## 3 Quasi-Steady-State Approach
### Classical Heat Conduction (\(\alpha=1\))  
The problem is described by the equations  
\[\frac{\partial T}{\partial t}=a\left(\frac{\partial^{2}T}{\partial r^{2}}+\frac{2 }{r}\,\frac{\partial T}{\partial r}\right)-bT,\quad R<r<\infty,\ \ 0<t<\infty, \tag{42}\]  
\[r=R:\quad\frac{\partial T}{\partial r}=-g_{0}\,\mathrm{e}^{i\omega t}, \tag{43}\]  
\[\lim_{r\to\infty}T(r,t)=0. \tag{44}\]  
under the assumption  
\[T(r,t)=U(r)\mathrm{e}^{i\omega t}. \tag{45}\]  
Hence,  
\[i\omega U=a\left(\frac{\mathrm{d}^{2}U}{\mathrm{d}r^{2}}+\frac{2}{r}\,\frac{ \mathrm{d}U}{\mathrm{d}r}\right)-bU,\quad R<r<\infty, \tag{46}\]  
\[r=R:\quad\frac{\mathrm{d}U}{\mathrm{d}r}=-g_{0}, \tag{47}\]  
\[\lim_{r\to\infty}U(r)=0. \tag{48}\]  
Employing the auxiliary function \(V=rU\) and the auxiliary spatial variable \(x=r-R\), we get  
\[\frac{\mathrm{d}^{2}V}{\mathrm{d}x^{2}}-\frac{b+i\omega}{a}\,V=0,\qquad 0<x<\infty, \tag{49}\]  
\[x=0:\quad\frac{\mathrm{d}V}{\mathrm{d}x}-\frac{1}{R}\,V=-g_{0}R, \tag{50}\]  
\[\lim_{x\to\infty}V(x)\text{ is bounded}. \tag{51}\]  
The general solution of Equation (49) has the form  
\[V(x)=A\,\mathrm{e}^{-x\sqrt{(b+i\omega)/a}}+B\,\mathrm{e}^{x\sqrt{(b+i\omega) /a}}, \tag{52}\]  
where the integration constants are determined from the boundary conditions (50) and (51):  
\[A=\frac{g_{0}R}{\sqrt{(b+i\omega)/a}+1/R},\qquad B=0. \tag{53}\]  
The final result reads  
\[T(r,t)=\frac{g_{0}R}{\sqrt{(b+i\omega)/a}+1/R}\,\frac{1}{r}\,\mathrm{e}^{-(r- R)\sqrt{(b+i\omega)/a}}\,\mathrm{e}^{i\omega t}. \tag{54}\]  
Now, we analyze the particular case of the solution (39) for \(\alpha=1\). Equation (35) for the value \(\alpha=1\) becomes:  
\[\tilde{v}^{*}(\xi,s)=ag_{0}R\,\frac{\xi}{\sqrt{\xi^{2}+1/R^{2}}}\cdot\frac{1} {s-i\omega}\cdot\frac{1}{s+a\xi^{2}+b}, \tag{55}\]  
and after inversion of the Laplace transform, we get  
\[\tilde{v}(\xi,t)=ag_{0}R\,\frac{\xi}{\sqrt{\xi^{2}+1/R^{2}}}\cdot\frac{1}{a \xi^{2}+b+i\omega}\Big{[}\mathrm{e}^{i\omega t}-\mathrm{e}^{-(a\xi^{2}+b)t} \Big{]}. \tag{56}\]  
Next, we invert the Fourier sine-cosine transform and use the partial fraction decomposition\[\frac{1}{(\xi^{2}+1/R^{2})[\xi^{2}+(b+i\omega)/a]}=\frac{1}{(b+i\omega)/a-1/R^{2}} \bigg{[}\frac{1}{\xi^{2}+1/R^{2}}-\frac{1}{\xi^{2}+(b+i\omega)/a}\bigg{]} \tag{57}\]  
and integrals (A1)-(A4) from Appendix A. After some transformations, we obtain  
\[T(r,t) = \frac{g_{0}R}{\sqrt{(b+i\omega)/a}+1/R}\,\frac{1}{r}\,\mathrm{e}^ {-(r-R)\sqrt{(b+i\omega)/a}}\,\mathrm{e}^{i\omega t} \tag{58}\] \[+ \frac{g_{0}}{(b+i\omega)/a-1/R^{2}}\,\frac{1}{r}\,\bigg{\{}\exp \bigg{(}\frac{r-R}{R}+\frac{at}{R^{2}}-bt\bigg{)}\mathrm{erfc}\bigg{(}\frac{ \sqrt{at}}{R}+\frac{r-R}{2\sqrt{at}}\bigg{)}\] \[- \frac{R}{2}\,\bigg{[}\sqrt{(b+i\omega)/a}-1/R\bigg{]}\mathrm{e} ^{-(r-R)\sqrt{(b+i\omega)/a}}\,\mathrm{erfc}\bigg{(}\sqrt{(b+i\omega)t}- \frac{r-R}{2\sqrt{at}}\bigg{)}\mathrm{e}^{i\omega t}\] \[- \frac{R}{2}\,\bigg{[}\sqrt{(b+i\omega)/a}+1/R\bigg{]}\mathrm{e} ^{(r-R)\sqrt{(b+i\omega)/a}}\,\mathrm{erfc}\bigg{(}\sqrt{(b+i\omega)t}+\frac{ r-R}{2\sqrt{at}}\bigg{)}\mathrm{e}^{i\omega t}\bigg{\}},\]  
where \(\mathrm{erfc}(z)\) is the complementary error function.  
The first summand in Equation (58) coincides with the quasi-steady-state expression (54) and the remaining terms describe the transition regime.  
Figure 5 compares the real parts of the solutions (54) and (58) for the values of heat absorption parameter \(\bar{b}=0\) and \(\bar{b}=2\).