## 3 Mathematical Models
### Inverse Heat Transfer Problem  
Let us consider the following problem, in which the measured temperature values are known in several places in the considered domain at different moments of time. In the case of the one-dimensional axisymmetric cylindrical domain, the sensors should be placed at different positions of the radial coordinate of the cylinder in order to capture the most diverse experimental measurement results. Let \(pos_{s}\), for \(s=1,\ldots,M\), denote the locations of \(M\) sensors on the radius axis \(r\). It is assumed that the exact positions \(pos_{s}\) are not known in advance and they will be estimated as a result of solving the inverse heat transfer problem. Let \(t_{f}\), for \(f=0,\ldots,F\), denote \(F+1\) moments of time that are known, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(k\) & \(\mu_{k}\) & \(k\) & \(\mu_{k}\) & \(k\) & \(\mu_{k}\) \\ \hline
1 & 2.404826 & 15 & 46.341188 & 200 & 627.533332 \\
2 & 5.520078 & 16 & 49.482610 & 300 & 941.692531 \\
3 & 8.653728 & 17 & 52.624052 & 400 & 1255.851763 \\
4 & 11.791534 & 18 & 55.765511 & 500 & 1570.011008 \\
5 & 14.930918 & 19 & 58.906984 & 600 & 1884.170260 \\
6 & 18.071064 & 20 & 62.048469 & 700 & 2198.329516 \\
7 & 21.211637 & 30 & 93.463719 & 800 & 2512.488774 \\
8 & 24.352472 & 40 & 124.879309 & 900 & 2826.648034 \\
9 & 27.493479 & 50 & 156.295034 & 1000 & 3140.807295 \\
10 & 30.634606 & 60 & 187.710827 & 2000 & 6282.399929 \\
11 & 33.775820 & 70 & 219.126658 & 3000 & 9423.992576 \\
12 & 36.917098 & 80 & 250.542513 & 4000 & 12565.585226 \\
13 & 40.058426 & 90 & 281.958384 & 5000 & 15707.177878 \\
14 & 43.199792 & 100 & 313.374266 & 10,000 & 31415.141142 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selected values of positive zeros of the Bessel function of order zero, i.e., \(J_{0}(\mu)=0\). The values of \(t_{f}\) can be arbitrary (but it is recommended that they are equidistant), while \(t_{F}\) is the final time. The measured temperature values can be tabulated as  
\[\hat{T}_{s,f}\equiv T_{measurement}(pos_{s},t_{f}),\quad\text{for }s=1,\ldots,M,\;\;f=0,1,\ldots,F. \tag{7}\]  
If we take a look at the solution (6), we can see that this function depends on many parameters, i.e., \(r\), \(t\), \(a\), \(T_{init}\), \(T_{b}\) and \(R\). In general, function \(T(r,t)\) can be replaced by the multi-argument function \(T(r,t,a,T_{init},T_{b},R)\), whose value depends on the above-mentioned arguments. In the inverse heat transfer problem, one can try to estimate each value of these unknown parameters. But, if in solution (6) we notice the occurrence of two independent relations \(a/R^{2}\) and \(r/R\), where the individual ratios of parameters may be the same, then it is worth setting one of the parameters as fixed, e.g., \(R\), which in this case means that the radius \(R\) of the cylinder is fixed (or was precisely measured). Therefore, the parameter \(R\) is excluded from the further considerations. So, in the problem under consideration, we will look for parameters \(r_{1}\), \(r_{2}\), \(\ldots\), \(r_{M}\), \(a\), \(T_{init}\), \(T_{b}\), where \(0\leq r_{s}\leq R\), for \(s=1,\ldots,M\), are searched sensor positions \(pos_{s}\). In Figure 2, the schematic position of thermocouple tips in the cross-section of the container with granular material and considered domain are shown. In principle, speaking in colloquial language, we are dealing with temperature measurements using several sensors placed in certain unknown positions inside any granular material at different moments of time, and we do not know the value of its thermal diffusivity coefficient; we also do not know the exact constant initial temperature and the constant temperature assumed at the outer surface of the cylindrical can. In the considered task, we want to estimate these unknown parameters by solving an optimization problem.