Крайова задача з м’якими межами для рівнянь параболічного типу з операторами Бесселя — Ейлера — Лежандра на сегменті кусково-однорідної полярної осі

Abstract

Composite materials are widely used in a variety of technological process-es, building, energy saving, in connection with which there is a need to pose and solve problems of thermal conductivity in environments that are heteroge-neous in their structure (multilayer bodies). At the same time, the heterogeneity of the medium leads to the consideration of boundary value problems with piecewise continuous or piecewise constant coefficients [3] and differential op-erators of the Bessel, Euler, Legendre, and Fouriertype, which model the het-erogeneity of the medium in terms of a geometric variable.In the classical setting, the processes of heat propagation were studied under the assumption that the boundary of the medium is rigid in relation to the reflection of waves. However, if we assume that wave absorption can occur at the boundary of the medium (soft boundary), we obtain a boundary value problem containing a time derivative in the operators of boundary conditions and conjugation conditions of the form (1).The analytical solution of the corresponding boundary value problem can be obtained using integral transformations with a spectral parameter, which work for problems with soft boundaries according to the same logi-cal scheme as integral transformations without a spectral parameter in problems with hard boundaries.This paper is devoted to the construction of one class of such hybrid in-tegral transformations generated by a hybrid differential operator of the Bessel-Euler-Legendre type on the polar axis.In this article an integral image of the exact analytical solution of a mixed problem for parabolic equations on a three-complex segment of the polar axis with soft boundaries is obtained by the method of a hybrid inte-gral transformation of the Bessel-Euler-Legendre type under the assump-tion that the boundary conditions and conjugation conditions contain a de-rivative with respect to the time variable