On the construction of solution of the heat equation in a multilayer medium with imperfect contact between the layers

The paper considers the solution of a one-dimensional homogeneous equation of heat conduction in a multilayer
$$ {a^{(i)}_2}(x)\frac{\partial }{{\partial x}}\left( {{a^{(i)}_1}(x)\frac{{\partial T^{(i)}}}{{\partial x}}} \right) = \frac{{\partial T^{(i)}}}{{\partial t}}, \quad i = \overline {1,n}$$
where the superscript in parentheses indicates the layer number, $a_1(x)$ and $a_2(x)$ depend on the geometrical and physical parameters of the layer. The flow is directed along the axis $ x $ Matching conditions of the third type are accepted at the contact points of the layers
$$ {T^{(i + 1)}}({x_{i + 1}},t) - {T^{(i)}}({x_{i + 1}},t) = - {r^{(i + 1)}}{J^{(i)}}({x_{i + 1}},t), $$

$$ {J^{(i)}}({x_{i + 1}},t) = {J^{(i + 1)}}({x_{i + 1}},t),\,\,\,i = \overline {1,n - 1} , $$
where $r^{(i + 1)}$ is a thermal resistance coefficients at the contact points of the layers $x_{i+1}$ and $J^{(i)}$ is the flow.
The initial temperature distribution is given
$$ {T^{(i)}}(x,0) = g(x),\quad x \in [{x_i},{x_{i + 1}}],\quad i = \overline {1,n} $$
and the first boundary value problem is posed
$$ {T^{(1)}}({x_1},t) = 0,\quad {T^{(n)}}({x_{n + 1}},t) = 0. $$

The solution is constructed by combining the Fourier method, the matrix method and the method of generalized powers of Bers. Previously, this approach was used to construct solutions of the heat equation under continuous matching conditions at the layer boundary.
The method of generalized powers makes it possible to obtain a unified analytical form for solving the problem for various geometries of a multilayer medium: translational, axial or central symmetry.
The essence of the matrix method is reduced to the sequential multiplication of functional matrices that depend on the physical and geometric parameters of the layers of the medium (the elements of these matrices are expressed in terms of generalized powers) and matrices describing the thermal resistance at the points of contact of the layers. Thus, it is possible to express the relationship between the value of the amplitude function at the point $x_1$ and the values of this function at any other point in the medium. Thus, it is possible to find uniform eigenvalues and the corresponding eigenfunctions for the entire medium for any finite number of layers by linking the values of the amplitude function at the boundary points $x_1$ and $x_{n+1}$ of the medium.
In this paper, the orthogonality of the obtained eigenfunctions is proved.