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SIGMA 1 (2005), 024, 12 pages nlin.SI/0512012
https://doi.org/10.3842/SIGMA.2005.024
Symmetry Properties of Autonomous Integrating Factors
Sibusiso Moyo a and P.G.L. Leach b
a) Department
of Mathematics, Durban Institute of Technology, PO Box 953, Steve Biko Campus, Durban 4000, Republic of South Africa
b) School of Mathematical Sciences, Howard College,
University of KwaZulu-Natal, Durban 4041, Republic of South Africa
Received September 27, 2005, in final form November 21,
2005; Published online December 05, 2005
Abstract
We study the symmetry properties of
autonomous integrating factors from an algebraic point of view.
The symmetries are delineated for the resulting integrals treated
as equations and symmetries of the integrals treated as functions
or configurational invariants. The succession of terms (pattern)
is noted. The general pattern for the solution symmetries for
equations in the simplest form of maximal order is given and the
properties of the associated integrals resulting from this
analysis are given.
Key words:
autonomous integrating factors; maximal symmetry.
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references
- Bluman G.W., Kumei S., Symmetries and differential equations,
New York, Springer-Verlag, 1989.
- Cheb-Terrab E.S., Roche A.D., Integrating factors for second
order ordinary differential equations, J. Symbolic Comput., 1999, V.27, 501-519,
math-ph/0002025.
- Leach P.G.L., Bouquet S.É., Symmetries and integrating factors,
J. Nonlinear Math. Phys., 2002, V.9, suppl. 2, 73-91.
- Abraham-Shrauner B., Hidden symmetries, first integrals and
reduction of order of nonlinear ordinary differential equations,
J. Nonlinear Math. Phys., 2002, V.9, suppl. 2, 1-9.
- Head A., LIE, a PC program for Lie analysis of differential equations,
Comput. Phys. Comm., 1993, V.77, 241-248.
- Ermakov V., Second-order differential equations. Conditions of
complete integrability, Universitetskie
Izvestiya, Kiev, 1880,
N 9, 1-25 (translated by A.O. Harin).
- Pinney E., The nonlinear differential equation y"(x) + p(x) y + cy-3 = 0, Proc. Amer. Math. Soc., 1950, V.1, 681.
- Lewis H.R., Classical and quantum systems with time-dependent
harmonic oscillator-type Hamiltonians, Phys. Rev. Lett., 1967, V.18, 510-512.
- Lewis H.R.Jr., Motion of a time-dependent harmonic oscillator and
of a charged particle in a time-dependent, axially symmetric, electromagnetic field,
Phys. Rev., 1968, V.172, 1313-1315.
- Mahomed F.M., Leach P.G.L., Symmetry Lie algebras of nth order
ordinary differential equations, J. Math. Anal. Appl., 1990, V.151, 80-107.
- Mubarakzyanov G.M., On solvable Lie algebras, Izv. Vys. Uchebn. Zaved. Matematika, 1963, N 1 (32), 114-123.
- Mubarakzyanov G.M., Classification of real structures of
five-dimensional Lie algebras, Izv. Vys. Uchebn. Zaved. Matematika, 1963, N 3 (34), 99-106.
- Mubarakzyanov G.M., Classification of solvable Lie
algebras of sixth order with a non-nilpotent basis element,
Izv. Vys. Uchebn. Zaved. Matematika, 1963, N 4 (35),
104-116.
- Patera, J., Sharp R.T., Winternitz P., Invariants of real low dimension Lie algebras,
J. Math. Phys., 1976, V.17, 986-994.
- Flessas G.P., Govinder K.S., Leach P.G.L., Remarks on the
symmetry Lie algebras of first integrals of scalar third order
ordinary differential equations with maximal symmetry,
Bull. Greek Math. Soc., 1994, V.36, 63-79.
- Flessas G.P., Govinder K.S., Leach P.G.L., Characterisation of
the algebraic properties of first integrals of scalar ordinary
differential equations of maximal symmetry, J. Math. Anal. Appl. 1997, V.212,
349-374.
- Moyo S., Leach P.G.L., Exceptional properties of second and third
order ordinary differential equations of maximal symmetry, J. Math. Anal. Appl., 2000, V.252, 840-863.
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