Domain-Theoretic Formulation of Linear Boundary Value Problems

  • Dirk Pattinson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)


We present a domain theoretic framework for obtaining exact solutions of linear boundary value problems. Based on the domain of compact real intervals, we show how to approximate both a fundamental system and a particular solution up to an arbitrary degree of accuracy. The boundary conditions are then satisfied by solving a system of imprecisely given linear equations at every step of the approximation. By restricting the construction to effective bases of the involved domains, we not only obtain results on the computability of boundary value problems, but also directly implementable algorithms, based on proper data types, that approximate solutions up to an arbitrary degree of accuracy. As these data types are based on rational numbers, no numerical errors are incurred in the computation process.


Linear Boundary Interval Analysis Fundamental Matrix Fundamental System Domain Theory 
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.
    Abramsky, S., Jung, A.: Domain Theory. In: Abramsky, S., Gabbay, D., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3. Clarendon Press, Oxford (1994)Google Scholar
  2. 2.
    Edalat, A.: Domain theory and integration. Theor. Comp. Sci. 151, 163–193 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Edalat, A.: Domains for computation in mathematics, physics and exact real arithmetic. Bulletin of Symbolic Logic 3(4), 401–452 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Edalat, A., Krznaric, M.: Numerical integration with exact arithmetic. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, p. 90. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Edalat, A., Krznarić, M., Lieutier, A.: Domain-theoretic solution of differential equations (scalar fields). In: Proceedings of MFPS XIX. Elect. Notes in Theoret. Comput. Sci., vol. 83 (2004)Google Scholar
  6. 6.
    Edalat, A., Lieutier, A., Pattinson, D.: A computational model for differentiable functions. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 505–519. Springer, Heidelberg (2005) (to appear)CrossRefGoogle Scholar
  7. 7.
    Edalat, A., Pattinson, D.: A domain theoretic account of euler’s method for solving initial value problems (sumitted), available at:
  8. 8.
    Edalat, A., Pattinson, D.: A domain theoretic account of picard’s theorem. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 494–505. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Edalat, A., Sünderhauf, P.: A domain theoretic approach to computability on the real line. Theoretical Computer Science 210, 73–98 (1998)CrossRefGoogle Scholar
  10. 10.
    Erker, T., Esacrdò, M., Keimel, K.: The way-below relation of function spaces over semantic domains. Topology and its Applications 89(1–2), 61–74 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)zbMATHCrossRefGoogle Scholar
  12. 12.
    Hansen, E.: On solving two-point boundary-value problems using interval arithmetic. In: Hansen, E. (ed.) Topics In Interval Analysis, pp. 74–90. Oxford University Press, Oxford (1969)Google Scholar
  13. 13.
    Iserles, A.: Numerical Analysis of Differential Equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1996)Google Scholar
  14. 14.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)zbMATHGoogle Scholar
  15. 15.
    Oliveira, F.A.: Interval analysis and two-point boundary value problems. SIAM J. Numer. Anal. 11, 382–391 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Stoltenberg-Hansen, V., Lindström, I., Griffor, E.: Mathematical Theory of Domains. Cambridge Tracts in Theoretical Computer Science, vol. 22. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Dirk Pattinson
    • 1
  1. 1.Department of ComputingImperial College LondonUK

Personalised recommendations