Parabolic Boundary Value Problems in a Piecewise Homogeneous Wedge-Shaped Cylindrical-Circular Layer with a Cavity

• Дата: 2022
• Автори: Громик, Андрій, Конет, Іван, Пилипюк, Тетяна
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2022
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/274033
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous by the radial variable z wedge-shaped by the angular variable cylindrical-circular layer with a cavity 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 the 1st kind (Dirichlet) and the 2nd kind (Neumann) and their possible combinations (Dirichlet — Neumann, Neumann — Dirichlet) are considered. Finite integral Fourier transform by an angular variable, a finite integral Fourier transform on the Cartesian segment by an applicative variable and a hybrid integral transform of the Weber type on the polar axis with n points of conjugation by a radial variable were used to construct solutions of investigated 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 consistent application of inverse integral transforms to the obtained solution in the space of images restores the solutions of the considered parabolic boundary value problems through their integral image in an explicit form in the space of the originals. At the same time, the main solutions to the problems were obtained in an explicit form.