Symmetry Solutions of a Non-Linear Elastic Wave Equation With Third Order Anharmonic Corrections

M. T. Mustafa; Khalid Masood

doi:10.3879/j.issn.1000-0887.2009.08.008

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M. T. Mustafa, Khalid Masood. Symmetry Solutions of a Non-Linear Elastic Wave Equation With Third Order Anharmonic Corrections[J]. Applied Mathematics and Mechanics, 2009, 30(8): 953-962. doi: 10.3879/j.issn.1000-0887.2009.08.008

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Symmetry Solutions of a Non-Linear Elastic Wave Equation With Third Order Anharmonic Corrections

doi: 10.3879/j.issn.1000-0887.2009.08.008

M. T. Mustafa¹,
Khalid Masood²

1.
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia;

Received Date: 2008-08-23
Rev Recd Date: 2009-03-16
Publish Date: 2009-08-15

Abstract

Abstract

Lie symmetry method was applied to analyze a non-linear elastic wave equation for longitudinal deformations with third order anharmonic corrections to the elastic energy. Symmetry algebra was found and reductions to second order ODEs were obtained through invariance under different symmetries. The reduced ODEs were further analyzed to obtain several exact solutions in explicit form. Apostol(Apostol B F. On a non-linear wave equation in elasticity. Phys Lett A, 2003, 318(6):545552) had observed that anharmonic corrections generally lead t o solutions with time-dependent singularities in finite time. Along with solutions with time-dependent singularities are obtained, also solutions which do not exhibit time-dependent singularities were obtained.
- group invariant solution,
- Lie symmetries,
- nonlinear elasticity equations,
- partial differential equations,
- ordinary differential equations

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References(26)

References

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