## 5 Conclusions  
A class of strongly nonlinear BVPs with singularity arising in the modeling of a reaction and diffusion process with Michaelis-Menten kinetics has been investigated in this research study. To obtain a polynomial solution for the model equation, the technique of spectral matrix collocation together with the quasilinearization strategy was proposed efficiently and accurately. In this respect, a new class of polynomials related to the well-known Bessel functions of Krall-Frink was used in the collocation procedure. In the \(L_{2}\) norm, we derived an upper bound for the error related to these polynomials theoretically. Several numerical test examples with all values of parameters were solved to acquire the (steady-state) substrate concentrations in a solid with planar and spherical shape, approximately. Comparisons with the outputs of two existing methods, namely, the Adomian decomposition approach and wavelet-based procedures, have been made to show the validity of the combined QLM-ChPs algorithm. The techniques of residual error functions and the error correction were employed to check the accuracy of the obtained solutions in the absence of the exact solutions. The latter technique further helped us to improve the quality of obtained approximations. The presented outcomes shown by tables and figures confirm that the QLM-ChPs technique is not only highly accurate and  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{\(\mathbf{n=0}\)} & \multicolumn{3}{c}{\(\mathbf{n=2}\)} \\ \cline{2-9}  & \multicolumn{3}{c}{\(\mathbf{h=0.5}\)} & \multicolumn{3}{c}{\(\mathbf{h=1}\)} & \multicolumn{3}{c}{\(\mathbf{h=0.5}\)} & \multicolumn{3}{c}{\(\mathbf{h=1}\)} \\ \cline{2-9} \(H\) & \(e_{\infty}\) & \(\text{Ord}_{\mathbf{H}}\) & \(e_{\infty}\) & \(\text{Ord}_{\mathbf{H}}\) & \(e_{\infty}\) & \(\text{Ord}_{\mathbf{H}}\) & \(e_{\infty}\) & \(\text{Ord}_{\mathbf{H}}\) \\ \hline \multirow{2}{*}{2} & \(3.7498\times\) & \multirow{2}{*}{\(-\)} & \(2.7776\times\) & \multirow{2}{*}{\(-\)} & \(1.2500\times\) & \multirow{2}{*}{\(-\)} & \(1.1905\times\) & \\  & \(10^{-5}\) & & & \(10^{-1}\) & & \(10^{-5}\) & & \(10^{-1}\) & \\ \multirow{2}{*}{4} & \(2.4301\times\) & \multirow{2}{*}{\(10.592\)} & \(6.8532\times\) & \multirow{2}{*}{\(5.3409\)} & \(2.2738\times\) & \multirow{2}{*}{\(12.425\)} & \multirow{2}{*}{\(1.5186\times\)} & \\  & \(10^{-8}\) & & & \(10^{-3}\) & & & & \\ \multirow{2}{*}{8} & \(3.2518\times\) & \multirow{2}{*}{\(6.0414\times\)} & \multirow{2}{*}{\(10.147\)} & \multirow{2}{*}{\(10.147\)} & \multirow{2}{*}{\(10^{-18}\)} & \multirow{2}{*}{\(29.091\)} & \multirow{2}{*}{\(10^{-7}\)} & \multirow{2}{*}{\(12.501\)} \\  & \(10^{-15}\) & & & \(10^{-6}\) & & & & \\ \multirow{2}{*}{16} & \(1.2902\times\) & \multirow{2}{*}{\(47.841\)} & \multirow{2}{*}{\(1.1835\times\)} & \multirow{2}{*}{\(22.284\)} & \multirow{2}{*}{\(2.3086\times\)} & \multirow{2}{*}{\(50.613\)} & \multirow{2}{*}{\(3.1188\times\)} & \\  & \(10^{-29}\) & & & \(10^{-12}\) & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: The maximum absolute value of REFs and the associated Ord\({}_{H}\) achieved via the QLM-ChPs procedure using \(\alpha,h,s_{h}=1\) and various \(H\).  
Figure 7: Approximate solutions using various \(\alpha=0.1,\ldots,0.5\), \(h=0.5\), \(s_{h}=2\), (**left**) and different \(s_{h}=3,4,6,8,15,h=1,\alpha=1\) (**right**) via the QLM-ChPs approach with \(H=10\) in the spherical case \(n=2\).  
efficient but also capable of solving various similar model problems with strong nonlinearity and singularity.  
Conceptualization, M.I. and H.M.S.; methodology, M.I. and H.M.S.; software, M.I.; validation, M.I. and H.M.S.; formal analysis, M.I. and H.M.S.; funding acquisition, H.M.S.; investigation, M.I. and H.M.S.; writing--original draft preparation, M.I.; writing--review and editing, M.I. and H.M.S. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.