Advertisement

Decomposition of Second-Order Operators

  • Fritz Schwarz
Chapter
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)

Abstract

This chapter deals with a genuine extension of Loewy’s theory. The ideals under consideration have differential type greater than zero. This means that the corresponding differential equations have a general solution involving not only constants but undetermined functions of varying numbers of arguments. Loewy’s results are applied to individual linear pde’s of second order in the plane with coordinates x and y, and the principal ideals generated by the corresponding operators. These equations have been considered extensively in the literature of the nineteenth century [14, 18, 28, 40, 44]. Like in the classical theory, equations with leading derivatives \({\partial }_{xx}\) or \({\partial }_{xy}\) are distinguished.

References

  1. 1.
    Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover, New YorkGoogle Scholar
  2. 2.
    Adams WW, Loustaunau P (1994) An introduction to Gröbner bases. American Mathematical Society, ProvidenceGoogle Scholar
  3. 3.
    Beke E (1894) Die Irreduzibilität der homogenen Differentialgleichungen. Math Ann 45: 278–294Google Scholar
  4. 4.
    Bluman GW, Kumei S (1990) Symmetries of differential equations. Springer, BerlinGoogle Scholar
  5. 5.
    Blumberg H (1912) Über algebraische Eigenschaften von linearen homogenen Differentialausdrücken. Inaugural-Dissertation, GöttingenGoogle Scholar
  6. 6.
    Bronstein M (1994) An improved algorithm for factoring linear ordinary differential operators. In: Proceedings of the ISSAC’94. ACM, New York, pp 336–340Google Scholar
  7. 7.
    Buchberger B (1970) Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequ Math 4:374–383Google Scholar
  8. 8.
    Buium A, Cassidy Ph (1999) Differential algebraic geometry and differential algebraic groups: from algebraic differential equations to diophantine geometry. In: Bass H, Buium A, Cassidy Ph (eds) Selected works of Ellis Kolchin. AMS, ProvidenceGoogle Scholar
  9. 9.
    Castro-Jiménez FJ, Moreno-Frías MA (2001) An introduction to Janet bases and Gröbner bases. In: Lecture notes in pure and applied mathematics, vol 221. Marcel Dekker, New York, pp 133–145Google Scholar
  10. 10.
    Chen G, Ma Y (2005) Algorithmic reduction and rational general solutions of first-order algebraic differential equations. In: Wang D, Zhen Z (eds) Differential equations with symbolic computation. Birkhäuser, BaselGoogle Scholar
  11. 11.
    Cohn PM (2006) Free ideal rings and localization in general rings. Cambridge University Press, Cambridge/New YorkGoogle Scholar
  12. 12.
    Coutinho SC (1995) A primer of algebraic D-modules. London mathematical society student texts, vol 33. Cambridge University Press, CambridgeGoogle Scholar
  13. 13.
    Cox D, Little J, O’Shea D (1991/1998) Ideals, varieties and algorithms. Springer, New York; Using algebraic geometry. Springer, New YorkGoogle Scholar
  14. 14.
    Darboux E (1972) Leçons sur la théorie générale des surfaces, vol II. Chelsea Publishing, New YorkGoogle Scholar
  15. 15.
    Davey BA, Priestley HA (2002) Introduction to lattices and order. Cambridge University Press, Cambridge/New YorkGoogle Scholar
  16. 16.
    Eremenko A (1998) Rational solutions of first-order differential equations. Ann Acad Scient FennicæMath 23L:181–190Google Scholar
  17. 17.
    Forsyth AR (1906) Theory of differential equations, vols I–VI. Cambridge University Press, CambridgeGoogle Scholar
  18. 18.
    Goursat E (1898) Leçon sur l’intégration des équation aux dérivées partielles, vol I and II. A. Hermann, ParisGoogle Scholar
  19. 19.
    Grätzer G (1998) General lattice theory. Birkhäuser, Basel/BostonGoogle Scholar
  20. 20.
    Greuel GM, Pfister G (2002) A singular introduction to commutative algebra. Springer, Berlin/New YorkGoogle Scholar
  21. 21.
    Grigoriev D (1990) Complexity of factoring and calculating the GCD of linear ordinary differential operators. J Symb Comput 7:7–37Google Scholar
  22. 22.
    Grigoriev D, Schwarz F (2004) Factoring and solving linear partial differential equations. Computing 73:179–197Google Scholar
  23. 23.
    Grigoriev D, Schwarz F (2005) Generalized Loewy decomposition of D-modules. In: Kauers M (ed) Proceedings of the ISSAC’05. ACM, New York, pp 163–170Google Scholar
