Mixed Problem for the Differential Equation of Parabolic Type with Measures

Abstract

We propose a scheme for the solution of a mixed problem for a parabolic differential equation with coefficients that are generalized derivatives of functions of bounded variation. We seek the solution of this problem by the method of reduction. According to this method, the solution of the proposed problem is reduced to the solution of two problems: (i) a quasistationary boundary-value problem with input boundary conditions and (ii) a mixed problem with trivial boundary conditions. The first of these problems is solved by introducing the quasiderivative. For the solution of the second problem, we use the Fourier method and the expansion in eigenfunctions of a certain boundary-value problem for a quasidifferential equation of the second-order. The obtained results can be used, in particular, for the investigation of the processes of heat transfer in multilayer plates, hollow cylinders, and spheres.