International Journal of Nonlinear Analysis and Applications

Infinitesimal generators of Lie symmetry group of parametric ordinary differential equations

Document Type : Research Paper

Authors

Department of Mathematics and Computer Sciences, Damghan University, Damghan, Iran

Abstract

Lie’s theory of symmetry groups plays an important role in analyzing and solving differential equations; for instance, by decreasing the order of equation. Moreover, there are some analytic methods to find the infinitesimal generators that span the Lie algebra of symmetries. In this paper, we first converted the problem of finding infinitesimal generators in to the problem of solving a system of polynomial equations in the context of computational algebraic geometry. Then, we used Gröbner basis a novel computational tool to solve this problem. As far as we know, when a differential equation contains some parameters, there is no linear algebraic algorithm up to our knowledge to deal with these parameters; so, we must apply the algorithms, which are based on Gröbner basis.

Keywords

[1] A. Abbasi Molai, A. Basiri and S. Rahmany, Resolution of a system of fuzzy polynomial equations using the Grobner basis, Inf. Sci. 220 (2013) 541–558.
[2] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, 1989.
[3] B. Buchberger, Ein algorithmus zum auffinden der basiselemente des restklassenringes nach einem nulldimensionalen polynomideal, PhD Thesis, Innsbruck, 1965.
[4] 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, Comp. Phy. Commun. (1997) 1–25.
[5] M. Cipu, Grobner bases and solutions to diophantine equations, Proc. 10th Int. Symp. Symb. Numeric Algorithms Sci. Comput., 2008, pp. 77–80.
[6] P. A. Clarkson and P. J. Olver, Symmetries and the Chazy equation, J. Diff. Eqns. 124 (1996) 225–246.
[7] D.A. Cox, J.B. Little and D. O’Shea. Ideals, Varieties, and Algorithms, Springer-Verlag, 3 edition, 2007.
[8] J.A. de Loera, Grobner bases and graph colorings, Cont. Alg. Geom. 36(1) (1995) 89–96.
[9] A. Hashemi and B.M.-Alizadeh. Computing minimal polynomial of matrices over algebraic extension fields, Bull. Math. Soc. Sci. Math. Roumanie Tome, 56(104) (2013) 217–228.
[10] P.E. Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, 2000.
[11] D. Kapur, Y. Sun, D.K. Wang, A new algorithm for computing comprehensive Grobner systems, Proc. ISSAC’2010. ACM Press, New York, 2010, pp. 29–36.
[12] D. Kapur, Y. Sun and D. Wang, An efficient algorithm for computing a comprehensive Grobner system of a parametric polynomial system, J. Symbolic Comput. 49 (2013) 27–44.
[13] S. Lie, On the integration of a class of linear partial differential equations by means of definite integrals, translation by N.H. Ibragimov, Arch. Math. 6 (1881) 328–368.
[14] E. L. Mansfield, and P. A. Clarkson, Applications of the differential algebra package diffgrob2 to classical symmetries of differential equations, J. Symb. Comput. 23 (1997) 517–533.
[15] P. J. Olver, Applications of Lie Groups to Differential Equations, (2nd ed), Springer-Verlag, New York, 1993.
[16] L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.
[17] M. Sala, T. Mora, L. Perret, S. Sakata and C. Traverso, Grobner Bases, Coding, and Cryptography, Springer, Berlin, 2009.
[18] H. Stephani, Differential Equations: Their solution Using Symmetries, Cambridge University Press, Cambridge, 1989.
[19] V. Weispfenning, Comprehensive Grobner bases, J. Symbolic Comput. 14 (1992) 1–29.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 877-891
  • Receive Date: 15 October 2018
  • Revise Date: 01 June 2019
  • Accept Date: 17 January 2020