  24. 24.
    Grigoriev D, Schwarz F (2008) Loewy decomposition of third-order linear PDE’s in the plane. In: Gonzales-Vega L (ed) Proceedings of the ISSAC 2008, Linz. ACM, New York, pp 277–286Google Scholar
  25. 25.
    Grigoriev D, Schwarz F (2010) Absolute factoring of non-holonomic ideals in the plane. In: Watt SM (ed) Proceedings of the ISSAC 2010, Munich. ACM, New YorkGoogle Scholar
  26. 26.
    Hillebrand A, Schmale W (2001) Towards an effective version of a theorem due to stafford. J Symb Comput 32L:699–716Google Scholar
  27. 27.
    Hubert E (2003) Notes on triangular sets and triangulation-decomposition algorithms; Part I: polynomial systems; Part II: differential systems. In: Winkler F, Langer U (eds) Symbolic and numerical scientific computing. LNCS 2630. Springer, Berlin/LondonGoogle Scholar
  28. 28.
    Imschenetzky VG (1872) Étude sur les méthodes d’intégration des équations aux dérivées partielles du second ordre d’une fonction de deux variables indépendantes. Grunert’s Archiv 54:209–360Google Scholar
  29. 29.
    Ince EL (1926) Ordinary differential equations. Longmans (Reprint by Dover, New York, 1960].Google Scholar
  30. 30.
    Janet M (1920) Les systèmes d’équations aux dérivées partielles. J Math 83:65–123Google Scholar
  31. 31.
    Kamke E (1962) Differentialgleichungen I. Partielle Differentialgleichungen. Akademische Verlagsgesellschaft, LeipzigGoogle Scholar
  32. 32.
    Kamke E (1964) Differentialgleichungen I. Gewöhnliche Differentialgleichungen. Akademische Verlagsgesellschaft, LeipzigGoogle Scholar
  33. 33.
    Kamke E (1965) Differentialgleichungen, Lösungsmethoden und Lösungen II. Partielle Differentialgleichungen. Akademische Verlagsgesellschaft, LeipzigGoogle Scholar
  34. 34.
    Kamke E (1967) Differentialgleichungen, Lösungsmethoden und Lösungen I. Gewöhnliche Differentialgleichungen. Akademische Verlagsgesellschaft, LeipzigGoogle Scholar
  35. 35.
    Kaplansky I (1957) An introduction to differential algebra. Hermann, ParisGoogle Scholar
  36. 36.
    Kolchin E (1964) The notion of dimension in the theory of algebraic differential equations. Bull AMS 70:570–573Google Scholar
  37. 37.
    Kolchin E (1973) Differential algebra and algebraic groups. Academic Press, New YorkGoogle Scholar
  38. 38.
    Kondratieva M, Levin A, Mikhalev A, Pankratiev E (1999) Differential and difference dimension polynomials. Kluwer, Dordrecht/BostonGoogle Scholar
  39. 39.
    Landau E (1902) Ein Satz über die Zerlegung homogener linearer Differentialausdrücke in irreduzible Faktoren. J Reine Angew Math 124:115–120Google Scholar
  40. 40.
    Laplace PS (1777) Mémoires de l’Aacademie royal des sciences (See also Œuvres complètes de Laplace, vol. IX, 5–68).Google Scholar
  41. 41.
    Li Z, Schwarz F (2001) Rational solutions of riccati-like partial differential equations. J Symb Comput 31:691–716Google Scholar
  42. 42.
    Li Z, Schwarz F, Tsarev S (2002) Factoring zero-dimensional ideals of linear partial differential operators. In: Mora T (ed) Proceedings of the ISSAC’02. ACM, New York, pp 168–175,Google Scholar
  43. 43.
    Li Z, Schwarz F, Tsarev S (2003) Factoring systems of linear PDE’s with finite-dimensional solution space. J Symb Comput 36:443–471Google Scholar
  44. 44.
    Lie S (1881) Über die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichungen. Arch Math VI:328–368. (Reprinted in Gesammelte Abhandlungen III (1922) Teubner, Leipzig, pp 492–523)Google Scholar
  45. 45.
    Liouville R (1889) Mémoire sur les invariants de certaines équations différentielles et sur leurs applications, Journal de l’Ecole Polytechnique 59:7–76Google Scholar
  46. 46.
    Loewy A (1906) Über vollständig reduzible lineare homogene Differentialgleichungen. Math Ann 56:89–117Google Scholar
  47. 47.
    Magid A (1994) Lectures on differential Galois theory. AMS university lecture series, vol 7. AMS Press, ProvidenceGoogle Scholar
  48. 48.
    Matiyasevich Y (1993) Hilbert’s tenth problem. MIT, CambridgeGoogle Scholar
  49. 49.
    Miller FH (1932) Reducible and irreducible linear differential operators. PhD Thesis, Columbia UniversityGoogle Scholar
  50. 50.
