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Conditional symmetries for ordinary differential equations and applications

机译：常微分方程及其应用的条件对称性

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摘要

We refine the definition of conditional symmetries of ordinary differential equations and provide an algorithm to compute such symmetries. A proposition is proved which provides criteria as to when the symmetries of the root system of ODEs are inherited by the derived higher-order system. We provide examples and then investigate the conditional symmetry properties of linear nth-order equations subject to root linear second-order equations. First this is considered for simple linear equations and then for arbitrary linear systems. We prove criteria when the symmetries of the root linear ODEs are inherited by the derived scalar linear ODEs and even order linear system of ODEs. Furthermore, we show that if a system of ODEs has exact solutions, then it admits a conditional symmetry subject to the first-order ODEs related to the invariant curve conditions which arises from the known solution curves. Moreover, we give examples of the conditional symmetries of non-linear third-order equations which are linearizable by second-order Lie linearizable equations. Applications to classical and fluid mechanics are presented.
机译：我们完善了常微分方程的条件对称性的定义，并提供了一种计算此类对称性的算法。证明了一个命题，该命题为导出的高阶系统何时继承ODE根系统的对称性提供了标准。我们提供示例，然后研究服从根线性二阶方程的线性n阶方程的条件对称性质。首先，对于简单的线性方程式，然后再对任意线性系统考虑。当根线性ODE的对称性被导出的标量线性ODE甚至ODE的偶数线性系统继承时，我们证明了准则。此外，我们表明，如果一个ODE系统具有精确解，则它接受条件对称性，该条件对称性服从与由已知解曲线产生的不变曲线条件相关的一阶ODE。此外，我们给出了非线性二阶方程的条件对称性的示例，这些条件可以通过二阶李线性化方程线性化。介绍了在经典和流体力学中的应用。

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  • 来源
    《International journal of non-linear mechanics》 |2014年第12期|95-105|共11页
  • 作者

    A Fatima; F.M. Mahomed;

  • 作者单位

    School of Computational and Applied Mathematics, DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa;

    School of Computational and Applied Mathematics, DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa,School of Mathematics and Statistics, University of New South Wales, Sydney NSW, 2052 Australia;

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  • 正文语种 eng
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  • 关键词

    Lie point symmetries; Conditional symmetries; Conditional first integrals; conditional linearizability;

    机译：谎言点对称性;条件对称;有条件的第一积分;条件线性化;

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