A Symbolic Framework for Operations on Linear Boundary Problems

  • Markus Rosenkranz
  • Georg Regensburger
  • Loredana Tec
  • Bruno Buchberger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5743)

Abstract

We describe a symbolic framework for treating linear boundary problems with a generic implementation in the Theorema system. For ordinary differential equations, the operations implemented include computing Green’s operators, composing boundary problems and integro-differential operators, and factoring boundary problems. Based on our factorization approach, we also present some first steps for symbolically computing Green’s operators of simple boundary problems for partial differential equations with constant coefficients. After summarizing the theoretical background on abstract boundary problems, we outline an algebraic structure for partial integro-differential operators. Finally, we describe the implementation in Theorema, which relies on functors for building up the computational domains, and we illustrate it with some sample computations including the unbounded wave equation.

Keywords

Linear boundary problem Green’s operator Integro-Differential Operator Ordinary Differential Equation Wave Equation 

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References

  1. 1.
    Stakgold, I.: Green’s functions and boundary value problems. John Wiley & Sons, New York (1979)MATHGoogle Scholar
  2. 2.
    Rosenkranz, M., Buchberger, B., Engl, H.W.: Solving linear boundary value problems via non-commutative Gröbner bases. Appl. Anal. 82, 655–675 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Rosenkranz, M.: A new symbolic method for solving linear two-point boundary value problems on the level of operators. J. Symbolic Comput. 39, 171–199 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Rosenkranz, M., Regensburger, G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. J. Symbolic Comput. 43, 515–544 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Regensburger, G., Rosenkranz, M.: An algebraic foundation for factoring linear boundary problems. Ann. Mat. Pura Appl. 188(4), 123–151 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Buchberger, B., Craciun, A., Jebelean, T., Kovacs, L., Kutsia, T., Nakagawa, K., Piroi, F., Popov, N., Robu, J., Rosenkranz, M., Windsteiger, W.: Theorema: Towards computer-aided mathematical theory exploration. J. Appl. Log. 4, 359–652 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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)MATHGoogle Scholar
  8. 8.
    Buchberger, B.: Introduction to Gröbner bases. In: Buchberger, B., Winkler, F. (eds.) Gröbner bases and applications, Cambridge Univ. Press, Cambridge (1998)CrossRefGoogle Scholar
  9. 9.
    Mora, T.: An introduction to commutative and noncommutative Gröbner bases. Theoret. Comput. Sci. 134, 131–173 (1994)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Köthe, G.: Topological vector spaces, vol. I. Springer, New York (1969)MATHGoogle Scholar
  11. 11.
    Brown, R.C., Krall, A.M.: Ordinary differential operators under Stieltjes boundary conditions. Trans. Amer. Math. Soc. 198, 73–92 (1974)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kamke, E.: Differentialgleichungen. Lösungsmethoden und Lösungen. Teil I: Gewöhnliche Differentialgleichungen. Akademische Verlagsgesellschaft, Leipzig (1967)Google Scholar
  13. 13.
    Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York (1955)MATHGoogle Scholar
  14. 14.
    Rosenkranz, M., Regensburger, G.: Integro-differential polynomials and operators. In: Jeffrey, D. (ed.) Proceedings of ISSAC 2008, pp. 261–268. ACM, New York (2008)Google Scholar
  15. 15.
    van der Put, M., Singer, M.F.: Galois theory of linear differential equations. Springer, Berlin (2003)MATHGoogle Scholar
  16. 16.
    Schwarz, F.: A factorization algorithm for linear ordinary differential equations. In: Proceedings of ISSAC 1989, pp. 17–25. ACM, New York (1989)Google Scholar
  17. 17.
    Tsarev, S.P.: An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator. In: Proceedings of ISSAC 1996, pp. 226–231. ACM, New York (1996)Google Scholar
  18. 18.
    Grigoriev, D., Schwarz, F.: Loewy- and primary decompositions of D-modules. Adv. in Appl. Math. 38, 526–541 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Tsarev, S.P.: Factorization of linear partial differential operators and Darboux integrability of nonlinear PDEs. SIGSAM Bull. 32, 21–28 (1998)CrossRefMATHGoogle Scholar
  20. 20.
    Regensburger, G., Rosenkranz, M., Middeke, J.: A skew polynomial approach to integro-differential operators. In: Proceedings of ISSAC 2009. ACM, New York (to appear, 2009)Google Scholar
  21. 21.
    Cohn, P.M.: Further algebra and applications. Springer, London (2003)CrossRefMATHGoogle Scholar
  22. 22.
    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, 135–137 (2008)Google Scholar
  23. 23.
    Buchberger, B.: Groebner rings and modules. In: Maruster, S., Buchberger, B., Negru, V., Jebelean, T. (eds.) Proceedings of SYNASC 2001, pp. 22–25 (2001)Google Scholar
  24. 24.
    Buchberger, B.: Groebner bases in Theorema using functors. In: Faugere, J., Wang, D. (eds.) Proceedings of SCC 2008, pp. 1–15. LMIB Beihang University Press (2008)Google Scholar
  25. 25.
    Windsteiger, W.: Building up hierarchical mathematical domains using functors in Theorema. Electr. Notes Theor. Comput. Sci. 23, 401–419 (1999)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Markus Rosenkranz
    • 1
  • Georg Regensburger
    • 1
  • Loredana Tec
    • 2
  • Bruno Buchberger
    • 2
  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  2. 2.Research Institute for Symbolic ComputationJohannes Kepler UniversitätCastle of HagenbergAustria

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