Параболічні крайові задачі в кусково-однорідному клиновидному циліндрично-круговому просторі з порожниною

Abstract

By the method of integral and hybrid integral transforms, in combination with the method of main solutions (matrices of influence and Green matrices) the only exact analytical solutions of the parabolic boundary value problems of mathematical physics in a piecewise homogeneous wedge-shaped cylindrical circular space with a cylindrical cavity were constructed for the first time.The cases of the Dirichlet and Neumann boundary conditions and their possible combinations (Dirichlet-Neumann, Neumann-Dirichlet) on the edges of the wedge are considered.The finite integral Fourier transform relative to the angular variable, Fourier integral transform on the Cartesian axis relative to the variable z and Weber-type hybrid integral transform on the polar axis with n conjugation points relative to the radial variable are used to construct solutions.The sequential application of integral transforms allows us to reduce the three-dimensional initial-boundary value problems to the Cauchy problem for the ordinary linear non-uniform differential equation of the 1st order, the only solution of which is written in a closed form.The use of inverse integral transforms restores the solutions of the considered problems in explicit form through their integral image.