## 4 Numerical Calculations
### Case Study II: Spherical Particle (\(n=2\))  
Similar to the case of \(n=0\), we start numerical evaluation by fixing the parameters \(s_{h}\), \(\alpha\) and using various \(h\) as [11]  
\[s_{h}=1,\quad\alpha=1,\quad h=0.1,0.2,\ldots,0.5.\]  
The approximate solutions by utilizing \(H=2\) and these values of \(h\) are listed as follows, respectively:  
\[M_{2}^{(5)}(x) =0.001662847242\,x^{2}+3.326531125\times 10^{-111}\,x+0.9950114583,\] \[M_{2}^{(5)}(x) =0.00660556069\,x^{2}+0.9801833179,\] \[M_{2}^{(5)}(x) =0.01469075649\,x^{2}+0.9559277305,\] \[M_{2}^{(5)}(x) =0.02569019993\,x^{2}+5.3224498\times 10^{-110}\,x+0.9229294002,\] \[M_{2}^{(5)}(x) =0.03928729831\,x^{2}+1.06448996\times 10^{-109}\,x+0.8821381051.\]  
The associated approximations obtained by the ADM [10] and the LWM [11] are also given for comparison purposes as follows:  
\[A_{ADM}(x) =0.001666666667\,x^{2}+0.995, U_{LWM}(x) =0.00161\,x^{2}+0.995172,\] \[A_{ADM}(x) =0.006666666667\,x^{2}+0.98, U_{LWM}(x) =0.00657\,x^{2}+0.98027,\] \[A_{ADM}(x) =0.015\,x^{2}+0.955, U_{LWM}(x) =0.01462\,x^{2}+0.956131,\] \[A_{ADM}(x) =0.02666666667\,x^{2}+0.92, U_{LWM}(x) =0.02569\,x^{2}+0.922925,\] \[A_{ADM}(x) =0.04166666667\,x^{2}+0.875, U_{LWM}(x) =0.03917\,x^{2}+0.882487.\]  
It seems that the agreement between our solutions and two other methods depends heavily on the parameter \(h\). For more insight, let us plot all three numerical solutions using \(h=0.1\) and \(h=0.5\). Figure 5 shows these approximations together. Clearly, for a small \(h=0.1\), the outputs of QLM-ChPs and ADM are very close together, while for a larger value \(h=0.5\), this closeness will be increased. On the other hand, the performance of LWM is poor and our method outperforms the LWM. To confirm the performance of the QLM-ChPs method, we fix \(h=0.5\) and vary \(H=2,4,8,16\). The outcomes presented in Figure 6 show the exponential convergence of our method, which is also simpler in implementation compared to existing ADM and LWM procedures.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(x\) & \(M_{4}^{(5)}(x)\) & \(|\mathfrak{R}_{4}^{(5)}(x)|\) & \(M_{4,6}^{(5)}(x)\) & \(|\mathfrak{R}_{4,6}^{(5)}(x)|\) \\ \hline
0.0 & 0.26557303 & \(3.1936\times 10^{-3}\) & 0.26550702 & \(4.7010\times 10^{-4}\) \\
0.1 & 0.26766356 & \(1.4356\times 10^{-3}\) & 0.26760887 & \(2.7671\times 10^{-4}\) \\
0.2 & 0.27396962 & \(3.2785\times 10^{-4}\) & 0.27393950 & \(6.9216\times 10^{-5}\) \\
0.3 & 0.28457805 & \(1.9296\times 10^{-4}\) & 0.28457571 & \(5.3198\times 10^{-5}\) \\
0.4 & 0.29962257 & \(2.4287\times 10^{-4}\) & 0.29964645 & \(5.1880\times 10^{-5}\) \\
0.5 & 0.31928378 & \(1.4663\times 10^{-2}\) & 0.31933216 & \(5.4292\times 10^{-5}\) \\
0.6 & 0.34378916 & \(2.8270\times 10^{-4}\) & 0.34386303 & \(1.9064\times 10^{-4}\) \\
0.7 & 0.37341307 & \(2.6150\times 10^{-4}\) & 0.37351652 & \(2.2601\times 10^{-4}\) \\
0.8 & 0.40847670 & \(5.1746\times 10^{-4}\) & 0.40861377 & \(3.1905\times 10^{-5}\) \\
0.9 & 0.44934817 & \(2.6407\times 10^{-3}\) & 0.44951528 & \(8.4719\times 10^{-4}\) \\
1.0 & 0.49644248 & \(6.8532\times 10^{-3}\) & 0.49661555 & \(2.5745\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Numerical evaluation of approximate solutions and REFs via the RC technique with \((H,H^{\prime})=(4,6)\) and \(\alpha,h,s_{h}=1\) at various \(x\in[0,1]\).  
Finally, we show the exponential convergence of the proposed QLM-ChPs approach for both case studies I and II related to \(n=0\) and \(n=2\) respectively. The outcomes of \(e_{\infty}\) and the corresponding numerical order of convergence Ord\({}_{H}\) (see (41)) for two different values of \(h=0.5\) and \(h=1\) are presented in Table 4. For these results, we used \(H=2^{i}\), \(i=1,2,3,4\) and \(\alpha,h,s_{h}=1\).