## 4 Solutions to Two Basic Subcases
### Case 2: Production Only  
In the case of absorption only when \(b=0\) and \(c=1\), using the result (18), we obtain  
\[\frac{d\delta^{2}}{dt}=D_{0}\frac{(n+1)(n+2)}{m+1}-\delta^{2}c\frac{(n+1)(n+2) }{(nq+1)(nq+2)}=A-C_{P}\delta^{2} \tag{32}\]  
where \(C_{P}=c\frac{(n+1)(n+2)}{(nq+1)(nq+2)}=cM_{P}\). Figure 6: Solution with absorption only (\(b=1\)): The front length measured as a function of the Fourier number \(Fo\) for various values of the exponent \(m\). Effects of the exponent \(p\). Note: The dashed lines mark only tentatively the range where the plot order inverts. (**a**) \(p=1\), \(m=1.5\); (**b**) \(p=1\), \(m=10\); (**c**) \(p=2\), \(m=1.5\); (**d**) \(p=2\), \(m=10\); (**e**) \(p=3\), \(m=1.5\); (**f**) \(p=3\), \(m=10\). The solution of (32), with the condition \(\delta(t=0)=0\), is  
\[\delta_{P}=\sqrt{\frac{A}{c}}\sqrt{1-e^{-B_{P}t}}=\sqrt{\frac{D_{0}}{c}}\sqrt{ \frac{N}{M_{P}}}\sqrt{1-e^{-M_{P}ct}}\sqrt{\frac{D_{0}}{c}}\sqrt{\frac{N}{M_{P} }}\sqrt{1-e^{-M_{P}Fo}} \tag{33}\]  
with a length scale of \(\sqrt{D_{0}/c}\), in this specific case, and the Fourier number defined as \(Fo=ct\), that is a specific formulation when \(b=0\) and \(c\
eq 0\). Then, the scaled penetration depth is  
\[\delta_{P}/\sqrt{\frac{D_{0}}{c}}\ =\sqrt{\frac{N}{M_{P}}}\sqrt{1-e^{-M_{P}Fo}} \tag{34}\]  
For a long time, we obtain a finite penetration depth \(\left(\delta_{P}/\sqrt{\frac{D_{0}}{c}}\ \right)_{t\rightarrow\infty}\rightarrow \sqrt{\frac{N}{M_{P}}}\) since the time-dependent term approaches unity. Figure 7: Solution with absorption only (\(b=1\)) as a function of the similarity variable \(\eta\) with simultaneous effects of the exponents \(m\) and \(p\). (**a**) \(m=1.5\); (**b**) \(m=2\); (**c**) \(m=3\); (**d**) \(m=4\); (**e**) \(m=5\); (**f**) \(m=1/2\). Now, with (34), we may construct the approximate solution as  
\[ \begin{split}& u_{a(\text{Prod})}=\left(1-\frac{x}{\sqrt{\frac{D_{0}}{c} }\sqrt{\frac{N}{M_{P}}}\sqrt{1-e^{-M_{P}Fo}}}\right)^{n}=\left(1-\frac{X_{P}}{ \sqrt{\frac{N}{M_{P}}}\sqrt{1-e^{-M_{P}Fo}}}\right)^{n}=\\ &\left(1-\frac{\eta p}{\sqrt{\frac{N}{M_{P}}}}\right)^{n}\end{split}  \tag{35}\]  
Now, we automatically defined a dimensionless spatial variable \(X_{P}=x/\sqrt{D_{0}/c}\), taking into account that \(A\equiv D_{0}\) and \(\sqrt{D_{0}/c}\)  is a length scale and the similarity variable \(\eta_{P}=\frac{x}{\sqrt{\frac{D_{0}}{c}}\sqrt{1-e^{-M_{P}Fo}}}\). Thus, the front is defined by \(X_{P(front)}=\sqrt{\frac{N}{M_{P}}}\sqrt{1-e^{-M_{P}Fo}}\) (and for long times, \(X_{P(front)(t\rightarrow\infty)}=\sqrt{\frac{N}{M_{P}}}\)) or by \(\eta_{P(front)}=\sqrt{\frac{N}{M_{P}}}\). Profiles of the approximate solutions as a function of the dimensionless spatial variable \(X_{P}\) (for different times, represented by the Fourier number) are shown in Figures 8 and 9, accounting for the effects of the exponents \(m\) and \(q\), respectively. The simultaneous effect of \(m\) and \(q\) is shown in Figure 10, where the profiles are presented as functions of the similarity variable \(\eta_{P}\). An interesting issue appears for a long time as an effect of the exponent \(q\). Precisely, for \(q>1\), there is an intersection of the profiles of \(X_{P}\) as functions of the Fourier number \(Fo\) (see Figure 9). For shorter and moderate times (moderate values of \(Fo\)), this effect does not appear. As a consequence, affecting the distributions of the approximate profiles, see Figure 10f, where penetration depth for \(m=1\) is shorter than that when \(m=5\). We only mark this behavior, but a more in-depth analysis is beyond the scope of this work and envisages new numerical and analytical experiments. Figure 8: Solution with production only (\(c=1\) and \(q=2\)) as a function of the dimensionless spatial variable \(X\) with effect of the exponent \(m\). (**a**) \(q=2\), \(Fo=0.1\); (**b**) \(q=2\), \(Fo=0.5\); (**c**) \(q=2\), \(Fo=1\); (**d**) \(q=2\), \(Fo=5\).