Conditional symmetries of nonlinear third-order ordinary differential equations

Aeeman Fatima; F. M. Mahomed; Chaudry Masood Khalique

doi:10.3934/dcdss.2018040

Article Contents

2018, Volume 11, Issue 4: 655-666. Doi: 10.3934/dcdss.2018040

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Conditional symmetries of nonlinear third-order ordinary differential equations

a.
International Institute for Symmetry Analysis and Mathematical Modeling, North-West University, Mafikeng Campus, P Bag X2046, Mafikeng, South Africa
b.
School of Computer Science and Applied Mathematics DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

* Corresponding author: Aeeman Fatima.
* Corresponding author: Aeeman Fatima.

Received: December 2016

Revised: April 2017

Published: August 2018

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Abstract

In this work, we take as our base scalar second-order ordinary differential equations (ODEs) which have seven equivalence classes with each class possessing three Lie point symmetries. We show how one can calculate the conditional symmetries of third-order non-linear ODEs subject to root second-order nonlinear ODEs which admit three point symmetries. Moreover, we show when scalar second-order ODEs taken as first integrals or conditional first integrals are inherited as Lie point symmetries and as conditional symmetries of the derived third-order ODE. Furthermore, the derived scalar nonlinear third-order ODEs without substitution are considered for their conditional symmetries subject to root second-order ODEs having three symmetries.

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Mathematics Subject Classification: 34A05, 22E05, 22E45, 22E60, 22E70.

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Table Ⅰ. Lie group classification of scalar second-order equations in the real plane

$p=\partial /\partial x$ and $q=\partial /\partial y$
Algebra	Canonical forms of generators	Representative equations
$L_{1}$	$X_1=p$	$y''=g(y,y')$
$L_{2;1}^I$	$X_1=p,X_2=q$	$y''=g(y')$
$L_{2;1}^{II}$	$X_1=q,X_2=xp+yq$	$xy''=g(y')$
$L_{3;3}^I$	$X_1=p, X_2=q, X_3=xp+(x+y)q$	$y''=Ae^{-y'}$
$L_{3;6}^I$	$X_1=p, X_2=q, X_3=xp+ayq$	$y''=Ay'^{(a-2)/(a-1)}$
$L_{3;7}^I$	$X_1=p, X_2=q, X_3=(bx+y)p+(by-x)q$	$y''=A(1+y'^{2})^{\frac{3}{2}}e^{b\arctan y'}$
$L_{3;8}^I$	$X_1=q, X_2=xp+yq, X_3=2xyp+y^2q$	$xy''=Ay'^{3}-\frac{1}{2}y'$
$L_{3;8}^{II}$	$X_1=q, X_2=xp+yq, X_3=2xyp+(y^2+x^2)q$	$xy''=y'+y'^{3}+A(1+y'^{2})^{\frac{3}{2}}$
$L_{3;8}^{III}$	$X_1=q, X_2=xp+yq, X_3=2xyp+(y^2-x^2)q$	$xy''=y'-y'^{3}+A(1-y'^{2})^{\frac{3}{2}}$
$L_{3;9}$	$X_1=(1+x^2)p+xyq, X_2=xyp+(1+y^2)q$,
	$X_3=yp-xq$	$y''=A[\displaystyle{1+y'+(y-xy')^2\over 1+x^2+y^2}]^{3/2}$

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Table Ⅱ. Inherited symmetries of derived scalar third-order equations

Representative 2nd-order ODE $A\ne0$ Derived 3rd-order ODE Inherited algebra
$y''=Ae^{-y'}$	$y'''+y''^{2}=0$	$L_{3;3}^{I}$
$y''=Ay'^{(a-2)/(a-1)}$	$y'y'''-\frac{a-2}{a-1}y''^2=0$	$L_{3;6}^I$
$y''=A(1+y'^{2})^{\frac{3}{2}}e^{b\arctan y'}$	$y'''-\frac{3y'+b}{1+y'^2}y''^2=0$	$L_{3;7}^I$
$xy''=Ay'^{3}-\frac{1}{2}y'$	$y'y'''-3y''^2=0$	$L_{3;8}^I$
$xy''=y'+y'^{3}+A(1+y'^{2})^{\frac{3}{2}}$	$y'''+y'''y'^2-3y'y''^2=0$	$L_{3;8}^{II}$
$xy''=y'-y'^{3}+A(1-y'^{2})^{\frac{3}{2}}$	$y'''-y'''y'^2-3y'y''^2=0$	$L_{3;8}^{III}$
$y''=AK^{3/2}$	$y'''=\frac32y''K^{-1}D_x K$	$L_{3;9}$
where $K=\displaystyle{1+y'+(y-xy')^2\over 1+x^2+y^2}$

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