Skip to main content
SpringerLink
Log in

Constructively Factoring Linear Partial Differential Operators in Two Variables

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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. E. Beke, Math. Ann., 45, 278–300 (1894).

    MATH  MathSciNet  Google Scholar 

  4. A. Loewy, Math. Ann., 62, 89–117 (1906); 56, 549–584 (1903).

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Landau, J. fur Math., 124, 115–120 (1901).

    MATH  Google Scholar 

  6. D. Grigoriev and F. Schwarz, Computing, 73, 179–197 (2004).

    MathSciNet  Google Scholar 

  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).

  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 

Download references

Author information

Authors and Affiliations

  1. Yale University, New Haven, Connecticut, USA

    R. Beals

  2. Johannes Kepler Universitat (JKU) Linz, Linz, Austria

    E. A. Kartashova

Authors
  1. R. Beals
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. A. Kartashova
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 165–180, November, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beals, R., Kartashova, E.A. Constructively Factoring Linear Partial Differential Operators in Two Variables. Theor Math Phys 145, 1511–1524 (2005). https://doi.org/10.1007/s11232-005-0178-7

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-005-0178-7

Keywords

Search

Navigation