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Math. Comput. Appl. 2010, 15(4), 585-600; doi:10.3390/mca15040585

Integration of Systems of ODEs via Nonlocal Symmetry-Like Operators

1
Center For Advanced Mathematics and Physics, National University of Sciences and Technology, Rawalpindi, Pakistan
2
Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa
*
Author to whom correspondence should be addressed.
Published: 1 December 2010
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Abstract

We apply nonlocal symmetry-like operators to systems of two first and two second-order ordinary differential equations to seek reduction to quadratures. The reduction of order of such systems is carried out with the help of analytic continuation of scalar equations in the complex plane. Examples are taken from the literature. Precisely it is shown how the reduction to quadratures of a system of two second-order ordinary differential equations that admits four Lie-like operators with certain structure is obtainable from a restricted complex ordinary differential equation possessing a connected two-dimensional complex Lie algebra. A direct method of integration for a system of two first and second-order equations which possess nonlocal symmetry-like operators are given. Moreover, we present the use of nonlocal Noether-like operators to effect double reduction of order of systems of two second-order equations that arise from the corresponding scalar complex Euler- Lagrange equations which admit nonlocal Noether symmetry.
Keywords: Nonlocal operators; systems; reductions; Noether integrals Nonlocal operators; systems; reductions; Noether integrals
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Farooq, M.U.; Mahomed, F.M.; Rashid, M.A. Integration of Systems of ODEs via Nonlocal Symmetry-Like Operators. Math. Comput. Appl. 2010, 15, 585-600.

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