Annali di Matematica Pura ed Applicata

, Volume 188, Issue 1, pp 123–151 | Cite as

An algebraic foundation for factoring linear boundary problems

  • Georg Regensburger
  • Markus Rosenkranz


Motivated by boundary problems for linear differential equations, we define an abstract boundary problem as a pair consisting of a surjective linear map (“differential operator”) and an orthogonally closed subspace of the dual space (“boundary conditions”). Defining the composition of boundary problems corresponding to their Green’s operators in reverse order, we characterize and construct all factorizations of a boundary problem from a given factorization of the defining operator. For the case of ordinary differential equations, the main results can be made algorithmic. We conclude with a factorization of a boundary problem for the wave equation.


Linear boundary value problems Factorization Green’s operators 

Mathematics Subject Classification (2000)

47A68 34B05 35G15 


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  1. 1.
    Brown R.C. and Krall A.M. (1974). Ordinary differential operators under Stieltjes boundary conditions. Trans. Am. Math. Soc. 198: 73–92 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Buchberger, B.: An algorithm for finding the bases elements of the residue class ring modulo a zero dimensional polynomial ideal (German) [English translation published in J. Symbolic Comput., 41(3-4), 475–511 (2006)]. Ph.D. Thesis, University of Innsbruck (1965)Google Scholar
  3. 3.
    Buchberger, B.: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems [English translation: an algorithmic criterion for the solvability of a system of algebraic equations. In: Gröbner Bases and Applications, Buchberger, B., Winkler, F. (eds.) London Math. Soc. Lecture Note Ser., vol. 251, pp. 535–545. Cambridge University Press (1998)]. Aequationes Math. 4, 374–383 (1970)Google Scholar
  4. 4.
    Conway, J.B.: A course in functional analysis. Graduate Texts in Mathematics, vol. 96, 2nd edn. Springer, New York (1990)Google Scholar
  5. 5.
    Eilenberg, S.: Automata, languages, and machines (vol. B). Pure and Applied Mathematics, vol. 59. Academic Press, New York (1976)Google Scholar
  6. 6.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of inverse problems. Mathematics and its Applications, vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996)Google Scholar
  7. 7.
    Erné, M., Koslowski, J., Melton, A., Strecker, G.E.: A primer on Galois connections. In: Papers on general topology and applications (Madison, WI, 1991), Annals of New York Academic Science, vol. 704, pp. 103–125. New York Academic Science, New York (1993)Google Scholar
  8. 8.
    Green, G.: An essay on the application of mathematical analysis to the theories of electricity and magnetism. Private, Nottingham (1828). Available at
  9. 9.
    Grigoriev D. and Schwarz F. (2004). Factoring and solving linear partial differential equations. Computing 73(2): 179–197 zbMATHMathSciNetGoogle Scholar
  10. 10.
    Grigoriev D. and Schwarz F. (2007). Loewy- and primary decompositions of D-modules. Adv. App. Math. 38: 526–541 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Grigoriev D.Y. (1990). Complexity of factoring and calculating the GCD of linear ordinary differential operators. J. Symbolic Comput. 10(1): 7–37 CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kamke, E.: Differentialgleichungen. Lösungsmethoden und Lösungen. Teil I: Gewöhnliche Differentialgleichungen. Mathematik und ihre Anwendungen in Physik und Technik A, vol. 18, 8th edn. Akademische Verlagsgesellschaft, Leipzig (1967)Google Scholar
  13. 13.
    Köthe, G.: Topological vector spaces. I. Die Grundlehren der mathematischen Wissenschaften, vol. 159. Springer, New York (1969)Google Scholar
  14. 14.
    Lang, S.: Real and functional analysis. Graduate Texts in Mathematics, vol. 142. Springer, New York (1993)Google Scholar
  15. 15.
    Nashed, M.Z., Votruba, G.F.: A unified operator theory of generalized inverses. In: Generalized Inverses and Applications (Proc. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1973), pp. 1–109. Publication of Mathematics Research Center University Wisconsin, No. 32. Academic Press, New York (1976)Google Scholar
  16. 16.
    Neumann C. (1861). Lösung des allgemeinen Problems über den stationären Temperaturzustand einer homogenen Kugel ohne Hilfe von Reihenentwicklungen, nebst einigen Sätzen zur Theorie der Anziehung. H.W. Schmidt, Halle Google Scholar
  17. 17.
    van der Put, M., Singer, M.F.: Galois theory of linear differential equations. Grundlehren der Mathematischen Wissenschaften, vol. 328. Springer, Berlin (2003)Google Scholar
  18. 18.
    Riemann, B.: Schwere, Electricität und Magnetismus. Nach den Vorlesungen von Bernhard Riemann bearbeitet. Carl Rümpler, Hannover (1880). Lectures delivered in the summer term of 1861 in Göttingen. Available at,_Elektricität_und_Magnetismus
  19. 19.
    Rosenkranz M. (2005). A new symbolic method for solving linear two-point boundary value problems on the level of operators. J. Symbolic Comput. 39(2): 171–199 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rosenkranz M., Buchberger B. and Engl H.W. (2003). Solving linear boundary value problems via non-commutative Gröbner bases. Appl. Anal. 82(7): 655–675 zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Rosenkranz M. and Regensburger G. (2007). Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. J. Symbolic Comput. doi: 10.1016/j.jsc.2007.11.007 Google Scholar
  22. 22.
    Schwarz, F.: A factorization algorithm for linear ordinary differential equations. In: ISSAC ’89: Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, pp. 17–25. ACM Press, New York (1989)Google Scholar
  23. 23.
    Stakgold I. (1979). Green’s functions and boundary value problems. Wiley, New York zbMATHGoogle Scholar
  24. 24.
    Tsarev, S.P.: An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator. In: ISSAC ’96: Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, pp. 226–231. ACM Press, New York (1996)Google Scholar
  25. 25.
    Tsarev S.P. (1998). Factorization of linear partial differential operators and darboux integrability of nonlinear pdes. SIGSAM Bull. 32(4): 21–28 zbMATHCrossRefGoogle Scholar
  26. 26.
    Wyler O. (1964). Green’s operators. Ann. Mat. Pura Appl. 66(4): 252–263 MathSciNetGoogle Scholar
  27. 27.
    Yosida, K.: Functional analysis. Grundlehren der Mathematischen Wissenschaften, vol. 123, 6th edn. Springer, Berlin (1980)Google Scholar

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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria

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