Abstract
This paper explains the developments on factorization and decomposition of linear differential equations in the last two decades. The results are applied for developing solution procedures for these differential equations. Although the subject is more than 100 years old, it has been rediscovered as recently as about 20 years ago. A fundamental ingredient has been the easy availability of symbolic computation systems to accomplish the extensive calculations usually involved in applications; to this end the interactive website http://www.alltypes.de has been provided. Although originally only developed for ordinary equations, it has been extended to large classes of partial equations as well. In the first part Loewy’s results for ordinary equations are outlined. Thereafter those results of differential algebra are summarized that are required for extending Loewy’s theory to partial equations. In the remaining part a fairly complete discussion of second- and some third-order partial differential equations in the plane is given; it is shown that Loewy’s result remains essentially true for these equations. Finally, several open problems and possible extensions are discussed.
Article PDF
Similar content being viewed by others
References
Adams, W.W., Loustaunau, P.: An Introduction to Gröbner Bases. American Mathematical Society, Providence (1994)
Anderson I.M., Fels M.E., Vassiliou P.J.: Superposition formulas for exterior differential systems. Adv. Math. 221, 1910–1963 (2009)
Beke E.: Die Irreduzibilität der homogenen Differentialgleichungen. Math. Ann. 45, 278–294 (1894)
Blumberg, H.: Über algebraische Eigenschaften von linearen homogenen Differential- ausdrücken. Inaugural-Dissertation, Göttingen (1912)
Bronstein, M.: An improved algorithm for factoring linear ordinary differential operators. In: Proceedings of the ISSAC’94, pp. 336–340. ACM Press (1994)
Bronstein, M., Lafaille, S.: Solutions of linear ordinary differential equations in terms of special functions. In: Mora, T. (ed.) Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, pp. 23–28. ACM, New York (2002)
Buchberger B.: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequ. Math. 4, 374–383 (1970)
Buium, A., Cassidy, Ph.: 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 Press (1999)
Cartan E.: Les systemes deifferentiell exterieur et leurs applications geometriques. Hermann, Paris (1971)
Castro-Jiménez, F. J., Moreno-Frías, M.A.: An introduction to Janet bases and Gröbner bases. In: Lecture Notes in Pure and Applied Mathematics, vol. 221, pp. 133–145, Marcel Dekker, New York (2001)
Cox D., Little J., Shea D.O.: Ideals, Varieties and Algorithms. Springer, Berlin (1991)
Cox D., Little J., Shea D.O.: Using Algebraic Geometry. Springer, Berlin (1998)
Davey B.A., Priestley H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)
Darboux E.: Leçons sur la théorie générale des surfaces, vol. II. Chelsea Publishing Company, New York (1972)
Forsyth A.R.: Theory of Differential Equations, vol. I,…,VI. Cambridge University Press, Cambridge (1906)
Goursat E.: Leçon sur l’intégration des équation aux dérivées partielles, vols. I and II. A. Hermann, Paris (1898)
Gratzer, G. (1998) General Lattice Theory. Birkhäuser, Basel
Grigoriev D.: Complexity of factoring and calculating the GCD of linear ordinary differential operators. J. Symb. Comput. 7, 7–37 (1990)
Grigoriev D., Schwarz F.: Factoring and solving linear partial differential equations. Computing 73, 179–197 (2004)
Grigoriev, D., Schwarz, F.: Generalized Loewy decomposition of D-modules. In: Kauers, M. (ed.) Proceedings of the ISSAC’05, pp. 163–170, ACM Press (2005)
Grigoriev, D., Schwarz, F.: Loewy decomposition of third-order linear PDE’s in the plane. In: Gonzales-Vega, L. (ed.) Proceedings of the ISSAC 2008, Linz, ACM Press, pp. 277–286 (2008)
van Hoeij M.: Factorization of differential operators with rational function coefficients. J. Symb. Comput. 24, 537–561 (1997)
Imschenetzky, V.G.: Etude sur les methodes d’integration des équations aux dérivées partielles du second ordre d’une fonction de deux variables indépendantes, Grunert’s Archiv LIV, pp. 209–360 (1872)
Ince, E.L.: Ordinary Differential Equations. Longmans, Green and Co., London (1926) [Reprint by Dover Publications Inc., 1960]
Janet M.: Les systemes d’équations aux dérivées partielles. Journal de mathématiques 83, 65–123 (1920)
Jurás M.: Generalized Laplace invariants and the method of darboux. Duke Math. J. 89, 351–375 (1997)
Kamke, E.: Differentialgleichungen I. Gewöhnliche Differentialgleichungen, Akademische Verlagsgesellschaft, Leipzig (1964)
