On some estimates of solutions to the problem of heat conduction in a multilayer medium by the matrix method

In the study of real thermal processes by mathematical modeling it is important to have analytical or approximate analytical methods of solution, which can simplify the analysis of processes, give an opportunity to predict the behavior of individual materials of the design, to identify possible undesirable phenomena, etc. One such analytical method can be the joint application of the matrix method, the Fourier method and the Bers generalized degree method. The paper considers a one-dimensional heat conduction process in a multilayer plate with continuous matching conditions at the contact points of the layers. The system consists of $n$ flat layers that make up the plate. In each layer, the basic system of equations, which determines the process in a multilayer plate, consists of equations of thermal conductivity, with specified heat conductity coefficient: tensity, specific heat, as well as conditions of agreement of type and contact, Consisting of continuous temperature and flow at the contact limits of the layers. At the initial moment, the temperature is set and zero temperature is always maintained in each layer and on the boundaries of the first and last layers. An algorithm for solving the problem is briefly described. The algorithm is based on a combination of the method of separation of variables (Fourier method) and the matrix method. This algorithm makes it relatively easy to find exact analytical solutions in the form of an infinite series for any finite number of layers of the medium. Calculations were carried out using this algorithm. Symmetric multilayer media were chosen for calculations, i.e. layers with the same thermophysical parameters were located in the plate symmetrically with respect to the middle of its thickness. The layers were chosen to be of equal thickness. The materials of the layers were chosen with a significant difference in the thermal conductivity, since it is this parameter that most affects the shape of the solution graph, and its sharp difference in neighboring layers makes it possible to qualitatively evaluate the result for compliance with the real physical process. The initial temperature distribution was also chosen to be symmetrical with respect to the middle of the plate thickness. In such media, the solution of the problem must also have a symmetrical form. Estimates were obtained for the convergence of the found solutions with respect to the space norm $L_2$
$${\Delta _{k,r}} = \frac{{{{\left\| {{T_k}(x,t) - {T_r}(x,t)} \right\|}_{{L_2}}}}}{{{{\left\| {{T_k}(x,t)} \right\|}_{{L_2}}}}} \cdot 100\% ,$$
and also estimates of the convergence of the initial temperature distribution $T(x,0)$
$${\Delta _{t = 0;{\text{ }}k,r}} = \frac{{{{\left\| {{T_k}(x,0) - {T_r}(x,0)} \right\|}_{{L_2}}}}}{{{{\left\| {{T_k}(x,0)} \right\|}_{{L_2}}}}} \cdot 100\% ,$$
where the indices $k$ and $r$ indicate how many terms of the Fourier series are taken in the solution.