### Abstract

Complete symmetry analysis is presented for non-linear Klein Gordon equations u_{tt} = u_{xx} + f (u). A group classification is carried out by finding f (u) that give larger symmetry algebra. One-dimensional optimal system is determined for symmetry algebras obtained through group classification. The subalgebras in one-dimensional optimal system and their conjugacy classes in the corresponding normalizers are employed to obtain, up to conjugacy, all reductions of equation by two-dimensional subalgebras. This is a new idea which improves the computational complexity involved in finding all possible reductions of a PDE of the form F (x, t, u, u_{x}, u_{t}, u_{xx}, u_{tt}, u_{xt}) = 0 to a first order ODE. Some exact solutions are also found.

Original language | English |
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Pages (from-to) | 1132-1147 |

Number of pages | 16 |

Journal | Communications in Nonlinear Science and Numerical Simulation |

Volume | 15 |

Issue number | 5 |

DOIs | |

Publication status | Published - May 2010 |

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### Keywords

- Group classification
- Invariant solutions
- Lie symmetries
- Nonlinear wave equation
- Optimal system

### ASJC Scopus subject areas

- Applied Mathematics
- Modelling and Simulation
- Numerical Analysis

### Cite this

*Communications in Nonlinear Science and Numerical Simulation*,

*15*(5), 1132-1147. https://doi.org/10.1016/j.cnsns.2009.05.045

**Group classification, optimal system and optimal reductions of a class of Klein Gordon equations.** / Azad, H.; Mustafa, M. T.; Ziad, M.

Research output: Contribution to journal › Article

*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 5, pp. 1132-1147. https://doi.org/10.1016/j.cnsns.2009.05.045

}

TY - JOUR

T1 - Group classification, optimal system and optimal reductions of a class of Klein Gordon equations

AU - Azad, H.

AU - Mustafa, M. T.

AU - Ziad, M.

PY - 2010/5

Y1 - 2010/5

N2 - Complete symmetry analysis is presented for non-linear Klein Gordon equations utt = uxx + f (u). A group classification is carried out by finding f (u) that give larger symmetry algebra. One-dimensional optimal system is determined for symmetry algebras obtained through group classification. The subalgebras in one-dimensional optimal system and their conjugacy classes in the corresponding normalizers are employed to obtain, up to conjugacy, all reductions of equation by two-dimensional subalgebras. This is a new idea which improves the computational complexity involved in finding all possible reductions of a PDE of the form F (x, t, u, ux, ut, uxx, utt, uxt) = 0 to a first order ODE. Some exact solutions are also found.

AB - Complete symmetry analysis is presented for non-linear Klein Gordon equations utt = uxx + f (u). A group classification is carried out by finding f (u) that give larger symmetry algebra. One-dimensional optimal system is determined for symmetry algebras obtained through group classification. The subalgebras in one-dimensional optimal system and their conjugacy classes in the corresponding normalizers are employed to obtain, up to conjugacy, all reductions of equation by two-dimensional subalgebras. This is a new idea which improves the computational complexity involved in finding all possible reductions of a PDE of the form F (x, t, u, ux, ut, uxx, utt, uxt) = 0 to a first order ODE. Some exact solutions are also found.

KW - Group classification

KW - Invariant solutions

KW - Lie symmetries

KW - Nonlinear wave equation

KW - Optimal system

UR - http://www.scopus.com/inward/record.url?scp=70449096298&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70449096298&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2009.05.045

DO - 10.1016/j.cnsns.2009.05.045

M3 - Article

VL - 15

SP - 1132

EP - 1147

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

SN - 1007-5704

IS - 5

ER -