An Automated Confluence Proof for an Infinite Rewrite System Parametrized over an Integro-Differential Algebra

  • Loredana Tec
  • Georg Regensburger
  • Markus Rosenkranz
  • Bruno Buchberger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6327)

Abstract

In our symbolic approach to boundary problems for linear ordinary differential equations we use the algebra of integro-differential operators as an algebraic analogue of differential, integral and boundary operators (Section 2). They allow to express the problem statement (differential equation and boundary conditions) as well as the solution operator (an integral operator called “Green’s operator”), and they are the basis for operations on boundary problems like solving and factoring [14,17]. A survey of the implementation is given in [18].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aichinger, E., Pilz, G.F.: A survey on polynomials and polynomial and compatible functions. In: Proceedings of the Third International Algebra Conference, pp. 1–16. Kluwer Acad. Publ., Dordrecht (2003)Google Scholar
  2. 2.
    Bergman, G.M.: The diamond lemma for ring theory. Adv. in Math. 29(2), 178–218 (1978)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Buchberger, B.: An algorithm for finding the bases elements of the residue class ring modulo a zero dimensional polynomial ideal (German). PhD thesis, Univ. of Innsbruck (1965); English Translation J. Symbolic Comput. 41(3-4), 475–511 (2006)Google Scholar
  4. 4.
    Buchberger, B.: Introduction to Gröbner bases. In: Buchberger, B., Winkler, F. (eds.) Gröbner Bases and Applications, pp. 3–31. Cambridge Univ. Press, Cambridge (1998)Google Scholar
  5. 5.
    Buchberger, B.: Gröbner rings and modules. In: Proceedings of SYNASC 2001, pp. 22–25 (2001)Google Scholar
  6. 6.
    Buchberger, B., et al.: Theorema: Towards computer-aided mathematical theory exploration. J. Appl. Log. 4(4), 359–652 (2006)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Buchberger, B., Loos, R.: Algebraic simplification. In: Computer Algebra, pp. 11–43. Springer, Vienna (1983)Google Scholar
  8. 8.
    Buchberger, B., Regensburger, G., Rosenkranz, M., Tec, L.: General polynomial reduction with Theorema functors: Applications to integro-differential operators and polynomials. ACM Commun. Comput. Algebra 42(3), 135–137 (2008)Google Scholar
  9. 9.
    Bueso, J., Gómez-Torrecillas, J., Verschoren, A.: Algorithmic methods in non-commutative algebra. Kluwer Academic Publishers, Dordrecht (2003)MATHGoogle Scholar
  10. 10.
    Guo, L., Keigher, W.: On differential Rota-Baxter algebras. J. Pure Appl. Algebra 212(3), 522–540 (2008)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kolchin, E.: Differential algebra and algebraic groups. Pure and Applied Mathematics, vol. 54. Academic Press, New York (1973)MATHGoogle Scholar
  12. 12.
    Lausch, H., Nöbauer, W.: Algebra of polynomials. North-Holland Publishing Co., Amsterdam (1973)MATHGoogle Scholar
  13. 13.
    Levandovskyy, V.: PLURAL, a non-commutative extension of SINGULAR: past, present and future. In: Iglesias, A., Takayama, N. (eds.) ICMS 2006. LNCS, vol. 4151, pp. 144–157. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Regensburger, G., Rosenkranz, M.: An algebraic foundation for factoring linear boundary problems. Ann. Mat. Pura. Appl. (4) 188(1), 123–151 (2009)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rosenkranz, M.: A new symbolic method for solving linear two-point boundary value problems on the level of operators. J. Symbolic Comput. 39(2), 171–199 (2005)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rosenkranz, M., Regensburger, G.: Integro-differential polynomials and operators. In: Proceedings of ISSAC 2008, pp. 261–268. ACM, New York (2008)CrossRefGoogle Scholar
  17. 17.
    Rosenkranz, M., Regensburger, G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. J. Symbolic Comput. 43(8), 515–544 (2008)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Rosenkranz, M., Regensburger, G., Tec, L., Buchberger, B.: A symbolic framework for operations on linear boundary problems. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 269–283. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Rosenkranz, M., Regensburger, G., Tec, L., Buchberger, B.: Symbolic analysis for boundary problems: From rewriting to parametrized Gröbner bases. Technical Report 2010-05, RICAM (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Loredana Tec
    • 1
  • Georg Regensburger
    • 2
  • Markus Rosenkranz
    • 3
  • Bruno Buchberger
    • 1
  1. 1.Research Institute for Symbolic ComputationJohannes Kepler UniversityCastle of HagenbergAustria
  2. 2.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  3. 3.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUnited Kingdom

Personalised recommendations