Kamke, E.: Differentialgleichungen, Lösungsmethoden und Lösungen, II. Partielle Differentialgleichungen. Akademische Verlagsgesellschaft, Leipzig (1965)
Kaplansky I.: An Introduction to Differential Algebra. Hermann, Paris (1957)
Kolchin E.: Notion of dimension in the theory of algebraic differential equations. Bull. AMS 70, 570–573 (1964)
Kolchin E.: Differential Algebra and Algebraic Groups. Academic Press, Dublin (1973)
Kondratieva, M., Levin, A., Mikhalev, A., Pankratiev, E.: Differential and difference dimension polynomial, Kluwer, Dordrecht (1999)
Landau E.: Ein Satz uber die Zerlegung homogener linearer Differentialausdrücke in irreduzible Faktoren. Journal für die reine und angewandte Mathematik 124, 115–120 (1902)
Laplace, P.S.: Méoires de l’Aacademie royal des sciences (1777) [see also (Euvres complètes de Laplace, vol. IX, pp. 5–68]
Lie, S.: Uber die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichungen. Arch. Math. VI, pp. 328–368 (1881) [Reprinted in Gesammelte Abhandlungen III, Teubner, pp. 492–523; Leipzig, 1922]
Li Z., Schwarz Z.F.: Rational solutions of Riccati like systems of partial differential equations. J. Symb. Comput. 31, 691–716 (2001)
Li, Z., Schwarz, Z.F., Tsarev, S. Factoring zero-dimensional ideals of linear partial differential operators. In: Mora, T. (ed.) Proceedings of the ISSAC’02, pp. 168–175. ACM Press (2002)
Li Z., Schwarz F., Tsarev S.: Factoring systems of linear PDE’s with finite-dimensional solution space. J. Symb. Comput. 36, 443–471 (2003)
Loewy A.: Uber vollständig reduzible lineare homogene Differentialgleichungen. Mathematische Annalen 56, 89–117 (1906)
Magid, A.: Lectures on Differential Galois Theory. AMS University Lecture Series 7 (1994)
Miller, F.H.: Reducible and irreducible linear differential operators. PhD Thesis, Columbia University (1932)
Oaku, T.: Some algorithmic aspects of D-module theory. In: Bony, J.M. Moritomo, M. (eds.) New Trends in Microlocal Analysis, Springer, Berlin (1997)
Olver P.: Application of Lie Groups to Differential Equations. Springer, Berlin (1986)
Ore, O.: Formale Theorie der linearen Differentialgleichungen. Journal für die reine und angewandte Mathematik 167:221–234 and 168:233–257 (1932)
Plesken W., Robertz D.D.: Janet’s approach to presentations and resolutions for polynomials and linear pdes. Arch. Math. 84, 22–37 (2005)
Polyanin, A.: Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, London (2002)
Renschuch, B., Roloff, H., Rasputin, G.G.: Vergessene Arbeit des Leningrader Mathematikers N.M. Gjunter zur Theorie der Polynomideale, Wiss. Zeit. Pädagogische Hochschule Potsdam 31, 111–126 (1987) [English translation in ACM SIGSAM Bulletin 37, 35–48 (2003)]
Schlesinger, L.: Handbuch der Theorie der linearen Differentialgleichungen, vols. I and II, Teubner, Leipzig (1897)
Schwarz, F.A.: Factorization Algorithm for Linear Ordinary Differential Equations. In: Gaston Gonnet (ed.) Proceedings of the ISSAC’89, pp. 17–25, ACM Press (1989)
Schwarz, F.: Janet bases for symmetry groups. In: Buchberger, B., Winkler, F. (eds.) Gröbner bases and applications. Lecture notes series, vol. 251. London Mathematical Society, pp. 221–234 (1998)
Schwarz, F.: Algorithmic Lie Theory for Solving Ordinary Differential Equations. Chapman & Hall/CRC, Boca Raton (2007)
Schwarz F.: ALLTYPES in the Web. ACM Commun. Comput. Algebra. 42(3), 185–187 (2008)
Schwarz, F.: Loewy decomposition of linear differential equations. Springer, Texts and Monographs in Symbolic Computation (in press)
Seiler, W.: Involution The Formal Theory of Differential Equations and its Applications in Computer Algebra. Springer, Berlin (2010)
Sit W.: Typical differential dimension of the intersection of linear differential algebraic groups. J. Algebra 32, 476–487 (1974)
Sit, W.: The Ritt-Kolchin theory for differential polynomials. In: Li Guo et al. (eds.) Differential Algebra and Related Topics. World Scientific (2002)
Tsarev S.: Factoring linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations. Theor. Math. Phys. 122, 121–133 (2000)
van der Put, M., Singer, M.: Galois theory of linear differential equation. In: Grundlehren der Math. Wiss., vol. 328, Springer, Berlin (2003)
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N. Trudinger.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Schwarz, F. Loewy decomposition of linear differential equations. Bull. Math. Sci. 3, 19–71 (2013). https://doi.org/10.1007/s13373-012-0026-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13373-012-0026-7