Theoretical and Mathematical Physics

, Volume 145, Issue 2, pp 1511–1524 | Cite as

Constructively Factoring Linear Partial Differential Operators in Two Variables

  • R. Beals
  • E. A. Kartashova


We study conditions under which a partial differential operator of arbitrary order n in two variables or an ordinary linear differential operator admits a factorization with a first-order factor on the left.The process of factoring consists of recursively solving systems of linear equations subject to certain differential compatibility conditions.In the general case of partial differential operators, it is not necessary to solve a differential equation. In special degenerate cases, such as an ordinary differential operator, the problem eventually reduces to solving some Riccati equation(s). We give the factorization conditions explicitly for the second and third orders and in outline form for higher orders.


differential operators factorization of differential operators algebraic factorization 


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  1. 1.
    J. Apel, J. Symbolic Comput., 25, 683–704 (1998); P. Olver, Applications of Lie Groups to Differential Equations (2nd ed.), Springer, New York (1993); J. F. Pommaret, Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, New York (1978).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    M. Bronstein, “An improved algorithm for factoring linear ordinary differential operators,” in: Proc. Intl. Symp. on Symbolic and Algebraic Computation (ISSAC'94) (Oxford, Great Britain, July 20–22, 1994, J. von zur Gathen and M. Giesbrecht, eds.), ACM Press, Baltimore, Md. (1994), pp. 336–340; F. Schwarz, “A factorization algorithm for linear ordinary differential equations,” in: Proc. Intl. Symp. on Symbolic and Algebraic Computation ACMSIGSAM 1989 (ISSAC'89) (Portland, Oregon, US, July 17–19, 1989, G. H. Gonnet, ed.), ACM Press, New York (1989), pp. 17–25; S. P. Tsarev, “An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator,” in: Proc. Intl. Symp. on Symbolic and Algebraic Computation (ISSAC'96) (Zurich, Switzerland, July 24–26, 1996, Y. N. Lakshman, ed.), ACM Press, New York (1996), pp. 226–231.Google Scholar
  3. 3.
    E. Beke, Math. Ann., 45, 278–300 (1894).zbMATHMathSciNetGoogle Scholar
  4. 4.
    A. Loewy, Math. Ann., 62, 89–117 (1906); 56, 549–584 (1903).CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    E. Landau, J. fur Math., 124, 115–120 (1901).zbMATHGoogle Scholar
  6. 6.
    D. Grigoriev and F. Schwarz, Computing, 73, 179–197 (2004).MathSciNetGoogle Scholar
  7. 7.
    S. P. Tsarev, “Factorization of linear partial differential operators and Darboux integrability of nonlinear PDEs,” Poster at ISSAC'98, Rostock, Germany, August 13–15, 1998 (1998); cs. SC/9811002 (1998).Google Scholar
  8. 8.
    S. P. Tsarev, “Factorization of overdetermined systems of linear partial differential equations with finite dimensional solution space,” in: Proc. 4th Intl. Workshop on Computer Algebra and Scientific Computing (CASC-2001) (Konstanz, Germany, September 22–26, 2001, V. Ganzha, E. Mayr, and V. Vorozhtsov, eds.), Springer, Berlin (2001), pp. 529–539; Z. Li, F. Schwarz, and S. P. Tsarev, J. Symbolic Comput., 36, 443–471 (2003).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • R. Beals
    • 1
  • E. A. Kartashova
    • 2
  1. 1.Yale UniversityNew HavenUSA
  2. 2.Johannes Kepler Universitat (JKU) LinzLinzAustria

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