## 3 Mathematical Models
### Inverse Heat Transfer Problem
#### 3.2.1 Optimization of Functional \(S\)
\begin{array}{cccccccc}\frac{\partial^{2}S}{\partial r_{ 1}^{2}}&\frac{\partial^{2}S}{\partial r_{1}\partial r_{2}}&\cdots&\frac{ \partial^{2}S}{\partial r_{1}\partial r_{M}}&\frac{\partial^{2}S}{\partial r _{1}\partial a}&\frac{\partial^{2}S}{\partial r_{1}\partial T_{init}}&\frac{ \partial^{2}S}{\partial r_{1}\partial T_{init}}\\ \frac{\partial^{2}S}{\partial r_{2}\partial r_{1}}&\frac{\partial^{2}S}{ \partial r_{2}^{2}}&\cdots&\frac{\partial^{2}S}{\partial r_{2}\partial r_{M}} &\frac{\partial^{2}S}{\partial r_{2}\partial a}&\frac{\partial^{2}S}{\partial r _{2}\partial T_{init}}&\frac{\partial^{2}S}{\partial r_{2}\partial T_{b}}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \frac{\partial^{2}S}{\partial r_{M}\partial r_{1}}&\frac{\partial^{2}S}{ \partial r_{M}\partial r_{2}}&\cdots&\frac{\partial^{2}S}{\partial r_{M}^{2}} &\frac{\partial^{2}S}{\partial r_{M}\partial a}&\frac{\partial^{2}S}{\partial r _{M}\partial T_{init}}&\frac{\partial^{2}S}{\partial r_{M}\partial T_{b}}\\ \frac{\partial^{2}S}{\partial a\partial r_{1}}&\frac{\partial^{2}S}{ \partial a\partial r_{2}}&\cdots&\frac{\partial^{2}S}{\partial a\partial r_{M }}&\frac{\partial^{2}S}{\partial a^{2}}&\frac{\partial^{2}S}{\partial a \partial T_{init}}&\frac{\partial^{2}S}{\partial a\partial T_{b}}\\ \frac{\partial^{2}S}{\partial T_{init}\partial r_{1}}&\frac{\partial^{2}S}{ \partial T_{init}\partial r_{2}}&\cdots&\frac{\partial^{2}S}{\partial T_{init }\partial r_{M}}&\frac{\partial^{2}S}{\partial T_{init}\partial a}&\frac{ \partial^{2}S}{\partial T_{init}}&\frac{\partial^{2}S}{\partial T_{init} \partial T_{b}}\\ \frac{\partial^{2}S}{\partial T_{b}\partial r_{1}}&\frac{\partial^{2}S}{ \partial T_{b}\partial r_{2}}&\cdots&\frac{\partial^{2}S}{\partial T_{b} \partial r_{M}}&\frac{\partial^{2}S}{\partial T_{b}\partial a}&\frac{ \partial^{2}S}{\partial T_{b}\partial T_{init}}&\frac{\partial^{2}S}{\partial T _{b}^{2}}\end{array} \right].\]  
The gradient vector and the Hessian matrix contain the first and second derivatives of \(S\), respectively. One can express these derivatives of function (8) in the following forms  
\[\frac{\partial S}{\partial p_{i}}=2\frac{1}{F+1}\frac{1}{M}\sum_{f=0}^{F}\sum_ {s=1}^{M}\left(T_{s,f}-\hat{T}_{s,f}\right)\frac{\partial T_{s,f}}{\partial p_ {i}},\]  
\[\frac{\partial^{2}S}{\partial p_{i}\partial p_{j}}=2\frac{1}{F+1}\frac{1}{M} \sum_{f=0}^{F}\sum_{s=1}^{M}\left(\frac{\partial T_{s,f}}{\partial p_{i}} \frac{\partial T_{s,f}}{\partial p_{j}}+\left(T_{s,f}-\hat{T}_{s,f}\right) \frac{\partial^{2}T_{s,f}}{\partial p_{i}\partial p_{j}}\right),\]  
where \(p_{i},p_{j}\in\{r_{1},r_{2},\ldots,r_{M},a,T_{init},T_{b}\}\). As can be seen, there are the partial derivatives of function \(T\) with respect to each considered differentiation variable that can be determined in the analytical way. The first-order partial derivatives of \(T\) are as follows  
\[\frac{\partial T_{s,f}}{\partial r_{q}}=\left\{ \begin{array}{ll}-(T_{init}-T_ {b})\frac{1}{R}\sum_{k=1}^{\infty}\frac{2}{J_{1}(\mu_{k})}\exp\left(-\mu_{k}^{2 }\frac{at_{f}}{R^{2}}\right)J_{1}\left(\mu_{k}\frac{r_{s}}{R}\right),&\mbox{if }q=s,\\ 0,&\mbox{if }q\
eq s,\end{array} \right.\]  
for \(q=1,2,\ldots,M\),  
\[\frac{\partial T_{s,f}}{\partial a}=-(T_{init}-T_{b})\frac{t_{f}}{R^{2}}\sum_ {k=1}^{\infty}\frac{2\mu_{k}}{J_{1}(\mu_{k})}\exp\left(-\mu_{k}^{2}\frac{at_{f }}{R^{2}}\right)J_{0}\left(\mu_{k}\frac{r_{s}}{R}\right),\]  
\[\frac{\partial T_{s,f}}{\partial T_{init}}=\sum_{k=1}^{\infty}\frac{2}{\mu_{k}J_{ 1}(\mu_{k})}\exp\left(-\mu_{k}^{2}\frac{at_{f}}{R^{2}}\right)J_{0}\left(\mu_ {k}\frac{r_{s}}{R}\right),\]  
\[\frac{\partial T_{s,f}}{\partial T_{b}}=1-\sum_{k=1}^{\infty}\frac{2}{\mu_{k}J_{ 1}(\mu_{k})}\exp\left(-\mu_{k}^{2}\frac{at_{f}}{R^{2}}\right)J_{0}\left(\mu_{k} \frac{r_{s}}{R}\right).\]  
While the second-order derivatives of \(T\) take the forms  
\[\frac{\partial^{2}T_{s,f}}{\partial r_{q}^{2}}=\left\{