A such that the operator K:xu is computable. We consider continuous operators and more generally closed operators A. For studying computability we use the representation approach to Computable Analysis (TTE) [7, 1] which is consistent with the model used in [6].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brattka, V.: Computable versions of the uniform boundedness theorem. In: Chatzidakis, Z., Koepke, P., Pohlers, W. (eds.) Logic Colloquium 2002. Lecture Notes in Logic, vol. 27, Urbana (2006), Association for Symbolic LogicGoogle Scholar
  2. 2.
    Gay, W., Zhang, B.-Y., Zhong, N.: Computability of solutions of the Korteweg-de Vries equation. Mathematical Logic Quarterly 47(1), 93–110 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kunkle, D.: Type-2 computability on spaces of integrable functions. Mathematical Logic Quarterly 50(4,5), 417–430 (2004)MATHMathSciNetGoogle Scholar
  4. 4.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983)MATHGoogle Scholar
  5. 5.
    Pour-El, M., Zhong, N.: The wave equation with computable initial data whose unique solution is nowhere computable. Mathematical Logic Quarterly 43(4), 499–509 (1997)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Pour-El, M.B., Ian Richards, J.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin (1989)MATHGoogle Scholar
  7. 7.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)MATHGoogle Scholar
  8. 8.
    Weihrauch, K., Zhong, N.: Is wave propagation computable or can wave computers beat the Turing machine? Proceedings of the London Mathematical Society 85(2), 312–332 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Weihrauch, K., Zhong, N.: An algorithm for computing fundamental solutions. In: Brattka, V., Staiger, L., Weihrauch, K. (eds.) Proceedings of the 6th Workshop on Computability and Complexity in Analysis. Electronic Notes in Theoretical Computer Science, vol. 120, pp. 201–215. Elsevier, Amsterdam (2004). 6th International Workshop, CCA 2004. Wittenberg, Germany, August 16–20 (2004)Google Scholar
  10. 10.
    Weihrauch, K., Zhong, N.: Computing the solution of the Korteweg-de Vries equation with arbitrary precision on Turing machines. Theoretical Computer Science 332(1–3), 337–366 (2005)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Klaus Weihrauch
    • 1
  • Ning Zhong
    • 2
  1. 1.University of HagenHagenGermany
  2. 2.University of CincinnatiCincinnatiUSA

Personalised recommendations