Factoring systems of linear PDEs with finite-dimensional solution spaces

Journal of Symbolic Computation

Volumn 36, Issue 3-4, 2003, Pages 443-471

  Li, Ziming ^a   Schwarz, Fritz ^b   Tsarev, Serguei P ^c  

^a UNIVERSITY OF WATERLOO   (Canada)

^b FRAUNHOFER INSTITUTE FOR ALGORITHMS AND SCIENTIFIC COMPUTING SCAI   (Germany)

^c Astaf'ev Krasnoyarsk State Pedagogical University   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042706384     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0747-7171(03)00090-7     Document Type: Article

References (27)

1. Beke, E., 1894. Die Irreduzibilität der homogenen linearen Differentialgleichungen. Math. Ann. 45, 278-294.
2. Borel, A., Grivel, P.P., Kaup, B., Haefliger, A., Malgrange, B., Ehlers, F., 1987. Algebraic D-Modules. Academic Press, Boston.
3. Boulier, F., Lazard, D., Ollivier, F., Petitot, M., 1995. Representation for the radical of a finitely generated differential ideal. In: Levelt, A.H.M. (Ed.), ISSAC'95, Proc. Int. Symp. on Symbolic and Algebraic Computation. ACM Press, Montreal, Canada, pp. 158-166.
4. Bronstein, M., 1992. Linear ordinary differential equations: breaking through the order 2 barrier. In: Wang, P. (Ed.), ISSAC'92, Proc. Int. Symp. on Symbolic and Algebraic Computation. ACM Press, Berkeley, pp. 42-48.
5. Bronstein, M., 1994. An improved algorithm for factoring linear ordinary differential operators. In: Giesbrecht, M. (Ed.), ISSAC'94, Proc. Int. Symp. on Symbolic and Algebraic Computation. ACM Press, Oxford, pp. 336-340.
6. Bronstein, M., 1997. Symbolic Integration I: Transcendental Functions. Springer-Verlag.
7. Chyzak, F., 2000. An extension of Zeilberger's fast algorithm to general holonomic functions. Discrete Math. 217, 115-134.
8. Chyzak, F., Salvy, B., 1998. Non-commutative elimination in Ore algebras proves multivariate identities. J. Symbolic Comput. 26, 187-227.
9. Galligo, A., 1985. Some algorithmic questions on ideals of differential operators. In: Buchberger B. (Ed.) EUROCAL' 85. Springer Lecture Notes in Computer Science, vol. 204. pp. 413-421.
10. Geddes, K., Czapor, S., Labahn, G., 1992. Algorithms for Computer Algebra. Kluwer Academic Publishers.
11. Janet, M., 1920. Les systéms d'équations aux dérivées partielles. J. de mathématiques 83, 65-123.
12. Kandru-Rody, A., Weispfenning, V, 1990. Non-commutative Gröbner bases in algebras of solvable type. J. Symbolic Comput. 9, 1-26.
13. Kolchin, E., 1973. Differential Algebra and Algebraic Groups. Academic Press, New York.
14. Li, Z., Schwarz, F., 2001. Rational solutions of Riccati-like partial differential equations. J. Symbolic Comput. 31, 691-716.
15. Li, Z., Schwarz, F., Tsarev, S.P., 2002. Factoring zero-dimensional ideals of linear partial differential operators. In: Mora, T. (Ed.), ISSAC'02, Proc. Int. Symp. on Symbolic and Algebraic Computation. ACM Press, Lille, France, pp. 168-175.
16. Lie, S. Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten III. Archiv for Mathematik VIII, pp. 371-458 (Gesammelte Abhandlungen V, pp. 362-427).
17. Oaku, T., Takayama, N., Tsai, H., 2001. Polynomial and rational solutions of holonomic systems. J. Pure Appl. Algebra 164, 199-220.
18. Rosenfeld, A., 1959. Specializations in differential algebra. Trans. Amer. Math. Soc 90, 394-407.
19. Saito, M., Sturmfels, B., Takayama, N., 2000. Gröbner Deformations of Hypergeometric Differential Equations. Springer-Verlag.
20. Schlesinger, L., 1895. Handbuch der Theorie der linearen Differentialgleichungen. Teubner, Leipzig.
21. Schwarz, F., 1989. A factorization algorithm for linear ordinary differential equations. In: Gonnet, G. (Ed.), ISSAC'89, Proc. Int. Symp. on Symbolic Algebraic Comput. ACM Press, Protland, pp. 17-25.
22. Schwarz, F., 1992. Reduction and completion algorithms for partial differential equations. In: Wang, P. (Ed.), ISSAC'92, Proc. Int. Symp. on Symbolic Algebraic Comput. ACM Press, Berkeley, pp. 49-56.
23. Schwarz, F., 1998. ALLTYPES: an algebraic language and type system. In: Calmet, J., Plaza, J. (Eds.), Artificial Intelligence and Symbolic Computation. Lecture Notes in Artificial Intelligence, vol. 1476. Springer, pp. 270-283.
24. Singer, M., 1991. Liouvillian solutions of linear differential equations with Liouvillian coefficients. J. Symbolic Comput. 11, 251-273.
25. Tsarev, S., 2001. Factorization of overdetermined systems of linear partial differential equations with finite dimensional solution space. In: Ganzha, V., Mayr, E., Vorozhtsov, V. (Eds.), CASC-2001, Proc. 4th Int. Workshop Computer Algebra Scient. Comput. Springer-Verlag, pp. 529-539.
26. van der Put, M., Singer, M., 2003. Galois Theory of Linear Differential Equations. Grundleheren der mathematischen Wissenschaften, vol. 328. Springer.
27. Wu, W.-t., 1989. On the foundation of algebraic differential geometry. Syst. Sci. Math. Sci. 2, 289-312.