Lie symmetry analysis of the time-variable coefficient B-BBM equation
© Molati and Khalique; licensee Springer 2012
Received: 19 September 2012
Accepted: 15 November 2012
Published: 11 December 2012
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.
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.
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  to generate and simplify the determining equations of the underlying Eq. (3) for maximal symmetry Lie algebra.
where the subscripts denote partial differentiation with respect to the indicated variables.
Upon solving (16), the coefficients of symmetry generator (12) become , and . Therefore, we obtain a one-dimensional principal Lie algebra which is spanned by the operator .
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: , , . In addition, we must have in order to extend the principal Lie algebra.
Case 1. .
In solving (17), we consider the subcases: and .
where and are arbitrary constants.
The condition: implies that and (this result is discarded).
Now, for , we obtain and . Therefore, the extra operator is given by .
We obtain for (this result is excluded).
The condition implies that and which corresponds to a constant coefficient case. The principal Lie algebra is extended by the operator .
Case 2. Next we consider the case . 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 , we have and thus,
where , and are nonzero arbitrary constants.
Subcase 2.2. : .
where , and are nonzero arbitrary constants.
where and 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
where a ‘prime’ denotes derivative with respect to z.
If , then the similarity variable takes the form , but the similarity solution and the reduced equation remain the same.
Remark It is noted that the reduced ordinary differential equations are still highly nonlinear to solve exactly.
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.
This paper is dedicated to Prof. Ravi P. Agarwal on his 65th birth anniversary.
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.
- Benjamin TB, Bona JL, Mahoney JJ: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. 1972, 272: 47–78. 10.1098/rsta.1972.0032View ArticleMathSciNetMATHGoogle Scholar
- Burgers JM: The Nonlinear Diffusion Equation. Reidel Publishing, Dordrecht; 1974.View ArticleMATHGoogle Scholar
- Bluman GW, Kumei S: Symmetries and Differential Equations. Springer, New York; 1989.View ArticleMATHGoogle Scholar
- Olver PJ: Applications of Lie Groups to Differential Equations. Springer, New York; 1986.View ArticleMATHGoogle Scholar
- Ovsiannikov LV: Group Analysis of Differential Equations. Academic, New York; 1982.MATHGoogle Scholar
- Johnpillai AG, Khalique CM: Group analysis of KdV equation with time dependent coefficients. Appl. Math. Comput. 2010, 216: 3761–3771. 10.1016/j.amc.2010.05.043MathSciNetView ArticleMATHGoogle Scholar
- Molati M, Ramollo MP: Symmetry classification of the Gardner equation with time-dependent coefficients arising in stratified fluids. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 1542–1548. 10.1016/j.cnsns.2011.09.002MathSciNetView ArticleMATHGoogle Scholar
- Zhao Y, Zhang WG: Qualitative analysis and solutions of bounded travelling wave for B-BBM equation. Acta Math. Appl. Sin. 2010, 26: 415–426. 10.1007/s10255-010-0007-0View ArticleMATHGoogle Scholar
- Sophocleous C: Transformation properties of a variable coefficient Burgers equation. Chaos Solitons Fractals 2004, 20: 1047–1057. 10.1016/j.chaos.2003.09.024MathSciNetView ArticleMATHGoogle Scholar
- Morrison PJ, Meiss JD, Cary JR: Scattering of regularized-long-wave solitary waves. Physica D, Nonlinear Phenom. 1984, 11: 324–336. 10.1016/0167-2789(84)90014-9MathSciNetView ArticleMATHGoogle Scholar
- Díaz, JM: Short guide to YaLie. Yet another Lie Mathematica package for Lie symmetries. http://library.wolfram.com/infocenter/MathSource/4231/Google Scholar
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