## 3 The Spectral Form of the Mathematical Model
According to [22], the spectral characteristic of the derivative \(\frac{\partial^{2}\theta(\xi,\tau)}{\partial\xi^{2}}\) in the case of the boundary conditions of the second kind has the form:  
\[Sp\bigg{[}\frac{\partial^{2}\theta(\xi,\tau)}{\partial\xi^{2}}\bigg{]}_{\xi \tau}=P_{20}\Phi_{\theta}+\Gamma_{20}^{10}+\Gamma_{20}^{11}\,, \tag{16}\]  
where \(\Phi_{\theta}\in R^{n_{1}n_{2}}(n_{1},n_{2}=\overline{1,\infty})\) is the vector of the spectral characteristic with respect to the spatial and temporal variables, the index \(i\) of this vector is calculated as\(l_{2}\big{(}h_{1}=\overline{1,n_{1}},h_{2}=\overline{1,n_{2}}\big{)}\); \(P_{20}\) is the operating differentiation matrix of second-order for the variable \(\xi\), defined as the tensor product:  
\[P_{20}=P_{2}^{\xi}\otimes I_{n_{2}} \tag{17}\]  
where \(P_{2}^{\xi}\) is the operating differentiation \(n_{1}\times n_{1}\) matrix of second-order \(\big{(}n_{1}=\overline{1,\infty}\big{)}\), with the elements calculated by the expression:  
\[ \begin{array}{c}P_{2}^{\xi}\Big{(}\overline{h}_{1},h_{1}\Big{)}=\int\limits \limits_{0}^{1}P\Big{(}\overline{h}_{1},\xi\Big{)}\frac{\partial^{2}P(h_{1}, \xi)}{\partial\xi^{2}}d\xi,\\ \overline{h}_{1},h_{1}=\overline{1,n_{1}},\,n_{1}=\overline{1,\infty};\end{array}  \tag{18}\]  
\(I_{n_{2}}\) is the identity \(n_{2}\times n_{2}\) matrix \(\big{(}n_{2}=\overline{1,\infty}\big{)}\); \(\Gamma_{20}^{10}\), \(\Gamma_{20}^{11}\) are vectors of boundary conditions of the second kind, determined by the expressions:  
\[ \begin{array}{c}\Gamma_{20}^{10}=\Big{\lfloor}-P(\xi)|_{\xi=0}\otimes I_{n _{2}}\Big{\rfloor}\Phi_{\theta_{0}^{1}},\\ \Gamma_{20}^{11}=\Big{\lfloor}P(\xi)|_{\xi=1}\otimes I_{n_{2}}\Big{\rfloor} \Phi_{\theta_{1}^{1}}.\end{array}  \tag{19}\]  
\(P(\xi)\) is the vector composed of functions of the System (9); \(\Phi_{\theta_{0}^{1}},\Phi_{\theta_{1}^{1}}\in R^{n_{1}},n_{1}=\overline{1,\infty}\) are vectors of spectral characteristics of the functions \(\theta_{0}^{1}(\tau),\theta_{1}^{1}(\tau)\). The spectral characteristic of the derivative \(\frac{\partial\theta(\xi,\tau)}{\partial\tau}\) with zero initial conditions has the form [22]:  
\[Sp\Big{[}\frac{\partial\theta(\xi,\tau)}{\partial\tau}\Big{]}_{\xi\tau}=P_{01 }\Phi_{\theta}, \tag{20}\]  
where:  
\[P_{01}=I_{n_{1}}\otimes P_{1}^{\tau}, \tag{21}\]  
\(P_{1}^{\tau}\) is the operating differentiation \(n_{2}\times n_{2}\) matrix of the first-order \(\big{(}n_{2}=\overline{1,\infty}\big{)}\), with the elements calculated by the expression:  
\[ \begin{array}{c}P_{1}^{\tau}\Big{(}\overline{h}_{2},h_{2}\Big{)}=\int\limits _{0}^{1}P_{2}\Big{(}\overline{h}_{2},\tau\Big{)}\frac{\partial P_{2}(h_{2}, \tau)}{\partial\tau}d\tau,\\ \overline{h}_{2},h_{2}=\overline{1,\,n_{2}},\,n_{2}=\overline{1,\infty}\,. \end{array}  \tag{22}\]  
Let us note that we have only one term in Expression (20) of the time derivative's spectral characteristic which corresponds to the spectral characteristic of \(\frac{\partial\theta_{0}(\xi,\tau)}{\partial\tau}\). The spectral characteristics of the terms \(\frac{\partial\theta_{0}^{0}(\tau)}{\partial\tau}1(0-\xi),\frac{\partial\theta _{1}^{0}(\tau)}{\partial\tau}1(\xi-1)\) are equal to zero. The spectral characteristic of the term \(k(\tau)\theta(\xi,\tau)\) has the form [22]:  
\[Sp[k(\tau)\,\theta(\xi,\tau)]_{\xi\tau}=P_{k}\Phi_{\theta}, \tag{23}\]  
where \(P_{k}\) is the operating matrix of the factor \(k(\tau)\), defined as the tensor product:  
\[P_{k}=I_{n_{1}}\otimes P_{k}^{\tau}, \tag{24}\]  
\(P_{k}^{\tau}\) is the \(n_{2}\times n_{2}\)-matrix \(\big{(}n_{2}=\overline{1,\infty}\big{)}\), with the elements calculated by the expression:  
\[ \begin{array}{c}P_{k}^{\tau}\Big{(}\overline{h}_{2},h_{2}\Big{)}=\int\limits _{0}^{1}k(\tau)P_{2}\Big{(}\overline{h}_{2},\tau\Big{)}P_{2}(h_{2},\tau)d\tau, \\ \overline{h}_{2},h_{2}=\overline{1,n_{2}},\,n_{2}=\overline{1,\infty}\,. \end{array}