## 3 Quasi-Steady-State Approach
### Ballistic Heat Conduction (\(\alpha=2\))  
The quasi-steady state approach is described by the equations  
\[\frac{\partial^{2}T}{\partial t^{2}}=a\bigg{(}\frac{\partial^{2}T}{\partial r ^{2}}+\frac{2}{r}\,\frac{\partial T}{\partial r}\bigg{)}-bT,\quad R<r<\infty, \ \ 0<t<\infty, \tag{59}\]  
\[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]  
Figure 5: Dependence of the solution to the parabolic heat conduction Equation (\(\alpha=1\)) on the radial coordinates. The results of computer simulations for the angular frequency \(\bar{\omega}=\pi/4\) and time \(\bar{t}=0.5\). the quasi-steady-state approach for \(\bar{b}=0\); the general solution describing the transition process for \(\bar{b}=0\); the quasi-steady-state approach for \(\bar{b}=2\); the general solution describing the transition process for \(\bar{b}=2\). \[\lim_{r\to\infty}T(r,t)=0.\] (61) under the assumption \[T(r,t)=U(r)\mathrm{e}^{i\omega t}.\] (62) Hence, \[-\omega^{2}U=a\Bigg{(}\frac{\mathrm{d}^{2}U}{\mathrm{d}r^{2}}+\frac{2}{r}\,\frac {\mathrm{d}U}{\mathrm{d}r}\Bigg{)}-bU,\quad R<r<\infty, \tag{63}\] \[r=R:\quad\frac{\mathrm{d}U}{\mathrm{d}r}=-i\omega\,g_{0},\] (64) \[\lim_{r\to\infty}U(r)=0. \tag{65}\]  
Making use of the auxilliary function \(V=rU\) and the auxilliary spatial variable \(x=r-R\), we get  
\[\frac{\mathrm{d}^{2}V}{\mathrm{d}x^{2}}-\frac{b-\omega^{2}}{a}\,V=0,\qquad 0<x <\infty, \tag{66}\]  
\[x=0:\quad\frac{\mathrm{d}V}{\mathrm{d}x}-\frac{1}{R}\,V=-i\omega\,g_{0}R, \tag{67}\]  
\[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]0  
We confine our consideration to the case \(b>\omega^{2}\). Then,  
\[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]1  
and from the boundary conditions (67) and (68), it follows that  
\[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]2  
and consequently,  
\[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]3  
The general solution for ballistic heat conduction can be obtained from Equation (35) with \(\alpha=2\):  
\[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]4  
After inverting the Laplace transform  
\[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]5  
and after evaluation of the integral in Equation (73), we obtain  
\[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]6 \[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]7 \[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]8Inversion of the Fourier sine-cosine transform gives  
\[r=R:\quad\frac{\partial T}{\partial r}=-i\omega\,g_{0}\,\mathrm{e}^{i\omega t}, \tag{60}\]9 Figure 5:0 Figure 5:1  
Partial fraction decomposition  
Figure 5:2  
after using integrals (A1) and (A2) from Appendix A leads to  
Figure 5:3 Figure 5:4 Figure 5:5 Figure 5:6  
Reverting in Equation (77) to the variable \(r\) and temperature \(T(r,t)\)  
Figure 5:7 Figure 5:8 Figure 5:9 \[\lim_{r\to\infty}T(r,t)=0.\]0  
we see that the first summand in the expression (78) coincides with the quasi-steady-state solution. For \(x<\sqrt{at}\), the integrals in Equation (78) cannot be evaluated analytically and should be calculated numerically. For \(x>\sqrt{at}\), the integrals in Equation (78) can be evaluated analytically using Equations (A5)-(A8) from Appendix A. In this case, Equation (78) leads to  
\[\lim_{r\to\infty}T(r,t)=0.\]1  
Hence, there is the wave front at \(r=R+\sqrt{at}\). Recall that in the Klein-Gordon equation (59), the coefficient \(a\) can be taken as \(a=c^{2}\), where \(c\) is the wave velocity. Figure 6 shows the real parts of solutions (71) and (78) taking the non-dimensional quantities (40) for \(\alpha=2\).