Approximate symmetries in nonlinear viscoelastic media
© Ruggieri and Valenti; licensee Springer. 2013
Received: 9 March 2013
Accepted: 20 May 2013
Published: 10 June 2013
Approximate symmetries of a mathematical model describing one-dimensional motion in a medium with a small nonlinear viscosity are studied. In a physical application, the approximate solution is calculated making use of the approximate generator of the first-order approximate symmetry.
MSC:35J25, 32A37, 43A15, 35A58, 42B20.
Keywordsexistence uniqueness and stability of solutions partial differential equations approximate method
where σ and λ are smooth functions, is the dependent variable and subscripts denote partial derivative with respect to the independent variables t and x.
Equation (1) can describe the behavior of a one-dimensional viscoelastic medium in which nonlinearities appear not only in the elastic part of the stress, but also in the viscoelastic one.
Some mathematical questions as the global existence, uniqueness and stability of solutions can be found in [1, 2]. Moreover, shear wave solutions are found in , where some explicit examples of blow-up for boundary value problems with smooth initial data are shown. A symmetry analysis and some exact solutions are shown in [4–6], while when , with a positive constant, a symmetry analysis can be performed in [7–9].
The combination of the Lie group theory and the perturbation analysis gives rise to the so-called approximate symmetry theories. The first paper on this subject is due to Baikov, Gazizov and Ibragimov . Successively another method for finding approximate symmetries was proposed by Fushchich and Shtelen . In the method proposed by Baikov, Gazizov and Ibragimov, the Lie operator is expanded in a perturbation series so that an approximate operator can be found. But the approximate operator does not reflect well an approximation in the perturbation sense; in fact, even if one uses a first-order approximate operator, the corresponding approximate solution could contain higher-order terms.
In the method proposed by Fushchich and Shtelen the dependent variables are expanded in a perturbation series; equations are separated at each order of approximation and the approximate symmetries of the original equations are defined to be the exact symmetries of the system coming out from equating to zero the coefficients of the smallness parameter. This method is consistent with the perturbation theory and yields correct terms for the approximate solutions but a ‘drawback’ is present: it is impossible to work in hierarchy, i.e., in the search for symmetries, there is a coupled system between the equations at several orders of approximation, therefore the algebra can increase enormously.
In this paper we work in the framework of the approximate method proposed in [12, 13], in which the expansions of the dependent variable are introduced also in the Lie group transformations so that one obtains an approximate Lie operator which permits to solve in hierarchy the invariance conditions starting from the classification of unperturbed equation (3). We obtain the symmetry classification of the functions and through which equation (2) is approximately invariant and search for approximate solutions.
The plan of the paper is the following. The approximate symmetry method is introduced in the next section; the group classification via approximate symmetries is performed in Section 3; in Section 4, in a physical application, the approximate solution is calculated by means of the approximate generator of the first-order approximate group of transformations.
2 Approximate symmetry method
where a is the group parameter.
where and are some smooth functions of t and x; and are some smooth functions of and .
are called the approximate Lie group, the approximate Lie algebra and the approximate Lie operator of equation (2), respectively.
and can be regarded as the infinitesimal operator of unperturbed nonlinear wave equation (9) (or (3)).
where and are the second and third extensions of the operators and X, respectively.
3 Group classification via approximate symmetries
where , are constants.
with , and being constants.
Classification of and with the corresponding extensions of
Forms of and
4 A physical application
where τ is the stress, x the position of a cross-section in the homogeneous rest configuration of the bar, the displacement at time t of the section from its rest position, the elastic tension ( is the strain), and is the viscosity component of the stress, reduces to (2).
which give the form of an invariant solution approximate at the first order in ε.
We have an unperturbed state represented by a stretching modified by the viscosity effect.
In this paper we perform the group analysis of the nonlinear wave equation with a small dissipation (2) in the framework of the approximate method proposed in [12, 13]. In order to remove the ‘drawback’ of the method proposed by Fushchich and Shtelen , we introduce, according to the perturbation theory, the expansions of the dependent variables in the one-parameter Lie group of infinitesimal transformations of equation (2). Equating to zero the coefficients of zero and first degree powers of ε, we obtain an approximate Lie operator which permits to solve in hierarchy the invariance condition of the system (9)-(10) starting from the classification of unperturbed nonlinear wave equation (3). The proposed strategy is consistent with the perturbation point of view and can be generalized in a simple way to the higher orders of approximation in ε.
MR acknowledges the support of GNFM through the project 2012 Metodologie di tipo analitico e numerico per lo studio di problemi iperbolici ed iperbolico-parabolici di natura ondosa.
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