Combined Boundary Problem of 2D in the Geometrical Middle -1D in the Middle-1D Multi-Point type in the Non-Classical Treatment for 4D Bianchi Equation with Non-Smooth Coefficients, Arising in the Modeling of Vibration Processes and Integral Representation

Abstract

In this paper substantiated  for a 4D Bianchi equation with non-smooth coefficients a  four dimensional  combined boundary problem - 4D combined boundary problem of  2D in the geometrical middle-1D in the middle-1D multi-point type with non-classical boundary conditions is considered, which requires no matching conditions. Equivalence of these conditions four 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 (Laplace equation, telegraph equation, string vibration equation) and also many models differential equations (3D and 4D telegraph equation, 3D Bianchi equation, 3D and 4D wave equations   and etc.).  It is grounded that the   2D in the geometrical middle -1D in the middle -1D multi-point combined boundary conditions in the classic and non-classic treatment are equivalent to each other. Thus, namely in this paper, the non-classic problem with 4D combined boundary problem of 2D in the geometrical middle -1D in the middle -1D multi-point type conditions is grounded for a  hyperbolic equation of   fourth-order. For simplicity, this was demonstrated for one model case in one of S.L. Sobolev   isotropic space    .