## 4 Example  
Let us calculate matrices of the spectral representation in Equation (27) for the System (2)-(4) with parameters \(L=0.7\) m, \(d=0.15\) m, \(F=1.18\cdot 10^{-2}\) m\({}^{2}\), \(a=1.26\cdot 10^{-5}\) m\({}^{2}\)/s, \(\alpha=14.54\) W\(/\big{(}\)m\({}^{2}\cdot\)K\(\big{)}\), \(\lambda=46.52\) W\(/\)(m\(\cdot\)K), \(q_{1}(t)=34.89\big{(}1-e^{-3600t}\big{)}\)W, \(q_{2}(t)=52.34\)\((1-e^{-3600t})\)W [23]. As scaling factors, we choose \(t^{*}=10^{5}\) s, \(T^{*}=1\) K. Then \(\xi=1.43x\), \(\tau=10^{-5}t\). Equation (5) of the system in dimensionless form is:  
\[ \begin{split}\frac{\partial\theta(\xi,\tau)}{\partial\tau}=2.57 \frac{\partial^{2}\theta(\xi,\tau)}{\partial\xi^{2}}-10.5\,\theta(\xi,\tau), \\ \xi\in(0,\,1),\,\,\tau\in(0,\,1)\end{split}  \tag{29}\]  
The boundary conditions (Equation (6)) are in the form:  
\[ \begin{split}\frac{\partial\theta(\xi,\tau)}{\partial\xi}\Big{|} _{\xi=0}&=-29.71\left(1-e^{-3.6\cdot 10^{8}\tau}\right),\\ \frac{\partial\theta(\xi,\tau)}{\partial\xi}\Big{|}_{\xi=1}& =44.56\left(1-e^{-3.6\cdot 10^{8}\tau}\right),\\ \tau\in[0,\,1]\end{split}  \tag{30}\]  
The initial condition:  
\[\theta(\xi,0)=0,\,\,\xi\in[0,1] \tag{31}\]  
As a decomposition system in the variable \(\xi\), we use the system of functions:  
\[P_{1}(h_{1},\xi)=\Big{\{}\,1\quad\sqrt{2}\cos(\pi\xi)\quad\sqrt{2}\cos(2\pi \xi)\quad\dots\Big{\}} \tag{32}\]  
orthonormal on the interval \(\xi\in[0,\,1]\). As a decomposition system in the variable \(\tau\), we use the system of functions:  
\[P_{2}(h_{2},\tau)=\big{\{}\,\sqrt{2}\sin\tfrac{\pi\tau}{2}\quad\sqrt{2}\sin \tfrac{3\pi\tau}{2}\quad\sqrt{2}\sin\tfrac{5\pi\tau}{2}\quad\dots\,\,\,\big{\}}\,, \tag{33}\]  
orthonormal on the interval \(\tau\in[0,\,1]\). The choice of expansion functions is determined by the intended form of decision and by the fact that vectors of boundary conditions of the second kind (Equation (19)) should be non-zero. The operating differentiation matrices \(P_{2}^{\xi},\;P_{1}^{\tau}\) computed by the Expressions (17) and (21), respectively, for the selected decomposition Systems (32), (33), are:  
\[P_{2}^{\xi}=diag\big{\{}\;0,\;\;-9.87,\;\;-39.48,\;\;-88.83,\;\;-157.91,\;\; \dots\;\big{\}}, \tag{34}\]  
\[P_{1}^{\tau}=\left[ \begin{array}{cccccc}1.00&-3.00&1.67&-2.33&1.80&\cdots\\ 1.00&1.00&-5.00&1.40&-3.00&\cdots\\ 0.33&3.00&1.00&-7.00&1.29&\cdots\\ 0.33&0.60&5.00&1.00&-9.00&\cdots\\ 0.20&1.00&0.71&7.00&1.00&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array} \right]. \tag{35}\]  
In the case of constant heat transfer coefficient, the operating matrix \(P_{k}^{\tau}\) is defined as \(P_{k}^{\tau}=ht^{*}E\). We obtain vectors \(\Gamma_{01}^{10},\Gamma_{01}^{11}\) by Expression (19). The matrices of the composed System (27) are determined by the calculated matrices \(P_{1}^{\tau},P_{2}^{\xi}\) and are not shown due to their high dimension. After computing the spectral characteristic vector \(\Phi_{\theta}\), according to Equation (28) it is possible to construct a function \(\theta(\xi,\tau)\) by the expression:  
\[\theta(\xi,\tau)=\sum\limits_{h_{1}=1}^{n_{1}}\sum\limits_{h_{2}=1}^{n_{2}} \Phi_{\theta}((h_{1}-1)n_{2}+h_{2})P_{1}(h_{1},\xi)P_{2}(h_{2},\tau)\;. \tag{36}\]  
The number of terms in Equation (36) should provide the required accuracy estimated by the expression:  
\[\Delta=\frac{\int\limits_{0}^{1}\int\limits_{0}^{1}\overline{\theta}(\xi, \tau)-\theta(\xi,\tau)\big{|}^{2}d\tau d\xi}{\int\limits_{0}^{1}\int \limits_{0}^{1}\overline{\theta}^{2}(\xi,\tau)d\tau d\xi}, \tag{37}\]  
where \(\theta^{-}(\xi,\tau)\) is the exact solution,\(\theta(\xi,\tau)\) is the solution in the form of Equation (36). In general, the exact solution is unknown but we can use the analytical solution defined for some special cases. For example, in the case of a constant heat transfer coefficient, we can use the expression for the analytical solution \(\overline{\theta}(\xi,\tau)\) defined via [PERSON]'s function [25]:  
\[