Applied Mathematics and Mechanics

, Volume 30, Issue 8, pp 1017–1026

Symmetry solutions of a nonlinear elastic wave equation with third-order anharmonic corrections


DOI: 10.1007/s10483-009-0808-z

Cite this article as:
Mustafa, M.T. & Masood, K. Appl. Math. Mech.-Engl. Ed. (2009) 30: 1017. doi:10.1007/s10483-009-0808-z


Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to second-order ordinary differential equations (ODEs) are obtained through invariance under different symmetries. The reduced ODEs are further analyzed to obtain several exact solutions in an explicit form. It was observed in the literature that anharmonic corrections generally lead to solutions with time-dependent singularities in finite times. Along with solutions with time-dependent singularities, we also obtain solutions which do not exhibit time-dependent singularities.

Key words

group invariant solutions Lie symmetries nonlinear elasticity equations partial differential equations 

Chinese Library Classification


2000 Mathematics Subject Classification

76M60 73C50 74J30 74H05 

Copyright information

© Shanghai University and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.Department of Mathematics, Hafr Al-Batin Community CollegeKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia