Approximate symmetries in nonlinear viscoelastic media
type: Item
instance of: scholarly article
title: Approximate symmetries in nonlinear viscoelastic media
zbMATH Open document ID: 1316.35279
DOI: 10.1186/1687-2770-2013-143
author: Marianna Ruggieri
author: A. Valenti
published in: Boundary Value Problems
publication date: 2014-07-11T00:00:00Z
review text: 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.
reviewed by: Luisa Fattorusso
Mathematics Subject Classification ID: 35Q74
Mathematics Subject Classification ID: 32A37
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 6316417
zbMATH Keywords: existence
zbMATH Keywords: uniqueness and stability of solutions
zbMATH Keywords: approximate method
zbMATH Keywords: partial differential equations
Wikidata QID: Q59300090
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1186/1687-2770-2013-143
OpenAlex ID: W2166781165
cites work: The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity
cites work: Shear Waves in a Class of Nonlinear Viscoelastic Solids
cites work: On approximate symmetry and approximate solutions of the nonlinear wave equation with a small parameter
description: scientific article