The article considers one boundary value problem of parabolic type with a divergent main part, when the boundary condition contains the time derivative of the desired function. Such non-classical problems arise in a number of applied problems, for example, when a homogeneous isotropic body is placed in the inductor of an induction furnace and an electromagnetic wave falls on its surface. Such problems have been little studied, so the study of problems of parabolic type, when the boundary condition contains the time derivative of the desired function, is relevant. The work defines a generalized solution to the problem under consideration in the space  The purpose of the study is to prove the theorem of the existence and uniqueness of an approximate solution of the Bubnov-Galerkin method for the considered non-classical parabolic problem with a divergent main part, when the boundary condition contains the time derivative of the desired function.

Mixed problems, quasilinear equation, boundary condition

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