Analytical approaches to the analysis of unsteady heat conduction for partially bounded regions

A mathematical theory is developed for the construction of integral transforms for the following partially bounded regions: a space with an internal cylindrical cavity in the cylindrical coordinates (radial heat flux); a space with an internal spherical cavity in the spherical coordinates (central symmetry); and a space bounded by a planar surface in the Cartesian coordinates. Expressions are proposed for the integral transforms, Laplace operator images, and inversions for images. The formulated approach differs from the classical theory of differential equations of mathematical physics for the construction of integral transforms with a continuous spectrum of eigenvalues based on the corresponding singular Sturm–Liouville problems. The proposed method is based on the operational solutions of the initial boundary-value problems of unsteady heat conduction with an inhomogeneous initial function and homogeneous boundary conditions. The formulated approach makes it possible to develop the Green’s function method and to construct integral representations of the analytical solutions of the boundary-value problems simultaneously based on the Green’s functions and inhomogeneities in the main equation and boundary conditions of the problem. The proposed functional relations can be used in numerous particular cases of practical thermal physics. Examples of the application of the obtained results in some fields of science and technology are presented.