×

zbMATH — the first resource for mathematics

A Maple package to find first order differential invariants of 2ODEs via a Darboux approach. (English) Zbl 1344.34002
Summary: Here we present an implementation of a semi-algorithm to find elementary first order differential invariants (elementary first integrals) of a class of rational second order ordinary differential equations (rational 2ODEs). The algorithm was developed in [L. G. S. Duarte and L. A. C. P. da Mota, J. Math. Phys. 50, No. 1, 013514, 17 p. (2009; Zbl 1200.34039)]; it is based on a Darboux-type procedure, and it is an attempt to construct an analog (generalization) of the method built by M. J. Prelle and M. F. Singer [Trans. Am. Math. Soc. 279, 215–229 (1983; Zbl 0527.12016)] for rational first order ordinary differential equations (rational 1ODEs). to deal, this time, with 2ODEs. The FiOrDi package presents a set of software routines in Maple for dealing with rational 2ODEs. The package presents commands permitting research investigations of some algebraic properties of the ODE that is being studied.

MSC:
34-04 Software, source code, etc. for problems pertaining to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
Software:
FiOrDi; FiOrDii; Maple
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Stephani, H., (MacCallum, M. A.H., Differential Equations: Their Solution Using Symmetries, (1989), Cambridge University Press New York, London) · Zbl 0704.34001
[2] Bluman, G. W.; Kumei, S., (Symmetries and Differential Equations, Applied Mathematical Sciences, vol. 81, (1989), Springer-Verlag) · Zbl 0698.35001
[3] Olver, P. J., Applications of Lie groups to differential equations, (1986), Springer-Verlag · Zbl 0588.22001
[4] Cheb-Terrab, E. S.; Duarte, L. G.S.; da Mota, L. A.C. P., Comput. Phys. Comm., 101, 254, (1997)
[5] Cheb-Terrab, E. S.; Duarte, L. G.S.; da Mota, L. A.C. P., Comput. Phys. Comm., 108, 90, (1998)
[6] Prelle, M.; Singer, M., Trans. Amer. Math. Soc., 279, 215, (1983)
[7] Shtokhamer, R., Solving first order differential equations using the Prelle-Singer algorithm, technical report 88-09, (1988), Center for Mathematical Computation, University of Delaware
[8] C.B. Collins, Algebraic invariants curves of polynomial vector fields in the plane, University of Waterloo, Canada, 1993. Preprint.; C.B. Collins, Quadratic vector fields possessing a centre, University of Waterloo, Canada, 1993. Preprint.
[9] Christopher, C., Electron. J. Differential Equations, 49, 7, (1999), (electronic)
[10] Christopher, C.; Llibre, J., Ann. Differential Equations, 16, 1, 5, (2000)
[11] Llibre, J., (Cañada, A.; Drábek, P.; Fonda, A., Handbook of Differential Equations, Ordinary Differential Equations, Vol. 1, (2004), Elsevier BV), 437, (Chapter 5)
[12] Duarte, L. G.S.; Duarte, S. E.S.; da Mota, L. A.C. P., J. Phys. A: Math. Gen., 35, 3899, (2002) · Zbl 1040.34006
[13] Duarte, L. G.S.; Duarte, S. E.S.; da Mota, L. A.C. P., J. Phys. A: Math. Gen., 35, 1001, (2002)
[14] Duarte, L. G.S.; Duarte, S. E.S.; da Mota, L. A.C. P.; Skea, J. F.E., Comput. Phys. Commun., 144, 1, 46, (2002), Holanda
[15] Avellar, J.; Duarte, L. G.S.; Duarte, S. E.S.; da Mota, L. A.C. P., J. Comput. Appl. Math., 182, 327, (2005)
[16] Duarte, L. G.S.; Duarte, S. E.S.; da Mota, L. A.C. P.; Skea, J. F.E., J. Phys. A: Math. Gen., 34, 3015, (2001)
[17] Davenport, J. H.; Siret, Y.; Tournier, E., Computer algebra: systems and algorithms for algebraic computation, (1993), Academic Press Great Britain · Zbl 0865.68064
[18] Duarte, L. G.S.; da Mota, L. A.C. P., J. Math. Phys., 50, 1, 013514, (2009)
[19] Avellar, J.; Duarte, L. G.S.; Duarte, S. E.S.; da Mota, L. A.C. P., (Simos, Theodore; Maroulis, George, Lecture Series on Computer and Computational Sciences, vol. 4B, (2005)), 1786
[20] Chandrasekar, V. K.; Senthilvelan, M.; Lakshmanan, M.; Chandrasekar, V. K.; Pandey, S. N.; Senthilvelan, M.; Lakshmanan, M.; Chandrasekar, V. K.; Senthilvelan, M.; Lakshmanan, M., J. Nonlinear Math. Phys., Chaos Solitons Fractals, Proc. R. Soc. A, 462, 2070, 1831, (2006) · Zbl 1111.34003
[21] Avellar, J.; Duarte, L. G.S.; Duarte, S. E.S.; da Mota, L. A.C. P., Appl. Math. Comput., 184, 2, (2007)
[22] A. Ghose Choudhury, Partha Guha, Barun Khanra, Solutions of some second order ODEs by the extended Prelle-Singer method and symmetries, 15-4, 2008, p. 365. · Zbl 1188.37058
[23] Duarte, L. G.S.; da Mota, L. A.C. P., J. Phys. A, 43, 6, (2010), Article number 065204
[24] Kamke, E., Differentialgleichungen: Lösungsmethoden und Lösungen, (1959), Chelsea Publishing Co. New York · JFM 68.0179.01
[26] Man, Y. K., J. Symbolic Comput., 16, 423, (1993)
[27] Avellar, J.; Duarte, L. G.S.; Duarte, S. E.S.; da Mota, L. A.C. P., Comput. Phys. Comm., 177, 584, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.