# The Stationary Thermal Field in a Multilayer Elliptic Cylinder
## 1 Introduction
Paper [23] provided an analytical solution to the heat conduction equation, which modeled the steady distribution of temperature in a multilayer spherical structure. The boundary conditions corresponded to different combinations of conduction, convection and radiation. The method of separation of variables was used. The coefficients in the obtained Fourier-Lagrange series were determined from the system of equations using the Thomson recurrent algorithm [23]. Studies devoted to three-dimensional (3D) multilayer systems are rarely found. An example could be paper [24]. Study [24] determined the transient three-dimensional thermal field in a cylinder composed of multiple radial layers. The eigenfunction expansion method was employed. In the angular and axial directions, an arbitrary combination of the homogeneous boundary conditions of the first and third kind was assumed. In the radial direction, in turn, the inhomogeneous boundary conditions of the third kind were assumed. An example of a different configuration was examined in reference [25]. A transient thermal field in two flat plates with a contact resistance between them was analyzed. On the upper and lower bases of the system, the boundary conditions of the first, second and third kind were assumed. On the vertical (lateral) walls, the boundary conditions of the first and second kind were assumed. The Green function method was employed. To the authors' knowledge, only homogeneous elliptic systems with Dirichlet and Neumann boundary conditions have been calculated by analytical methods in the literature. The following papers [26; 27; 28; 29; 30; 31] fall within this scope. In [26], an analysis of thermally developing forced convection flow of water in aluminum metal-aluminum foam elliptic annulus was presented. The stationary heat equation with velocity term was solved. The following boundary conditions were assumed: adiabatic, the Dirichlet (first-type) and the Neumann (second-type). The boundary problem was solved using the superposition method with separation of variables. In [27], an elliptical duct heat exchanger model with periodically varying inlet temperature was considered. The heat equation with velocity terms was solved. Adiabatic conditions and conditions of the first kind (Dirichlet) were assumed. The generalized integral transform technique (GITT) method was used. In the paper [28], the transient heating of an elliptical annulus was considered. On its inner surface, boundary conditions of the first (Dirichlet) or second (Neumann) type were assumed. A constant temperature (i.e., conditions of the first type) was assumed on the outer surface of the elliptical annulus. The heat equation was solved using Fourier series and separation of variables methods. The textbook [29] considered elliptically shaped systems (an elliptical hole in a wide plate and a vacuum diode). In the first case, the Laplace equation was solved and in the second case, the Poisson equation. These equations also described the stationary heat flow. In the first example, a condition analogous to adiabatic was assumed. In turn, in the case of the diode, boundary conditions of the first type (Dirichlet) were used. The separation of variables method was used. In the paper [30], the hydrodynamic pressure distribution exerted on a hollow elliptical cylinder by water during an earthquake was calculated analytically. The pressure distribution was described in an elliptical coordinate system using the Laplace equation (analogous to the stationary heat equation). The right-hand side of the boundary condition equations does not depend on the pressure distribution sought. The problem was solved using the separation of variables method. The paper [31] presented the derivation of the Green's function of the wave equation in an elliptic region with boundary conditions of the first type. The mentioned function was presented in the form of a double series of Mathew functions. They were eigenfunctions of the Helmholtz operator in the elliptic region under consideration.