Obtaining a Priori Error Estimate An Approximate Solution for a Parabolic Type Problem with a Divergent Principal Part

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. Such nonclassical problems with boundary conditions containing the time derivative of the required function arise in a number of applied problems, for example, when a homogeneous isotropic body is placed in an inductor of an induction furnace and an electromagnetic wave is incident on its surface, or its points, washed off with a well-mixed liquid. Similar linear problems of parabolic type with boundary conditions containing the time derivative of the required function arise in the study of the thermal regime of the bed of a wide high-water river. Such problems have been little studied, therefore, the study and solution of problems of parabolic type, when the boundary condition contains the time derivative of the desired function, is relevant and in demandIn this article, a generalized solution to the problem under consideration is defined in the space . The aim of the study is to obtain an a priori estimate of the error of the approximate solution in the norm. The proposed boundary value problem is considered under certain conditions for the function involved in the equation and the boundary condition, which allow the existence and uniqueness of the generalized solution.  For the numerical solution of the problem under consideration, an approximate solution was constructed by the Bubnov-Galerkin method for the considered nonclassical parabolic problem with a divergent principal part, when the boundary condition contains the time derivative of the desired function.  Under certain conditions for the boundary of the domain, as well as for the coefficients and functions involved in the problem under consideration, we obtained an a priori estimate of the error in the approximate solution of the Bubnov-Galerkin method.