3D Goursat Problem in the Non-Classical Treatment for Manjeron Generalized Equation with Non-Smooth Coefficients

Abstract

In this paper substantiated for a Manjeron generalized equation with non-smooth coefficients a three dimensional Goursat problem -3D Goursat problem with non-classical boundary conditions is considered, which requires no matching conditions. Equivalence of these conditions three dimensional boundary condition is substantiated classical, in the case if the solution of the problem in the isotropic S. L. Sobolev's space is found. The considered equation as a hyperbolic equation generalizes not only classic equations of mathematical physics (heat-conductivity equations, string vibration equation) and also many models differential equations (telegraph equation, Aller's equation, moisture transfer generalized equation, Manjeron equation, Boussinesq - Love equation and etc.). It is grounded that the 3D Goursat boundary conditions in the classic and non-classic treatment are equivalent to each other. Thus, namely in this paper, the non-classic problem with 3D Goursat conditions is grounded for a hyperbolic equation of sixth order. For simplicity, this was demonstrated for one model case in one of S.L. Sobolev isotropic space.W_p^((2,2,2) ) (G)