Parabolic Boundary Value Problems in Unlimited Piecewise Homogeneous Wedge-Shaped Hollow Cylinder

Abstract

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in unbounded by variable z piecewise-homogeneous by radially variable r wedge-shaped by an angularly variable φ hollow cylinder were constructed at first time by the method of classical integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices) in the proposed article. The cases of assigning on the verge of the wedge the boundary conditions of Dirichlet and Neumann and their possible combinations (Dirichlet — Neumann, Neumann — Dirichlet) are considered. Finite integral Fourier transform by an angular variable, a Fourier integral transform on a Cartesian axis by an applicative variable and a hybrid integral transform of the Hankel type of the second kind on a segment of the polar axis with n points of conjugation were used to construct classic solutions of investigated initial-boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The application of inverse integral transforms restores explicitly the solution of the considered problems through their integral image.