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# Approximate symmetries in nonlinear viscoelastic media

- Marianna Ruggieri
^{1}Email author and - Antonino Valenti
^{2}

**2013**:143

https://doi.org/10.1186/1687-2770-2013-143

© Ruggieri and Valenti; licensee Springer. 2013

**Received:**9 March 2013**Accepted:**20 May 2013**Published:**10 June 2013

## Abstract

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.

## Keywords

- existence
- uniqueness and stability of solutions
- partial differential equations
- approximate method

## 1 Introduction

where *σ* and *λ* are smooth functions, $w(t,x)$ 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 [3], 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 $\lambda ({w}_{x})={\lambda}_{0}$, with ${\lambda}_{0}$ a positive constant, a symmetry analysis can be performed in [7–9].

*ε*, namely

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 [10]. Successively another method for finding approximate symmetries was proposed by Fushchich and Shtelen [11]. 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 $f({w}_{x})$ and $\lambda ({w}_{x})$ 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.

*ε*, can be expanded in power series of

*ε*,

*i.e.*,

where ${w}_{0}$ and ${w}_{1}$ are some smooth functions of *t* and *x*; ${\stackrel{\u02c6}{w}}_{0}$ and ${\stackrel{\u02c6}{w}}_{1}$ are some smooth functions of $\stackrel{\u02c6}{t}$ and $\stackrel{\u02c6}{x}$.

*ε*, we arrive at the following system of PDEs:

Hence, ${w}_{0}$ is a solution of nonlinear wave equation (9) which we call *unperturbed equation*, while ${w}_{1}$ can be determined from the linear equation (10).

*ε*, we get the following one-parameter Lie group of infinitesimal transformations in the $(t,x,{w}_{0},{w}_{1})$-space

*approximate symmetries*of equation (2) the (exact) symmetries of the system (9)-(10) through the one-parameter Lie group of infinitesimal transformations (11)-(14). Consequently, the one-parameter Lie group of infinitesimal transformations (11)-(14), the associated Lie algebra and the corresponding infinitesimal operator

are called the approximate Lie group, the approximate Lie algebra and the approximate Lie operator of equation (2), respectively.

and ${X}_{0}$ can be regarded as the infinitesimal operator of unperturbed nonlinear wave equation (9) (or (3)).

where ${X}_{0}^{(2)}$ and ${X}^{(3)}$ are the second and third extensions of the operators ${X}_{0}$ and *X*, respectively.

## 3 Group classification via approximate symmetries

where ${a}_{i}$, $i=1,2,\dots ,10$ are constants.

with ${a}_{11}$, ${a}_{12}$ and ${a}_{13}$ being constants.

*f*and

*λ*arbitrary functions, we obtain

*approximate principal Lie algebra*of equation (2). We denote it by $\mathcal{A}\mathit{pprox}{\mathcal{L}}_{\mathcal{P}}$ and it is spanned by the seven operators

**Classification of**
$\mathit{f}\mathbf{(}{{\mathit{w}}_{\mathbf{0}}}_{\mathit{x}}\mathbf{)}$
**and**
$\mathit{\lambda}\mathbf{(}{{\mathit{w}}_{\mathbf{0}}}_{\mathit{x}}\mathbf{)}$
**with the corresponding extensions of**
$\mathcal{A}\mathit{pprox}{\mathcal{L}}_{\mathcal{P}}$

Case | Forms of $\mathit{f}\mathbf{(}{{\mathit{w}}_{\mathbf{0}}}_{\mathit{x}}\mathbf{)}$ and $\mathit{\lambda}\mathbf{(}{{\mathit{w}}_{\mathbf{0}}}_{\mathit{x}}\mathbf{)}$ | Extensions of $\mathcal{A}\mathit{p}\mathit{p}\mathit{r}\mathit{o}\mathit{x}{\mathcal{L}}_{\mathcal{P}}$ |
---|---|---|

| $f({{w}_{0}}_{x})={f}_{0}{e}^{\frac{2}{p}{{w}_{0}}_{x}}$ | ${X}_{8}=x\frac{\partial}{\partial x}+({w}_{0}+px)\frac{\partial}{\partial {w}_{0}}+s{w}_{1}\frac{\partial}{\partial {w}_{1}}$ |

$\lambda ({{w}_{0}}_{x})={\lambda}_{0}{e}^{\frac{1+s}{p}{{w}_{0}}_{x}}$ | ||

| $f({{w}_{0}}_{x})={f}_{0}{({{w}_{0}}_{x}+q)}^{\frac{2}{p}}$ | ${X}_{8}=x\frac{\partial}{\partial x}+[(1+p){w}_{0}+pqx]\frac{\partial}{\partial {w}_{0}}+s{w}_{1}\frac{\partial}{\partial {w}_{1}}$ |

$\lambda ({{w}_{0}}_{x})={\lambda}_{0}{({{w}_{0}}_{x}+q)}^{\frac{1+s}{p}-1}$ |

## 4 A physical application

where *τ* is the stress, *x* the position of a cross-section in the homogeneous rest configuration of the bar, $w(t,x)$ the displacement at time *t* of the section from its rest position, $\sigma ({w}_{x})$ the elastic tension (${w}_{x}$ is the strain), and $\lambda ({w}_{x}){w}_{xt}$ is the viscosity component of the stress, reduces to (2).

which was suggested by Capriz [15, 16].

*II*of Table 1 with the following identifications:

*w*:

which give the form of an invariant solution approximate at the first order in *ε*.

*ψ*and

*χ*must satisfy the following system of ODEs to which, after (44), the system (9)-(10) is reduced through (47) and (48) (we have chosen $s=-5$):

*ε*is

We have an unperturbed state represented by a stretching modified by the viscosity effect.

## 5 Conclusions

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 [11], 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*.

## Declarations

### Acknowledgements

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*.

## Authors’ Affiliations

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