Travelling wave solutions of a one-dimensional viscoelasticity model

ABSTRACT

In this paper a one-dimensional viscoelasticity model describing the behaviour of a one-dimensional viscoelastic medium is studied through Lie symmetry approach. The infinitesimals of the group of transformations leaving the equation invariant are determined. The optimal systems of one-dimensional subalgebras of the Lie symmetry algebras are calculated. Afterwards, using the Lie symmetry approach, we transformed the nonlinear partial differential equation into a nonlinear ordinary differential equation. Furthermore, we applied the $\left(\frac{G^{\prime }}{G}\right)$-expansion method to find explicitly new travelling wave solutions. Moreover, some conservation laws are constructed by applying the multiplier method.

KEYWORDS:

• Viscoelasticity
• Lie symmetries
• $\left(\frac{G^{\prime }}{G}\right)$-expansion method
• conservation laws
• multiplier method

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 35Q74

Acknowledgments

The authors acknowledge the support of Junta de Andalucía FQM-201 group. A. P. Márquez expresses its sincere gratitude to the Plan Propio de Investigación y Transferencia of the University of Cadiz. We are also very thankful to the reviewers for their valuable suggestions to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

A. P. Márquez  http://orcid.org/0000-0001-7067-4906

M. S. Bruzón  http://orcid.org/0000-0002-3599-6106