Geodesic
Inverse Problem of the Theory of Compatibility and Functional-Invariant Solutions Wave Equation in Two-Dimensional Space
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 14 (2012), pp. 99-107 Cet article a éte moissonné depuis la source Math-Net.Ru

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We investigated the system of equations with variable coefficients, which describe functional-invariant solutions of wave equation in space $\mathbb{R}^3(t,x,y)$. It is well known that in the case of identity matrix of coefficients we can describe all functional-invariant solutions by Sobolev formula. In this paper we prove that if solutions of considered systems have maximal arbitrariness (in the sense of the theory of compatibility overdetermined systems of differential equations in partial derivatives) then coefficients of the wave equation are connected by algebraic relation of the second order (hyperbolic or elliptic type) and in addition by differential relation of the second order. Group of transformations induced by change of space variables acts on the all set of differential equations naturally. We obtain complete classification of the considered systems with respect to this group. More precisely, we prove that there are exactly three classes of equivalence. In this paper we use classical methods Requier theory of investigation of overdetermined systems of differential equations in partial derivatives.
Keywords: Wave equation, theory of compatibility, functional-invariant solutions. 
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     title = {Inverse {Problem} of the {Theory} of {Compatibility} and {Functional-Invariant} {Solutions} {Wave} {Equation} in {Two-Dimensional} {Space}},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
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M. V. Neshchadim. Inverse Problem of the Theory of Compatibility and Functional-Invariant Solutions Wave Equation in Two-Dimensional Space. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 14 (2012), pp. 99-107. http://geodesic.mathdoc.fr/item/VYURU_2012_14_a9/

[1] Sobolev S. L., “Functionally invariant solutions to the wave equations”, Trudy Math. Inst. Steklov, 5, 1934, 259–264 | Zbl

[2] Pommaret J. F., Systems of Partial Differential equations and Lie Pseudogroups, Gordon and Breach, N. Y., 1978, 398 pp. | MR | MR | Zbl | Zbl

[3] Finikov S. P., The Method of Cartan's Exterior Forms, Gostehizdat, M.–L., 1948, 432 pp.

[4] Sidorov A. F., Shapeev V. P., Yanenko N. N., The Method of Differential Relations and its Applications in Gas Dynamics, Izdatel'stvo Nauka, Novosibirsk, 1984, 272 pp. | MR