Abstract
This paper studies a non-linear viscoelastic wave equation, with non-linear damping and source terms, from the point of view of the Lie groups theory. Firstly, we apply Lie’s symmetries method to the partial differential equation to classify the Lie point symmetries. Afterwards, we reduce the partial differential equation to some ordinary differential equations, by using the symmetries. Therefore, new analytical solutions are found from the ordinary differential equations. Finally, we derive low-order conservation laws, depending on the form of the damping and source terms, and discuss their physical meaning.
Highlights
Many viscoelastic wave equations have been considered in the literature
In this paper we present new results for the model. It is found a complete classification of Lie point symmetries with its associated reductions, new soliton-type solutions, and a complete classification of multipliers and conservation laws with a discussion of their physical meaning
Afterwards, we present the reductions obtained from the different symmetries, transforming the partial differential equations (PDEs) into ordinary differential equations (ODEs)
Summary
Many viscoelastic wave equations have been considered in the literature. The single viscoelastic wave equation of the form. In this paper we present new results for the model It is found a complete classification of Lie point symmetries with its associated reductions, new soliton-type solutions, and a complete classification of multipliers and conservation laws with a discussion of their physical meaning. Afterwards, we present the reductions obtained from the different symmetries, transforming the PDE into ODEs. we obtain traveling wave solutions by comparing Equation (1) and similar equations studied previously [31,32,33]. The structure of the paper is as follows: In Section 2, we study the Lie point symmetries of Equation (1), and, we obtain the symmetry reductions, the symmetry variables, and the reduced ODEs. in Section 4, we construct traveling wave solutions using the reduced equations.
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