Geodesic
Two approaches to solving the potential equation in self-similar variables
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 7 (2015) no. 3, pp. 30-38

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The authors, using the method previously proposed by them, investigate the general velocity potential equation for the case of three self-similar variables. Two approaches of this method are used. The first approach assumes that the solution depends only on one variable, which, in turn, is an unknown function of all independent variables, and thus potential equation is reduced to the ODE. Finding unknown function is based on a study of the corresponding overdetermined system of partial differential equations. A number of compatibility conditions for the system are found. Some exact solutions are constructed. It is shown how the solutions obtained can be used in considering the problem of shock-free compression of the gas. The second approach assumes that the function is known and coincides with the function that gives a solution of the potential equation. It is also received a number of exact solutions that can be used to solve some initial and boundary value problems.
Keywords: nonlinear partial differential equations; geometric method of research; reducing a partial differential equation to ODE; exact solutions. 
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L. I. Rubina; O. N. Ul'yanov. Two approaches to solving the potential equation in self-similar variables. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 7 (2015) no. 3, pp. 30-38. http://geodesic.mathdoc.fr/item/VYURM_2015_7_3_a4/