Advertisement

Programming and Computer Software

, Volume 36, Issue 2, pp 97–102 | Cite as

Ideals of differential operators and transformations of linear partial differential equations

  • O. V. KaptsovEmail author
Article
  • 35 Downloads

Abstract

A ring of linear differential operators with smooth coefficients generated by two differentiations is considered. Concepts of operators closed with respect to commutation, a resultant of two operators, and a two-dimensional analogue of Wronskian are introduced. Sufficient conditions that two differential operators are generators of a left ideal annihilating a finite-dimensional space of functions are found. Differential operators annihilating given functions are constructed. The operators obtained transform solutions of one secondorder differential equation into solutions of another equation of the same order.

Keywords

Left Ideal Domain Versus Darboux Transformation Order Operator Finite Dimensional Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Darboux, G., Le ons sur la Ge’ne’rale des Surfaces et les Applications Ge’ome’triques du Calcul Infinite’simal, vol. 2, Paris: Gauthier-Villars, 1915.Google Scholar
  2. 2.
    Tsarev, S.P., Generalized Laplace Transformations and Integration of Hyperbolic Systems of Linear Partial Differential Equations, Proc. of the 2005 Int. Symp. on Symbolic and Algebraic Computation, Beijing: ACM, 2005, pp. 325–331.CrossRefGoogle Scholar
  3. 3.
    Zhiber, A.V. and Sokolov, V.V., Exactly Integrable Equations of Liouville Type, Usp. Mat. Nauk, 2001, vol. 56, no. 1 (337), pp. 63–106.MathSciNetGoogle Scholar
  4. 4.
    Startsev, S.Ya., Method of Cascade Laplace Integration for Systems of Linear Hyperbolic Equation, Mat. Zametki, 2008, vol. 83, no. 1, pp. 107–118.MathSciNetGoogle Scholar
  5. 5.
    Euler, L., Integral Calculus, Moscow: GIFML, 1985, vol. 3.Google Scholar
  6. 6.
    Lamb, G.L., Jr., Elements of Soliton Theory, New York: Wiley, 1980. Translated under the title Vvedenie v teoriyu solitonov, Moscow: Mir, 1983.zbMATHGoogle Scholar
  7. 7.
    Matveev, V.B. and Salle, M.A., Darboux Transformations and Solitons, New York: Springer, 1991.zbMATHGoogle Scholar
  8. 8.
    Rogers, C. and Schief, W.K., Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, Cambridge: Cambridge Univ. Press, 2002.zbMATHCrossRefGoogle Scholar
  9. 9.
    Kaptsov, O.V., Equivalence of Linear Partial Differential Equations and Euler-Darboux Transformations, Vychislitel’nye tekhnologii, 2007, vol. 12, no. 4, pp. 59–72.Google Scholar
  10. 10.
    Berest, Y. and Veselov, A., On the Structure of Singularities of Integrable Schrödinger Operators, Lett. Math. Phys., 2000, no. 52, pp. 103–111.Google Scholar
  11. 11.
    Lévy, L., Sur quelques équations Linéaires aux dé rivées partielles du second order, J. l’Éc. Polytechnique, 1886, vol. 37, Cah. LVI, pp. 63–77.Google Scholar
  12. 12.
    Cox, D., Little, J., and O’shea, D., Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, New York: Springer, 1998. Translated under the title Idealy, mnogoobraziya i algoritmy, Moscow: Mir, 2000.Google Scholar
  13. 13.
    Van der Waerden, B.L., Algebra, Moscow: Nauka, 1979.zbMATHGoogle Scholar
  14. 14.
    Kondratieva, M.V., Levin, A.B., Mikhalev, A.V., and Pankratiev, E.V., Differential and Difference Dimension Polynomials, Kluwer, 1998.Google Scholar
  15. 15.
    Walker, R.J., Algebraic Curves, Princeton: Princeton Univ. Press, 1950. Translated under the title Algebraicheskie krivye, Moscow: Kom. Kniga, 2006.zbMATHGoogle Scholar
  16. 16.
    Tsarev, S.P. and Shemyakova, E., Differential Transformations of Parabolic Second-Order Operators on the Plane, Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk, 2009, vol. 266, pp. 227–236.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Institute of Computational Modeling, Siberian BranchRussian Academy of SciencesAkademgorodok, KrasnoyarskRussia

Personalised recommendations