On the Estimation of the Generalized Solution of a Certain Problem of Parabolic type with a Boundary Condition Containing by Time from the Required Function.

Abstract

The article considers a parabolic-type boundary value problem with a divergent principal part, when the boundary condition contains the time derivative of the required function.

In such non-classical problems with boundary conditions, in which the electromagnetic wave is located, or if the surface of the body is the same, the temperature is the same at all its points, it is washed off with a good mixed liquid.Such problems have been little studied, therefore, the study of problems of parabolic type, when the boundary condition contains the time derivative of the desired function, is relevant.In this paper, a generalized solution of the problem under consideration is defined in the space The aim of the study is to obtain conditions under which the estimate of the error of the approximate solution in the norm has order The article first considers an auxiliary problem of elliptic type.Under certain conditions, the conditions are involved in the equation and the boundary condition, estimates are obtained for the solution of the auxiliary elliptic problem under consideration.Further, under the condition that the problem is elliptic, inequalities are obtained for the difference of the generalized solution of the considered parabolic problem with a divergent principal part, when the boundary condition contains the time derivative of the desired function and the solution of the auxiliary elliptic problem.These estimates allow obtaining estimates of the error of the approximate solution of the Bubnov-Galerkin method for the solution in the norm having order for the considered nonclassical parabolic problem with divergent principal part, when the boundary condition contains the time derivative from the required function.