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Approximate symmetries in nonlinear viscoelastic media

Marianna Ruggieri1* and Antonino Valenti2

Author Affiliations

1 Facoltà di Ingegneria e Architettura, Università di Enna ‘Kore’, Cittadella Universitaria, Enna, 94100, Italy

2 Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria 6, Catania, 95125, Italy

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Boundary Value Problems 2013, 2013:143  doi:10.1186/1687-2770-2013-143


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/143


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

© 2013 Ruggieri and Valenti; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

We consider the third-order partial differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M1">View MathML</a>

(1)

where σ and λ are smooth functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M2">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M3">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M4">View MathML</a> a positive constant, a symmetry analysis can be performed in [7-9].

It is well known that a small dissipation is able to prevent the breaking of the wave profile allowing to study the so called ‘far field’, and a technique widely used is the perturbation analysis performed by expanding the dependent variables in power series of a small parameter (may be a physical parameter or often artificially introduced). Having in mind to perform an ‘approximate symmetry analysis’, we introduce in (1) a small parameter ε, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M5">View MathML</a>

(2)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M6">View MathML</a> (hereafter, a prime denotes derivative of a function with respect to the only variable upon which it depends). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M7">View MathML</a>, we recover the nonlinear wave equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M8">View MathML</a>

(3)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M10">View MathML</a> 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

In general, any solution of (2) will be of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M11">View MathML</a> and the one-parameter Lie group of infinitesimal transformations in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M12">View MathML</a>-space of equation (2) can be considered in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M13">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M14">View MathML</a>

(5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M15">View MathML</a>

(6)

where a is the group parameter.

Let us suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M17">View MathML</a>, analytic in ε, can be expanded in power series of ε, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M18">View MathML</a>

(7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M19">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M21">View MathML</a> are some smooth functions of t and x; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M23">View MathML</a> are some smooth functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M25">View MathML</a>.

Upon formal substitution of (7) in (2), equating to zero the coefficients of zero and first degree powers of ε, we arrive at the following system of PDEs:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M26">View MathML</a>

(9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M27">View MathML</a>

(10)

where we have set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M28">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M20">View MathML</a> is a solution of nonlinear wave equation (9) which we call unperturbed equation, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M21">View MathML</a> can be determined from the linear equation (10).

In order to have a one-parameter Lie group of infinitesimal transformations of the system (9)-(10), which is consistent with the expansions of the dependent variables (7) and (8), we introduce these expansions in the infinitesimal transformations (4)-(6). Upon formal substitution, equating to zero the coefficients of zero and first degree powers of ε, we get the following one-parameter Lie group of infinitesimal transformations in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M31">View MathML</a>-space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M32">View MathML</a>

(11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M33">View MathML</a>

(12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M34">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M35">View MathML</a>

(14)

where we have set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M36">View MathML</a>

(15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M37">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M38">View MathML</a>

(17)

We give the following definition: We call 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M39">View MathML</a>

(18)

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

Moreover, after putting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M40">View MathML</a>

(19)

the approximate Lie operator (18) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M41">View MathML</a>

(20)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M42">View MathML</a> can be regarded as the infinitesimal operator of unperturbed nonlinear wave equation (9) (or (3)).

It is worthwhile noticing that, thanks to the functional dependencies of the coordinates of the approximate Lie operator (18) (or (20)), now we are able to work in hierarchy in finding the invariance conditions of the system (9)-(10): firstly, by classifying unperturbed nonlinear wave equation (9) through the operator (19) and after by determining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M44">View MathML</a> from the invariance condition that follows by applying the operator (20) to the linear equation (10). In fact the invariance condition of the system (9)-(10) reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M45">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M46">View MathML</a>

(22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M48">View MathML</a> are the second and third extensions of the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M42">View MathML</a> and X, respectively.

3 Group classification via approximate symmetries

The classification of equation (9) is well known (see for details Ibragimov [14] and bibliography therein). From (21), we arrive at the following result:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M50">View MathML</a>

(23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M51">View MathML</a>

(24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M52">View MathML</a>

(25)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M53">View MathML</a>

(26)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M54">View MathML</a>

(27)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M56">View MathML</a> are constants.

Taking (23)-(27) into account, from (22) we obtain the following additional conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M57">View MathML</a>

(28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M58">View MathML</a>

(29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M59">View MathML</a>

(30)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M60">View MathML</a>

(31)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M63">View MathML</a> being constants.

After observing that conditions (28) impose restrictions upon <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M42">View MathML</a>, summarizing we have to manage the following relations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M65">View MathML</a>

(32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M66">View MathML</a>

(33)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M67">View MathML</a>

(34)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M68">View MathML</a>

(35)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M69">View MathML</a>

(36)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M70">View MathML</a>

(37)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M71">View MathML</a>

(38)

For f and λ arbitrary functions, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M72">View MathML</a>

(39)

and we call the associate seven-dimensional Lie algebra the approximate principal Lie algebra of equation (2). We denote it by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M73">View MathML</a> and it is spanned by the seven operators

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M74">View MathML</a>

(40)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M75">View MathML</a>

(41)

The classification of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M77">View MathML</a> with the corresponding extensions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M73">View MathML</a> arising from (32)-(38) is reported in Table 1.

Table 1. Classification of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M76">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M77">View MathML</a>with the corresponding extensions of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M73">View MathML</a>

4 A physical application

Let us consider a homogeneous viscoelastic bar of uniform cross-section and assume the material to be a nonlinear Kelvin solid. The classical equation of motion (the constant density is normalized to 1 and the mass forces are neglected)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M93">View MathML</a>

(42)

assuming a stress-strain relation of the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M94">View MathML</a>

(43)

where τ is the stress, x the position of a cross-section in the homogeneous rest configuration of the bar, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M2">View MathML</a> the displacement at time t of the section from its rest position, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M96">View MathML</a> the elastic tension (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M97">View MathML</a> is the strain), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M98">View MathML</a> is the viscosity component of the stress, reduces to (2).

Let us consider the following form of the tension <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M96">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M100">View MathML</a>

(44)

which was suggested by Capriz [15,16].

So, we fall in Case II of Table 1 with the following identifications:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M101">View MathML</a>

(45)

In this case, the approximate Lie operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M102">View MathML</a> assumes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M103">View MathML</a>

(46)

and from the corresponding invariant surface conditions, we obtain the following representation for the different terms in the expansion of w:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M104">View MathML</a>

(47)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M105">View MathML</a>

(48)

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

The functions 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M106">View MathML</a>):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M107">View MathML</a>

(49)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M108">View MathML</a>

(50)

After solving (49)-(50) and taking (47)-(48) into account, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M109">View MathML</a>

(51)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M110">View MathML</a>

(52)

Therefore, the invariant solution up to the first order in ε is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/143/mathml/M111">View MathML</a>

(53)

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.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors wrote this paper in collaboration and with the same responsibility. All authors read and approved the final version of the manuscript.

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.

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