    Oaku T (1997) Some algorithmic aspects of D-module theory. In: Bony JM, Moritomo M (eds) New trends in microlocal analysis. Springer, Tokyo/New YorkGoogle Scholar
  51. 51.
    Olver P (1986) Application of Lie groups to differential equations. Springer, BerlinGoogle Scholar
  52. 52.
    Ore Ö (1932) Formale theorie der linearen differentialgleichungen. J Reine Angew Math 167:221–234; 168:233–257Google Scholar
  53. 53.
    Petrén L (1911) Extension de la méthode de Laplace aux équations \(\sum\limits_{i=0}^{n=1} A_{{1}{i}}(x, y) \frac{\partial^{i+1}z}{\partial{x}\partial{y}^{i}} + \sum\limits_{i=0}^{n} A_{{0}{i}}(x, y)\frac{\partial^{i}Z}{\partial{y}^{i}} = 0.\) Lunds Universitets Arsskrift. N. F. Afd.2, Bd 7. Nr 3, LundGoogle Scholar
  54. 54.
    Plesken W, Robertz D (2005) Janet’s approach to presentations and resolutions for polynomials and linear pdes. Arch Math 84:22–37Google Scholar
  55. 55.
    Polyanin A 2002 Handbook of linear partial differential equations for engineers and scientists. Chapman and Hall/CRC, Boca RatonGoogle Scholar
  56. 56.
    Prelle MJ, Singer M (1983) Elementary integrals of differential equations. Trans Am Math Soc 279:215–229Google Scholar
  57. 57.
    Ritt JF (1950) Differential algebra. Dover, New YorkGoogle Scholar
  58. 58.
    Schlesinger L (1897) Handbuch der Theorie der linearen Differentialgleichungen. Teubner, LeipzigGoogle Scholar
  59. 59.
    Schwarz F (1989) A factorization algorithm for linear ordinary differential equations. In: Gonnet G (ed) Proceedings of the ISSAC’89. ACM, New York, pp 17–25Google Scholar
  60. 60.
    Schwarz F (1998) Janet bases for symmetry groups. Buchberger B, Winkler F (eds) In: Gröbner bases and applications. London mathematical society lecture notes series 251. Cambridge University Press, Cambridge/New York, pp 221–234Google Scholar
  61. 61.
    Schwarz F (2007) Algorithmic Lie theory for solving ordinary differential equations. Chapman and Hall/CRC, Boca Raton/LondonGoogle Scholar
  62. 62.
    Schwarz F (2008) ALLTYPES in the Web. ACM Commun Comput Algebra 42(3):185–187Google Scholar
  63. 63.
    Schwarz F (2011) Ideal intersections in rings of partial differential operators. Adv Appl Math 47:140–157Google Scholar
  64. 64.
    Shemyakova E, Winkler F (2011) Linear partial differential equations and linear partial differential operators in computer algebra In: Progress and Prospects in Numerical and Symbolic Scientific Computing, P. Paule et al. (eds), Springer, Texts & Monographs in Symbolic Computation, pp. 333–358Google Scholar
  65. 65.
    Sit W (1974) Typical differential dimension of the intersection of linear differential algebraic groups. J Algebra 32:476–487Google Scholar
  66. 66.
    Sit W (2002) The Ritt-Kolchin theory for differential polynomials. In: Guo L, et al. (eds) Differential algebra and related topics. World Scientific, Singapore/Hong KongGoogle Scholar
  67. 67.
    Stafford JT (1978) Module structure of Weyl algebras. J Lond Math Soc 18(Ser II):429–442Google Scholar
  68. 68.
    Tsarev S (1999) On Darboux integrable nonlinear partial differential equations. Proc Steklov Inst Math 225:389–399Google Scholar
  69. 69.
    Tsarev S (2000) Factoring linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations. Theor Math Phys 122:121–133Google Scholar
  70. 70.
    Tsarev S (2005) Generalized Laplace transformations and integration of hyperbolic systems of linear partial differential equations. In: Kauers M (ed) Proceedings of the ISSAC’05. ACM, New York, pp 163–170Google Scholar
  71. 71.
    van der Put M, Singer M 2003 Galois theory of linear differential equations. Grundlehren der mathematischen Wissenschaften, vol 328. Springer, New YorkGoogle Scholar
  72. 72.
    van Hoeij M (1997) Factorization of differential operators with rational function coefficients. J Symb Comput 24:537–561Google Scholar
  73. 73.
    Yiu-Kwong Man (1993) Computing closed form solutions of first order ODEs using the Prelle-Singer procedure. J Symb Comput 16:423–443Google Scholar
  74. 74.
    Yiu-Kwong Man, MacCallum AH (1997) A rational approach to the Prelle-Singer algorithm. J Symb Comput 24:31–43Google Scholar

Copyright information

© Springer-Verlag Wien 2012

Authors and Affiliations

  • Fritz Schwarz
    • 1
  1. 1.Institute SCAI Fraunhofer GesellschaftSankt AugustinGermany

Personalised recommendations