Дисертація на здобуття наукового ступеня доктора технічних наук за спеціальністю 01.05.02 – математичне моделювання та обчислювальні методи. –Національна металургійна академія України, Днiпро, 2020.У дисертації вирішено важливу науково-технічну проблему підвищення точності моделювання явища теплопровідності в тілах, що обертаються, з урахуванням релаксації теплового потоку, що дає змогу підвищити точність розрахунку температурних полів у цих тілах. У роботі вперше отримано диференційне узагальнене рівняння переносу енергії для рушійного елемента суцільного середовища з урахуванням кінцевої швидкості поширення тепла в криволінійній системі координат. Також вперше побудовано нове кінцеве інтегральне перетворення для рівняння Лапласа в довільній області , обмеженій декількома замкненими кусково-гладкими контурами, і нове кінцеве інтегральне перетворення для рівняння Лапласа в циліндричній системі координат для області .Отримані результати дозволили вперше побудувати нові математичні моделі процесу теплопровідності в тілах обертання у вигляді узагальнених (із урахуванням релаксації теплового потоку) крайових задач для гіперболічного рівняння теплопровідності і розробити нові методи розв'язування відповідних крайових задач.^UA thesis for obtaining a scientific degree of the Doctor of Technical Sciences in the specialty 01.05.02 – Mathematical Modeling and Computational Methods. – Dnipro University of Technology. – National Metallurgical Academy of Ukraine, Dnipro, 2020.The work is devoted to the pressing scientific and technical problem of improving the accuracy of modeling the thermal conductivity phenomenon in rotating bodies through taking into consideration a heat flow relaxation that enables to increase accuracy of temperature fields computation in rotating bodies. For the first time a differential generalized energy transfer equation for the driving element of a continuous medium, taking into account the finite velocity of heat propagation in a curvilinear coordinate system, was obtained in the thesis. A new finite integral transformation for the Laplace equation in the arbitrary domain , bounded by several closed partially-smooth contours, and a new integral transformation for the Laplace equation in a cylindrical coordinate system for the domain were developed for the first time as well. Finding the nuclei of the constructed finite integral transformations by the finite element method and the Galerkin method for first- and second-order simplex elements were reduced to the solution of a system of algebraic equations. The obtained results allowed to build new mathematical models of the thermal conductivity process in rotating bodies in the form of generalized boundary value problems (taking into account the heat flow relaxation) for the hyperbolic equation of thermal conductivity, and to develop new methods for solving corresponding boundary value problems. Mathematical models for calculating temperature fields in solid and hollow cylinders, as well as in solid and empty two-component cylinders of finite length rotating with a constant angular velocity, taking into account the finite velocity of heat propagation, were built in the form of generalized boundary value problems for hyperbolic equations of thermal conductivity with the Dirichlet and Neumann boundary conditions. Using finite integral transformations for piecewise homogeneous media, as well as the Hankel, Fourier, and Laplace transforms, the temperature fields in the cylinders were found in the form of converging orthogonal series by Bessel and Fourier functions.Mathematical models for calculating temperature fields in an isotropic and hollow isotropic body rotating with a constant angular velocity were constructed, taking into account the finite velocity of heat propagation as boundary value problems for hyperbolic equations of thermal conductivity with the Dirichlet and Neumann boundary conditions and mixed conditions for a hollow isotropic body. Using the developed new integral transformation for a two-dimensional finite space, the temperature fields in the isotropic and empty isotropic rotating bodies in the form of convergent series by the Fourier functions depending on the boundary conditions were found.For the first time mathematical models for calculating temperature fields in a paraboloid, a hemispherical body, a thin-walled single-cavity hyperboloid and in a straight circular cone, which rotate with a constant angular velocity, were developed, taking into account the finite velocity of heat propagation in the form of boundary value problems for the thermal conductivity hyperbolic equation with the Dirichlet boundary conditions. With the help of a new integral transformation developed for the 2D finite space, temperature fields were found in these bodies in the form of convergent series by using the Fourier functions.For the first time a mathematical model for calculating temperature fields in arbitrary domains during the electron beam welding was constructed in the form of a boundary value problem of mathematical physics with the Dirichlet boundary conditions. Heat flow in the body during the welding was simulated by a point source of heat moving along the contour of the body with a constant velocity and known intensity by using the Dirac function. With the help of a new integral transformation developed for the Laplace equation and the finite element method in the Galerkin form, a temperature field in the form of a convergent series was found.