## 3 The Spectral Form of the Mathematical Model  
We obtain the spectral form representation of the system (5)-(7). Representing the function \(\theta(\xi,\tau)\) in the form of a Fourier series in the system of orthonormal differential functions \(\{P_{1}(h_{1},\xi)\}\), \(\{P_{2}(h_{2},\tau)\}\):  
\[\theta(\xi,\tau)=\sum_{h_{1}}^{\infty}\sum_{h_{2}}^{\infty}\Phi_{\theta}(h_{1}, h_{2})P_{1}(h_{1},\xi)P_{2}(h_{2},\tau) \tag{10}\]  
Representing the function \(\theta(\xi,\tau)\) with zero initial conditions (Equation (7)) in the form:  
\[\theta(\xi,\tau)=\theta_{0}(\xi,\tau)+\left.\theta(\xi,\tau)\right|_{\xi=0..}1 (0-\xi)+\left.\theta(\xi,\tau)\right|_{\xi=1..}1(\xi-1), \tag{11}\]  
where the function \(\theta_{0}(\xi,\tau)\) coincides with the function \(\theta(\xi,\tau)\) in the domain \(\xi\in(0,\ 1),\ \ \tau\in(0,\ 1);\ \theta(\xi,\tau)\big{|}_{\xi=0..},\ \theta(\xi,\tau) \big{|}_{\xi=1..}\) are amplitudes of step functions, operating on the boundaries \(\xi=0_{-}\) and \(\xi=1_{+}\), respectively. The first-time derivative of Function (11) is determined by the expression [24]:  
\[\frac{\partial\theta(\xi,\tau)}{\partial\tau}=\frac{\partial\theta_{0}(\xi, \tau)}{\partial\tau}+\frac{\partial\theta_{0}^{0}(\tau)}{\partial\tau}1(0- \xi)+\frac{\partial\theta_{1}^{0}(\tau)}{\partial\tau}1(\xi-1), \tag{12}\]  
where \(\theta_{0}^{0}(\tau)=\left.\theta(\xi,\tau)\right|_{\xi=0..}\theta_{1}^{0}( \tau)=\left.\theta(\xi,\tau)\right|_{\xi=1..}\). To provide the correct expression for the second spatial derivative, the first spatial derivative is represented in the form:  
\[ \begin{array}{c}\frac{\partial\theta(\xi,\tau)}{\partial\xi}=\left.\left( \frac{\partial\theta(\xi,\tau)}{\partial\xi}\right)_{0}+\frac{\partial\theta( \xi,\tau)}{\partial\xi}\right|_{\xi=0..}1(0-\xi)+\left.\frac{\partial\theta( \xi,\tau)}{\partial\xi}\right|_{\xi=0..}1(\xi-1)+\\ +\left.\theta(\xi,\ \tau)\right|_{\xi=0..}\frac{\partial 1\left(0-\xi \right)}{\partial\xi}+\left.\theta(\xi,\ \tau)\right|_{\xi=1..}\frac{\partial 1 \left(\xi-1\right)}{\partial\xi}\,,\end{array}  \tag{13}\]  
where \(\left(\frac{\partial\theta(\xi,\tau)}{\partial\xi}\right)_{0}\) coincides with the function \(\frac{\partial\theta(\xi,\tau)}{\partial\xi}\) in the domain \(\xi\in(0,\ 1),\ \ \tau\in(0,\ 1)\). Thus, the second spatial derivative has the form:  
\[ \begin{array}{c}\frac{\partial^{2}\theta(\xi,\tau)}{\partial\xi^{2}}=\left. \left(\frac{\partial^{2}\theta(\xi,\tau)}{\partial\xi^{2}}\right)_{0}+\theta _{0}^{1}(\tau)\delta(0-\xi)+\theta_{1}^{1}(\tau)\delta(\xi-1)+\\ +\theta_{0}^{0}(\tau)\frac{\partial\delta(0-\xi)}{\partial\xi}+\theta_{1}^{0}( \tau)\frac{\partial\delta\left(\xi-1\right)}{\partial\xi}\,,\end{array}  \tag{14}\]  
where \(\theta_{0}^{1}(\tau)=\left.\frac{\partial\theta(\xi,\tau)}{\partial\xi} \right|_{\xi=0..}\theta_{1}^{1}(\tau)=\left.\frac{\partial\theta(\xi,\tau)}{ \partial\xi}\right|_{\xi=1..}\) are the amplitudes of the jumps of the first derivative of the functions operating on the boundaries \(\xi=0_{-}\) and \(\xi=1_{+}\), respectively. When we have the boundary conditions of the second kind, Expression (14) is transformed as:  
\[\frac{\partial^{2}\theta(\xi,\tau)}{\partial\xi^{2}}=\left.\frac{\partial^{2} \theta_{0}(\xi,\tau)}{\partial\xi^{2}}+\theta_{0}^{1}(\tau)\delta(0-\xi)+ \theta_{1}^{1}(\tau)\delta(\xi-1)\,. \tag{15}\]  
For each of the components of Equation (5), the spectral form is obtained considering Expressions (12) and (15).