# The Stationary Thermal Field in a Multilayer Elliptic Cylinder
## 1 Introduction  
Multilayer systems composed of different materials are often found in technology. By combining different physical and materials properties of individual layers, elements of improved parameters as compared to the homogeneous structure are obtained. The multilayer systems under discussion are found in the automotive and building industries, as well as in thermal and nuclear engineering and many other applications. Multilayer systems can be investigated using analytical and numerical methods. The latter form the basis for numerous commercial software programs (such as Comsol Multiphysics [1], Nisa [2]). With their help, the nonlinearity and inhomogeneity of the medium can be easily considered. In the study of multilayer systems, analytical methods are also willingly used. Their main advantages include: the results in the form of formulas, higher accuracy, no need for a discretizing grid and better physical interpretation, compared to numerical methods. In the present study, a computer-aided analytical method was developed. For this reason, the literature review below applies exclusively analytical methods employed in the calculation of multilayer systems. Many authors have analyzed one-dimensional (1D) distributions of temperature in multilayer flat, cylindrical and spherical systems. For solving one-dimensional problems, the orthogonal expansion technique [3, 4, 5, 6], the Green function method [7, 8, 6], the Laplace transform technique [9, 10, 11], as well as other methods [12, 13, 14] were used. For example,in [15], an analytical solution of the heat conduction equation in a one-dimensional bar was presented. A combination of the method of separation of variables with the orthogonal expansion technique was used therein. Reference [16] examined a multilayer flat structure with the asymmetric boundary conditions of the third kind and with heat sources. The multilayer structure was reduced to a single layer, whose properties were continuous by sections. Study [17] also examined a one-dimensional thermal field in a layered flat solid body. An arbitrary heat source and time-dependent boundary conditions were considered. The method of separation of variables was employed. In turn, the multilayer structure of a hollow cylinder was analyzed in [18]. The coefficients of the one-dimensional heat conduction equation were assumed to be constant in each layer. Heat exchange on the inner and outer surfaces of the system was described using the convection condition. The Fourier method, the modified method of eigenfunctions and the concept of quasi derivative were employed. More mathematically advanced methods are the analyses of the two-dimensional (2D) thermal field in multilayer systems. In this respect, a flat configuration was dealt with in reference [19]. In [19], the method of separation of variables and the boundary conditions of the third kind were used. The solution was the product of two separate one-dimensional solutions. In turn, two-dimensional cylindrical configurations, examined in polar coordinates (\(r\), \(\varphi\)), were shown in references [20; 21]. The subject of investigation in [20] was a multilayer sector of an annulus, while [21] dealt with a solid multilayer annulus. Both of the aforementioned studies employed the method of separation of variables and the superpositions of states, and different boundary conditions were assumed. Multilayer spherical configurations were examined in papers [22; 23]. In [22], transient heat flow was modelled in spherical coordinates (\(r\), \(\Theta\)). A hollow spherical body of an arbitrary number of layers was considered. The inhomogeneous boundary conditions of the first, second and third kind, respectively, were applied at the inner and outer borders of concentric layers. The method of separation of variables was employed. The solution was given for a hemisphere and full-sphere multilayer configuration.