Lie and Noether symmetries for a class of fourth-order Emden–Fowler equations

Abstract

A group classification of a fourth-order ordinary differential equation is carried out. The Noether symmetries are considered and some first integrals are established. Solutions for special Lane-Emden systems are also obtained from the invariant solutions of the investigated equation.

Symmetry analysis of a class of autonomous even-order ordinary differential equations

“…We, however, will not proceed with further information on these transformations in this paper once we are interested in understanding the onedimensional version of the results obtained in [Svirshchevskii (1993), Bozhkov(2006)]. Moreover, it is well known that for semilinear equations such as (1), in order to carry out a complete group classification, up to equivalence transformations, one must take the functions f as considered in the Theorem 1, see [Bokhari et al(2010), Bozhkov(2006), Bozhkov and Freire (2008a), Bozhkov and Freire (2010), Freire et al (2013), Svirshchevskii (1993)]. They, in fact, arise from a compatibility condition that will be deduced on section 4.…”

“…In a previous paper ([ Freire et al (2013)], see also [Freire et al (2012)]) we, jointly with M. Torrisi, considered the fourth-order equation y ′′′′ + ax γ y p = 0.…”

An operational approach to the Emden–Fowler equation

“…These equations appear in several problems in mathematics, physics, and engineering. See [3] and the references therein. Suppose that˛2 N and r ¤ 0, 1.…”

“…where ,˛, and are real numbers and > 0. Equation (1) appears in several problems in mathematics and physics, and many different methods for its solution have been proposed; see, for example, [3][4][5][6][7][8][9]. For an extensive survey on the Emden-Fowler equation, see [10] and the references therein.…”

Symmetries and conserved quantities of constrained mechanical systems

“…The physical meaning of the conservation law in Lagrangian mechanics is less clear than that in Newtonian mechanics, but the conserved quantities deduced by Lagrangian mechanics are more than that deduced by Newtonian mechanics. Since Noether published her well-known paper [1], the Noether symmetry method has become a modern method for seeking the conservation law of mechanical systems [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The physical meaning of the conservation law in the Noether symmetry method is less than that in Lagrangian mechanics, but the conserved quantities deduced by the Noether symmetry method are more than that deduced by Lagrangian mechanics.…”