Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 1017-1026 |
| Number of pages | 10 |
| Journal | Applied Mathematics and Mechanics |
| Volume | 30 |
| Issue number | 8 |
| DOIs | |
| State | Published - Aug 2009 |
Bibliographical note
Funding Information:Acknowledgements The authors acknowledge the support of King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. The authors are also grateful to Prof. Hassan Azad for many useful discussions related to the Lie symmetry method.
Keywords
- Group invariant solutions
- Lie symmetries
- Nonlinear elasticity equations
- Partial differential equations
ASJC Scopus subject areas
- Mechanics of Materials
- Mechanical Engineering
- Applied Mathematics