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

H. Azad; M. T. Mustafa; M. Ziad

doi:10.1016/j.cnsns.2009.05.045

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

H. Azad
, M. T. Mustafa^*
, M. Ziad

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

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
Pages (from-to)	1132-1147
Number of pages	16
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	15
Issue number	5
DOIs	https://doi.org/10.1016/j.cnsns.2009.05.045
State	Published - May 2010
Externally published	Yes

Bibliographical note

Funding Information:
The authors acknowledge KFUPM for funding Research Project #IN080397. This paper is based on work done in the project.

Keywords

Group classification
Invariant solutions
Lie symmetries
Nonlinear wave equation
Optimal system

ASJC Scopus subject areas

Numerical Analysis
Modeling and Simulation
Applied Mathematics

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10.1016/j.cnsns.2009.05.045

Cite this

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title = "Group classification, optimal system and optimal reductions of a class of Klein Gordon equations",

abstract = "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.",

keywords = "Group classification, Invariant solutions, Lie symmetries, Nonlinear wave equation, Optimal system",

author = "H. Azad and Mustafa, \{M. T.\} and M. Ziad",

note = "Funding Information: The authors acknowledge KFUPM for funding Research Project \#IN080397. This paper is based on work done in the project.",

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doi = "10.1016/j.cnsns.2009.05.045",

language = "English",

volume = "15",

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AU - Azad, H.

AU - Mustafa, M. T.

AU - Ziad, M.

N1 - Funding Information: The authors acknowledge KFUPM for funding Research Project #IN080397. This paper is based on work done in the project.

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

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DO - 10.1016/j.cnsns.2009.05.045

M3 - Article

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JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

IS - 5

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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