Abstract
In this paper, by applying Valenti’s theory for the approximate symmetry, we introduce and define the concept of a one-dimensional optimal system of approximate subalgebras for a generalized Ames’s equation; furthermore, the algebraic structure of the approximate Lie algebra is discussed. New approximately invariant solutions to the equation are found.
Highlights
IntroductionAs in the exact symmetries, even in the approximate one, an important task in determing approximately invariant solutions is to employ the concept of an optimal system of approximate subalgebras in order to obtain all the essentially different approximate invariant solutions
In [1], Ames et al performed the symmetries classification of the model utt = [ f (u)u x ] x (1)that can describes the flow of one-dimensional gas, longitudinal wave propagation on a moving threadline, dynamics of a finite nonlinear string
As in the exact symmetries, even in the approximate one, an important task in determing approximately invariant solutions is to employ the concept of an optimal system of approximate subalgebras in order to obtain all the essentially different approximate invariant solutions
Summary
As in the exact symmetries, even in the approximate one, an important task in determing approximately invariant solutions is to employ the concept of an optimal system of approximate subalgebras in order to obtain all the essentially different approximate invariant solutions. In this manuscript, in the context of Valenti’s theory [3], we define the definition of one-dimensional optimal system of approximate subalgebras for Equation (2). The plan of the manuscript is the following: in Section 2, after a brief introduction of the main concepts of Lie theory, we introduce the definition of Approximate Subalgebra and, recall the main results of the approximate symmetry analysis of Equation (2). Symmetry 2019, 11, 1230 construct new approximate non-invariant solutions for two other models linked to Equation (2) by a nonlocal transformation
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