## 4 Computational Aspects and Applications
### Applications to Some Real-World Problems
Note that, during heat-denaturation, the protein molecule passed through several stages (usually four or five) of structural changes before turning into gel, and thence into a food product (cheese, yellow cheese, etc.). In the optical spectra, this is observed as maximums (see Figure 4), which correspond to the maximal values of the heat capacity._  
_The endothermic temperature characteristic of the thermo-denaturation process during all stages can be easily and quickly found by means of a non-destructive turbidimetric method performed on a specially constructed fiber-optic measuring device [36] (\[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]41a)._  
_Using the OrginePro \(8.5\) software product, a correlation analysis of the dependence of the optical density on temperature was made at the used wavelengths in the range from \(650\,nm\) to \(850\,nm\) in the temperature interval from \(60^{\circ}\) to \(80^{\circ}\). The function of the dependence between the optical density and the heating temperature during the kinetic process was approximated by the following polynomial (see Figure 4):_  
\[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]5 \[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]6  
_In this example, we applied the above mentioned members of the family (10) for the simultaneous computing of all zeros of \(f\) starting from [PERSON]'s initial approximation (48) with \(r=160\)._  
\[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]7 Dependence between the optical density and the temperature. \[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]8 Basins of attraction for Example 1. (**a**) [PERSON]’s method (\(\alpha=0\)). (**b**) [PERSON]’s method (\(\alpha=1\)). (**c**) The method by \(\alpha=1/2\). (**d**) The method by \(\alpha=0.7669+0.4847i\). The data in Table 2 show that [PERSON]'s method and the random one give better results than the other two methods. It is interesting to note that, for this example, the random method behaves better than [PERSON]'s method and much better than [PERSON]'s method. The approximation trajectories for [PERSON]'s method and the random method are plotted in Figure 5a,b. Despite the similarity of the convergence behaviors, the trajectories of the approximations of the two methods are highly different. It is worth noting that, in Figure 5a, the coordinate starting from the initial point \(-138.564-79.999i\) 'jumps away' to the point \(-849.322+1560.620i\) before turning back towards the root. **Example 3** (Schrodinger wave equation for a hydrogen atom). : _In quantum mechanics, the position of the electron relative to the core has a probability distribution that is related to the solution of the Schrodinger wave equation for a charged particle moving in a Coulomb potential. The famous Schrodinger equation for a single particle of mass \(\mu\) moving in a central potential can be written in the following form:_  
\[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]9  
_where \(\mathcal{E}\) is the energy and \(r\) is the distance of the electron from the core. In spherical coordinates, Equation (50) takes the form_  
\[\|\,x^{(k)}-\xi\,\|_{\infty}\leq\varepsilon_{k}=\alpha(E_{f}(x^{(k)}))\,\|W_{ f}(x^{(k)})\|_{\infty}\]0 \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \(k\) & \(E_{f}(\alpha^{(k)})\) & \(\tau\) & \(\varepsilon_{k}\) & \(\varepsilon_{k+1}\) & \(r_{k}\) \\ \hline \(\alpha=0\) & 23 & \(7.876\times 10^{-13}\) & 0.06822 & \(1.003\times 10^{-11}\) & \(1.244\times 10^{-35}\) & 3.000619 \\ \hline \(\alpha=1\) & 14 & \(1.043\times 10^{-20}\) & \(-\) & \(4.847\times 10^{-18}\) & \(1.107\times 10^{-59}\) & 3.002078 \\ \hline \(\alpha=1/2\) & 17 & \(7.045\times 10^{-13}\) & \(-\) & \(8.978\times 10^{-12}\) & \(6.633\times 10^{-36}\) & 3.016032 \\ \hline \(\alpha=0.766+0.484i\) & 13 & \(3.968\times 10^{-12}\) & \(-\) & \(5.057\times 10^{-11}\) & \(1.023\times 10^{-33}\) & 3.002426 \\ \hline \end{tabular}
\end{table} 
Table 2: Numerical results for Example 2. Figure 5: Trajectories of approximations for Example 2.