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Towards a new ode solver based on cartan's equivalence method

Published: 29 July 2007 Publication History

Abstract

The aim of the present paper is to propose an algorithm for a new ODE-solver which should improve the abilities of current solvers to handle second order differential equations. The paper provides also a theoretical result revealing the relationship between the change of coordinates, that maps the generic equation to a given target equation, and the symmetry D-groupoid of this target.

References

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F. Boulier. Réécriture algébrique dans les systèmes d'équations différentielles en vue d 'applications dans les Sciences du Vivant. Habilitation, Univ. Lille I, 2006, URL: http://www2.lifl.fr/~boulier/.
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F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Representation for the radical of a ?nitely generated differential ideal. In proc. ISSAC '95, pages 158--166, Montréal, Canada, 1995.
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F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Representation for the radical of a ?nitely generated differential ideal. In proc. ISSAC '95, pages 158--166, Montréal, Canada, 1995.
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E. S. Cheb-Terrab, L. G. S. Duarte, and L. A. C. P. da Mota. Computer algebra solving of second order ODEs using symmetry methods. Comput. Phys. Comm., 108(1):90--114, 1998.
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R. Dridi. Utilisation de la mèthode d'équivalence deCartan dans la construction d'un solveur d'équations différentielles. PhD thesis in preparation, Univ. Lille I.
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L. Hsu and N. Kamran. Classification of second-order ordinary differential equations admitting Lie groups of fiber-preserving symmetries. Proc. London Math. Soc., 58:387--416, 1989.
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E. Hubert. Factorization free decomposition algorithms in differential algebra. Journal of Symbolic Computations, 29(4-5), 2000.
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E. Kolchin. Differential algebra and algebraic groups. Academic press, New-York and London, 1973.
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B. Malgrange. Le groupoïde de Galois d'unfeuilletage. In Monographies del'Enseignement mathématique, volume 38, pages 465--501. 1902.
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S. Neut. Implantation et nouvelles applications de la méthode d'équivalence de Cartan. PhD thesis, Univ. Lille I, 2003, URL: http://www2.lifl.fr/~neut/.
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P. Olver and Pohjanpelto. Differential invariants for lie pseudo-groups, preprint. 2006.
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P. J. Olver. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics. Springer-Verlag, 1993.
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P. J. Olver. Equivalence, invariants, and symmetry. Cambridge University Press, Cambridge, 1995.
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G. J. Reid, D. T. Weih, and A. D. Wittkopf. A point symmetry group of a differential equation which cannot be found using in ?nitesimal methods. In Modern group analysis: advanced analytical and computational methods in mathematical physics (Acireale, 1992), pages 311--316. Kluwer Acad. Publ., Dordrecht, 1993.
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Cited By

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  • (2009)On the geometry of the first and second Painlevé equationsJournal of Physics A: Mathematical and Theoretical10.1088/1751-8113/42/12/12520142:12(125201)Online publication date: 24-Feb-2009

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cover image ACM Conferences
ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation
July 2007
406 pages
ISBN:9781595937438
DOI:10.1145/1277548
  • General Chair:
  • Dongming Wang
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Association for Computing Machinery

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Publication History

Published: 29 July 2007

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Author Tags

  1. cartan's equivalence method
  2. differential algebra
  3. equivalence problems
  4. ode-solver

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ISSAC07
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ISSAC07: International Symposium on Symbolic and Algebraic Computation
July 29 - August 1, 2007
Ontario, Waterloo, Canada

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Overall Acceptance Rate 395 of 838 submissions, 47%

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  • (2009)On the geometry of the first and second Painlevé equationsJournal of Physics A: Mathematical and Theoretical10.1088/1751-8113/42/12/12520142:12(125201)Online publication date: 24-Feb-2009

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