Abstract
All generators of the optimal algebra associated with a generalization of the Endem-Fowler equation are showed; some of them allow to give invariant solutions. Variational symmetries and the respective conservation laws are also showed. Finally, a representation of Lie symmetry algebra is showed by groups of matrices.
Highlights
It is known that the class of Emden-Fowler equations yxx = Axnym, such that A, m, n are real constants, have applications in physics, astronomy and chemistry [1],[2],[3],[4]
The Lie symmetry group associated to this equation is presented by Arrigo in [6], the computations used to obtain this result are not given in detail
The Lie symmetry group associated to (1) is an 8-dimensional Lie group presented by Arrigo in [6], the computations used to obtain this result are not given in detail
Summary
It is known that the class of Emden-Fowler equations yxx = Axnym, such that A, m, n are real constants, have applications in physics, astronomy and chemistry [1],[2],[3],[4]. In [5] Polyanin and Zaitsev present a generalized Emden-Fowler equation yxx = Axnym(yx)l with A, m, n, l real constants. They proposed for this equation a big amount of solutions for multiple combinations of the parameters A, n, m, l. Since the symmetry group of (1) is an 8-dimensional group and following the ideas of citas [7],[8],[9], we search for its algebraic characteristics and some invariant solutions of (1). The goal of this work is: i) to calculate the 8-dimensional Lie symmetry group in all detail, ii) to present the optimal algebra (optimal system) for (1), iii) to use some elements of the optimal algebra to propose invariant solutions for (1), iv) to construct the Lagrangian with which we could determine the variational symmetries and to present conservation laws associated, and iv) to classify the Lie algebra associated to (1) by groups of matrices
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