## Appendix A  
Equation (7) can be converted into Equation (A1) by replacing \(r(T(t,r)-T_{in})\) with \(U(t,r)\):  
\[Ut(t,r)-D_{0}\partial^{2}U/\partial r^{2}=\frac{r\Phi(r)F(t)}{\rho c_{0}}+ \varepsilon_{0}\frac{\partial}{\partial t}\int_{0}^{t}exp\left(-\frac{t-\tau} {\tau_{0}}\right)Ut(\tau,r)d\tau, \tag{10}\]  
Thus, we obtain a one-dimensional problem with uniform boundary and initial conditions: \(U(t,0)=0\), \(U(t,R_{0})=0\), and \(U(t,r)=0\) at \(t\leq 0\).  
This boundary value problem is satisfied by the following series:  
\[U(t,r)=\sum_{n=1}\psi_{n}(t)sin(\tau nr/R_{0}), \tag{11}\]  
where the functions \(\psi_{n}(t)\) are the solutions of Equation (9) [69]. Thus, for a heat source uniformly distributed in a volume of radius \(r_{0}\) with power density \(\Phi_{0}\), and for the heating pulse \(\Phi_{0}sin(\pi t/\tau_{p})\), acting during the time interval \(\tau_{p}\), we obtain solutions \(\psi_{n}(t)\) to Equation (9) represented by Equations (A3) and (A4) for \(0\leq t\leq\tau_{p}\) and \(\tau_{p}<t\), respectively.  
\[\psi_{n}(t)=\frac{\Phi_{n}}{\rho c_{0}}\frac{\tau_{p}\tau_{p}\mu_{n}}{(\tau_{n }-\mu_{n})}\Bigg{[}\frac{(\tau_{n}\tau_{0}-1)\left[\
u_{n}sin(\pi t/\tau_{p})+ \frac{\pi}{\tau_{p}}(exp(-\tau_{n}t)-cos(\pi t/\tau_{p}))\right]}{(\tau_{n})^ {2}+\left(\frac{\pi}{\tau_{p}}\right)^{2}}+\frac{(1-\mu_{n}\tau_{0})\left[\
u _{n}sin(\pi t/\tau_{p})+\frac{\pi}{\tau_{p}}(exp(-\mu_{n}t)-cos(\pi t/\tau_{p }))\right]}{(\mu_{n})^{2}+\left(\frac{\pi}{\tau_{p}}\right)^{2}}\Bigg{]}, \tag{12}\]  
\[\psi_{n}(t)=\frac{\Phi_{n}}{\rho c_{0}}\frac{\pi\tau_{p}\mu_{n}}{\tau_{p}\mu_{n }}\Bigg{[}\frac{(\tau_{n}\tau_{0}-1)\left[exp(-\tau_{n}t)+exp(\tau_{n}(\tau_{ p}-t))\right]}{(\tau_{n})^{2}+\left(\frac{\pi}{\tau_{p}}\right)^{2}}+ \frac{(1-\mu_{n}\tau_{0})\left[exp(-\mu_{n}t)+exp(\mu_{n}(\tau_{p}-t))\right] }{(\mu_{n})^{2}+\left(\frac{\pi}{\tau_{p}}\right)^{2}}\Bigg{]}, \tag{13}\]  
where \(-\gamma_{n}\) and \(-\mu_{n}\) are the roots of the polynomial \((1-\varepsilon_{0})p^{2}+p\left(\tau_{n}^{-1}+\tau_{0}^{-1}\right)+\tau_{n}^{- 1}\tau_{0}^{-1}\). The parameters \(\gamma_{n}\) and \(\mu_{n}\) are real, positive, and \((\gamma_{n}-\mu_{n})\
eq 0\) for \(0<\varepsilon_{0}<1\) and \(0<\tau_{n}\), \(\tau_{0}\). The series in Equation (8) converges as \(1/n^{2}\) for \(\tau_{p}<t\). In fact, it is enough to calculate the sum in Equation (8) to about a thousand terms to obtain a result with an error of less than 1%.  
Similarly, for a heating pulse of a square shape with power density \(\Phi_{0}\), acting on a time interval \(\tau_{p}\), we obtain \(\psi_{n}(t)=\phi_{n}(t)\) and \(\psi_{n}(t)=\left[\phi_{n}(t)-\phi_{n}(\tau_{p}-t)\right]\) for \(\tau_{p}<t\) and \(0\leq t\leq\tau_{p}\), respectively, where \(\phi_{n}(t)\) are represented by Equation (A5):  
\[\phi_{n}(t)=\frac{\Phi_{n}}{\rho c_{0}}\tau_{n}\Big{[}1+\frac{\frac{\gamma_{n} \tau_{0}\mu_{n}(exp(-\mu_{n}t)-exp(-\gamma_{n}t))}{(\tau_{n}-\mu_{n})}+\frac{ \mu_{n}exp(-\gamma_{n}t)-\gamma_{n}exp(-\mu_{n}t)}{(\tau_{n}-\mu_{n})}\Big{]}. \tag{14}\]  
Note that the solutions \(\phi_{n}(t)\) continuously transform into solutions of the classical Fourier heat equation: \(\phi_{n}(t)\rightarrow\frac{\Phi_{n}}{\rho c_{0}}\tau_{n}[1-exp(-t/\tau_{n})]\) as \(\varepsilon_{0}\to 0\) or/and \(\tau_{0}\to 0\).  
Similarly, the functions \(\psi_{n}(t)\) represented by Equations (A3) and (A4) continuously transform into solutions of the classical Fourier heat equation as \(\varepsilon_{0}\to 0\) or/and \(\tau_{0}\to 0\)--see Equations (A6) and (A7)--for \(0\leq t\leq\tau_{p}\) and \(\tau_{p}<t\), respectively.  
\[\psi_{n}(t)=\frac{\Phi_{n}}{\rho c_{0}}\frac{\frac{1}{n}sin(\pi t/\tau_{p})+ \frac{\pi}{\tau_{p}}\left[exp(-t/\tau_{n})-cos(\pi t/\tau_{p})\right]}{\left( \frac{1}{\tau_{n}}\right)^{2}+\left(\frac{\pi}{\tau_{p}}\right)^{2}}, \tag{15}\]\[\psi_{n}(t)=\frac{\Phi_{n}}{\rho c_{0}}\frac{\pi}{\tau_{p}}\frac{\left[exp(-t/\tau_ {n})+exp((\tau_{p}-t)/\tau_{n}))\right]}{\left(\frac{1}{\tau_{n}}\right)^{2}+ \left(\frac{\pi}{\tau_{p}}\right)^{2}}. \tag{10}\]