Approximate symmetries in nonlinear viscoelastic media
In this paper the authors study with an appoximate symmetry method a mathematical model describing the behavior of a one-dimensional viscoelastic medium in which both in elastic part of the stress and also in the viscoelastic one appear nonlinearities, performing the group analysis of this nonlinear wave equation with a small dissipation. To give the perturbation analysis they expand the dependent variable applying also the Lie group transformation and so they obtain an approximate Lie operator and, from which, they solve the invariance conditions starting from the classification of unperturbed equation of the model. They obtain so the symmetry classification of the functions of the dependent variable which are present in the perturbed equation, that is in the equation obtained from the equation of the model by introducing a small parameter \(\epsilon \); through these functions the perturbed equation is approximately invariant and the authors search for approximate solutions. Moreover they consider a physical application: they study the case of a homogeneous viscoelastic bar of uniform cross section assuming that the material is a nonlinear Kelvin solid; the approximate solution is studied by means of the approximate generator of the first order approximate group of trasformations.
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