Loewy’s Results for Ordinary Differential Equations

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

Abstract

The idea of factoring an ordinary differential operator, or the corresponding linear ordinary differential equation (ode), into components of lower order originated from the analogous problem for algebraic polynomials. In the latter case, it is the first step when the solutions of the corresponding algebraic equation are to be determined. It turns out that a similar strategy is the key for understanding the structure of the solution space of a linear ode. Good introductions into the subject are the books by Ince [29] or Kamke [32, 34], and the book by van der Put and Singer [71].

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

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