Summary
Herein the physically and mathematically significant semi-linear hyperbolic equationu xt = f(u), where f(u) is an arbitrary smooth function ofu and which encompasses the Liouville equation, Phi-four equation, Sine-Gordon equation, Klein-Gordon equation, Mikhailov equation and double Sine-Gordon equation, has been analysed via the symmetry method as developed by Steinberg (Symmetry methods in differential equations, Technical Report No. 367, University of New Mexico, 1979). The infinitesimals, similarity variables, dependent variables and reduction to quadrature or exact solutions have been tabulated for the mentioned physical forms of f(u). Some interesting outcomes of this study are the deductions of the new exact solutions that does not seem to have been reported in the literature.
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References
Birkhoff G.,Hydrodynamics (Princeton University Press, Princeton, N.J.) 1960.
Bhutani O. P. andMital P.,J. Met. Soc. Jpn.,64 (1986) 593.
Bhutani O. P., Vijayakumar K., Mital P. andChandrasekaran G.,Int. J. Engng. Sci.,27 (1989) 921.
Steinberg S.,Symmetry methods in differential equations, Technical Report No. 367, University of New Mexico, 1979.
Calogero F.,Stud. Appl Math.,70 (1984) 189.
Shiff L. I.,Phys. Rev.,84 (1951) 1.
Perring J. K. andSkyrme T. H. R.,Nucl. Phys.,31 (1962) 550.
Barone A., Esposito F., Magee C. G. andScott A. C.,Riv. Nuovo Cimento, Vol.1, No. 2 (1971) 227.
Esenhart L. P.,A Treatise on the Differential Geometry of Curves and Surfaces (Dover, New York, N.Y.) 1972.
Shadwick E. F.,J. Math. Phys.,19 (1987) 2312.
Clarkson P. A.,Olver P. J.,Mcleod J. B. andRamani A.,Math. Rep., University of Minnesota, 83–159 (1986).
Bhutani O. P. andVijayakumar K.,Int. J. Engng. Sci.,30 (1992) 1049.
Moussa M. H. M.,New Similarity Solutions of Klein-Gordon Type Equations, PhD Thesis, HT Delhi (1991).
Tamizhmani K. M. andLakshmanan M.,J. Math. Phys.,27 (1986) 2257.
Ince L. E.,Ordinary Differential Equations (Dover, New York, N.Y.) 1956.
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Moussa, M.H.M., Sallam, S.N. On the similarity solutions of the semi-linear hyperbolic equationu xt =f(u) via symmetry method. Nuov Cim B 111, 995–1003 (1996). https://doi.org/10.1007/BF02743295
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DOI: https://doi.org/10.1007/BF02743295