## 4 Concluding Remarks  
We have studied time-fractional heat conduction with heat absorption in an infinite medium with a spherical hole under time-harmonic heat flux at its boundary. In numerical simulations, we have utilized different values of non-dimensional parameters to show the distinguishing properties of solutions.  
It should be pointed out that the quasi-steady-state approach can be considered only for integer orders of the time derivative \(\alpha=1\) and \(\alpha=2\) and cannot be studied for fractional orders of the Caputo derivative because for a non-integer \(\alpha\)[42],  
\[\frac{\mathrm{d}^{\alpha}}{\mathrm{d}t^{\alpha}}\,\mathrm{e}^{i\omega t}\
eq( i\omega)^{\alpha}\mathrm{e}^{i\omega t}. \tag{80}\]  
It is observable from the figures that at different times, the absorption parameter \(\bar{b}\) can cause a change in the sign of the temperature on the cavity surface. Figure 1 shows that for \(0\leq\alpha<1\), the temperature at the cavity surface is larger and for \(1<\alpha\leq 2\), it is smaller than that in the case of the standard heat conduction with \(\alpha=1\). The situation changes and inverts at larger distances from the cavity, and there is a point at some distance \(\bar{r}\) where the temperature \(\overline{T}\) is approximately the same for all values of \(\alpha\). In medical applications, it is possible to choose the necessary combination of the values of \(\omega\), \(b\), and \(\alpha\) according to the goals of the procedure. In the literature, there is a discussion about the optimal choice of the order of the time-derivative \(\alpha\) in the fractional heat conduction model for bilayered spherical tissue in the hyperthermia experiment [43].  
It is evident from Figure 5 that for the parabolic Equation (\(\alpha=1\)), the process is dissipative itself and the absorption parameter \(\bar{b}\) significantly decreases the difference between the quasi-steady-state solution and the general solution involving the transition process. In the case of the hyperbolic Klein-Gordon Equation (\(\alpha=2\)), the difference between the quasi-steady state approach and the general solution describing the transition regime is more visible (see Figure 6). In particular, the quasi-steady-state solution does not have a wave front. For \(\alpha=2\), the wave front at \(r=R+\sqrt{a}t\) (or in terms of non-dimensional quantities, at \(\bar{r}=1+\bar{t}\)) is also seen in Figure 1 for different values of non-dimensional time \(\bar{t}\).  
Our future work will focus on studying the space-time-fractional equations with the Caputo time derivative and Riesz fractional operator (fractional Laplacian).  
Figure 6: Dependence of the solution to the hyperbolic Klein–Gordon Equation (\(\alpha=2\)) on the spatial variable \(\bar{r}\). The computer simulation results for the angular frequency \(\hat{\omega}=\pi/4,\ \bar{b}=2\), and \(\bar{t}=1\). the general solution describing the transition process; the quasi-steady-state approach (71).  
**Author Contributions:** Conceptualization, Y.P. and T.K.; methodology, Y.P. and T.K.; validation, B.W.-S. and A.Y.; formal analysis, A.Y.; investigation, Y.P. and T.K.; software, T.K., B.W.-S. and A.Y.; supervision, B.W.-S.; writing--original draft preparation, T.K. and A.Y; writing--review and editing, Y.P. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data are contained within the article.  
**Conflicts of Interest:** The authors declare no conflicts of interest.