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# Lie symmetry analysis of the time-variable coefficient B-BBM equation

- Motlatsi Molati
^{1, 2}and - Chaudry Masood Khalique
^{1}Email author

**2012**:212

https://doi.org/10.1186/1687-1847-2012-212

© Molati and Khalique; licensee Springer 2012

**Received:**19 September 2012**Accepted:**15 November 2012**Published:**11 December 2012

## Abstract

We perform Lie group classification of a time-variable coefficient combined Burgers and Benjamin-Bona-Mahony equations (B-BBM equation). The direct analysis of the determining equations is employed to specify the forms of these time-dependent coefficients also known as arbitrary parameters. It is established that these model parameters have time-dependent functional forms of linear, power and exponential type.

## Keywords

- B-BBM equation
- group classification
- classifying relations

## 1 Introduction

in its different forms has been studied extensively as a model of shock-waves in applications ranging from traffic flow to turbulence. The diverse approaches, both analytical and numerical, have been employed in the extensive studies of these equations in their various forms. Many of these investigations involve assuming the forms of the arbitrary parameters which appear in the mathematical models. However, the Lie group classification is a systematic approach through which the parameters assume their forms naturally. This is the essence of Lie group analysis of differential equations [3–5]. The direct method of group classification [6, 7] is used to analyze the *classifying relations*, *i.e.*, the equations which contain the arbitrary model parameters.

*q*is a positive constant. We deduce the following from Eq. (3): If $q=1$ and $h(t)=0$, then we obtain the variable coefficient Burgers equation

A thorough investigation of the Lie group properties and some exact solutions of Eq. (4) can be found in [9].

The outline of this work is as follows. The determining equations for the arbitrary elements (classifying relations) are generated in Section 2. In Section 3, the arbitrary elements are specified via the direct method of group classification. The results of group classification are utilized for symmetry reductions solutions in Section 4. Finally, we summarize our findings in Section 5.

## 2 Classifying equations and principal Lie algebra

The manual task of generating determining equations is tedious, but nowadays the Lie’s algorithm is implemented using a lot of various computer software packages for symbolic computation. We use the Mathematica software package YaLie [11] to generate and simplify the determining equations of the underlying Eq. (3) for maximal symmetry Lie algebra.

*η*, satisfy the determining equations

where the subscripts denote partial differentiation with respect to the indicated variables.

*t*. Then from the last equations (13)-(15), we have

Upon solving (16), the coefficients of symmetry generator (12) become ${\xi}^{1}=0$, ${\xi}^{2}={c}_{2}$ and $\eta =0$. Therefore, we obtain a one-dimensional principal Lie algebra which is spanned by the operator ${X}_{1}={\partial}_{x}$.

## 3 Lie group classification

This section deals with specifying the forms of the arbitrary parameters through the direct analysis of the classifying relations with the aim of extending the principal symmetry Lie algebra.

We analyze the classifying relations (13)-(15) for the cases: $f(t)\ne 0$, $g(t)\ne 0$, $h(t)\ne 0$. In addition, we must have ${\xi}^{1}(t)\ne 0$ in order to extend the principal Lie algebra.

Case 1. $f(t)\ne 0$.

In solving (17), we consider the subcases: $m\ne 0$ and $m=0$.

where ${f}_{0}\ne 0$ and ${c}_{4}$ are arbitrary constants.

The condition: $\beta =0$ implies that $g(t)=0$ and $h(t)=0$ (this result is discarded).

Now, for ${c}_{4}=0$, we obtain $g(t)={g}_{0}$ and $h(t)={h}_{0}$. Therefore, the extra operator is given by ${X}_{2}=q{\partial}_{t}-mu{\partial}_{u}$.

We obtain $g(t)=h(t)=0$ for $\mu =0$ (this result is excluded).

The condition ${c}_{3}=0$ implies that $g(t)={\overline{g}}_{0}$ and $h(t)={\overline{h}}_{0}$ which corresponds to a constant coefficient case. The principal Lie algebra is extended by the operator ${X}_{2}={\partial}_{t}$.

Case 2. Next we consider the case $g(t)\ne 0$. Proceeding as in the previous case, here we base our analysis on the classifying equation (14). As a result, we obtain the following classification results.

Subcase 2.1. For some constant $n\ne 0$, we have $g(t)={\tilde{g}}_{0}{e}^{nt}$ and thus,

where ${\tilde{f}}_{0}$, ${\tilde{g}}_{0}$ and ${\tilde{h}}_{0}$ are nonzero arbitrary constants.

Subcase 2.2. $n=0$: $g(t)={\tilde{g}}_{0}$.

where ${\tilde{f}}_{0}$, ${\tilde{g}}_{0}$ and ${\tilde{h}}_{0}$ are nonzero arbitrary constants.

*k*, the analysis of the classifying equation (15) leads to $h(t)={\stackrel{\u02c6}{h}}_{0}{e}^{kt}$, where ${\stackrel{\u02c6}{h}}_{0}\ne 0$. The corresponding forms of $f(t)$ and $g(t)$ together with the operator, which extends the principal Lie algebra, read

where ${\stackrel{\u02c6}{f}}_{0}\ne 0$ and ${\stackrel{\u02c6}{g}}_{0}\ne 0$ are arbitrary constants. The last set of classification results is a particular case of 2.1.1 for some choice of arbitrary constants. It can also be shown by similar calculations that the remaining subcases are duplication of some subcases in Cases 1 and 2.

## 4 Symmetry reductions

*c*is an arbitrary constant. For illustration, we consider Subcase 2.2. Firstly, we look at the Subcase 2.2.1. Thus, for $c\ne 0$, we have the linear combination

where a ‘prime’ denotes derivative with respect to *z*.

If $c=0$, then the similarity variable takes the form $z=x/\sqrt{2t+\mu}$, but the similarity solution and the reduced equation remain the same.

*F*is an arbitrary function of its argument. The corresponding reduced equation is given by

**Remark** It is noted that the reduced ordinary differential equations are still highly nonlinear to solve exactly.

## 5 Conclusion

The direct analysis of the classifying relations was employed to obtain the functional forms of the time-dependent arbitrary parameters for the B-BBM equation. These time-dependent functional forms are of linear, power and exponential type. The maximal symmetry Lie algebra of dimension two is obtained for each set of classification results. Consequently, the symmetries which span the symmetry Lie algebra are utilized to perform symmetry reductions.

## Dedication

This paper is dedicated to Prof. Ravi P. Agarwal on his 65th birth anniversary.

## Declarations

### Acknowledgements

CMK would like to thank the Organizing Committee of ‘International Conference on Applied Analysis and Algebra’, (ICAAA 2012) for their kind hospitality during the conference. MM thanks the North-West University, Mafikeng campus for the post-doctoral fellowship.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.