SOLUTION OF MULTILAYER PROBLEMS FOR THE HEAT EQUATION BY THE FOURIER METHOD - Barmagambetov, Sagynysh; Koilyshov, Umbetkul (Vestnik KazNU. Serii͡a︡ matematika, mekhanika, informatika, 2025)
The multilayer problems for the heat equation arise in many areas of heat and mass transfer applications. There are two main approaches to finding exact solutions to multilayer diffusion problems: separation of variables and integral transformations. The difficulty of applying the Laplace transform method is redoubled by the difficulty of finding the inverse transform. The inverse Laplace transform is often performed numerically. The most popular analytical approach to multilayer problems for the heat equation is the method of separation of variables. It is very important to obtain analytical solutions to such problems as they provide a higher level of understanding of the solution behavior and can be used for comparative analysis of numerical solutions. In this paper, the solution of the multilayer problem for the heat equation by the Fourier method is substantiated. The solution of the initial-boundary value problem for the heat equation with discontinuous coefficients by the method of separation of variables is reduced to the corresponding non-self-adjoint spectral Sturm-Liouville eigenvalue problem. Such eigenvalue problems do not belong to the ordinary type of Sturm-Liouville problems due to the discontinuity of the heat conductivity coefficients. In addition, the non-self-adjointness of the corresponding spectral problem also complicates the solution of the problem. Using the replacement, the problem is reduced to a self-adjoint spectral problem and the eigenfunctions of this problem forming an orthonormal basis are constructed. The considered problem models the process of heat propagation of the temperature field in a thin rod of finite length, consisting of several sections with different thermal-physical characteristics. In this problem, in addition to the boundary conditions of the Sturm type, the conditions of conjugation at the point of contact of different media are specified. The existence and uniqueness of the classical solution of the considered multilayer problem for the heat conduction equation are proved.