## 4 Calculation Results
### Effect of Auxiliary Parameter \(\varepsilon\) on Convergence
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(\boldsymbol{\alpha}\)} & \multicolumn{2}{c}{Exact} & \multirow{2}{*}{\(\boldsymbol{\varepsilon}=0\) (HAM [48])} & \multirow{2}{*}{\(\boldsymbol{\varepsilon}=8/\boldsymbol{\pi}^{2}\) (IHAM)} & \multirow{2}{*}{\(\boldsymbol{\varepsilon}=0\) (IHAM)} \\ \cline{2-2} \cline{5-8}  & \(\boldsymbol{\Theta_{b}}\) & & & & & & \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
0.1 & 0.031802 & 0.031831 & 0.0916\% & 0.031807 & 0.0172\% & 0.031802 & 0.000178\% \\ \hline
0.2 & 0.063430 & 0.063662 & 0.365\% & 0.063473 & 0.0669\% & 0.063429 & 0.00284\% \\ \hline
0.3 & 0.094719 & 0.095493 & 0.817\% & 0.094854 & 0.143\% & 0.094705 & 0.0143\% \\ \hline
0.4 & 0.12551 & 0.12732 & 1.44\% & 0.12581 & 0.237\% & 0.12546 & 0.0446\% \\ \hline
0.5 & 0.15567 & 0.15915 & 2.24\% & 0.15620 & 0.336\% & 0.15551 & 0.107\% \\ \hline
0.6 & 0.18509 & 0.19099 & 3.19\% & 0.18588 & 0.426\% & 0.18468 & 0.219\% \\ \hline
0.7 & 0.21366 & 0.22282 & 4.29\% & 0.21470 & 0.488\% & 0.21281 & 0.399\% \\ \hline
0.8 & 0.24132 & 0.25465 & 5.52\% & 0.24254 & 0.505\% & 0.23971 & 0.667\% \\ \hline
0.9 & 0.26801 & 0.28648 & 6.89\% & 0.26924 & 0.458\% & 0.26521 & 1.05\% \\ \hline \end{tabular}
\end{table} 
Table 2: First-order numerical solutions \(\Theta_{b}^{[n=1]}\) by IHAM vs. exact solutions \(\Theta_{b}\). When the relative error \(\Delta\leq 1\%\), the convergent region is:  
\[\alpha\in[0,1.3],\ \Theta_{b}\in[0,0.36] \tag{86}\]  
From Table 2, when \(\varepsilon=0\) the convergence region of \(\Theta_{b}\) is iteratively calculated to be \(0<\alpha\leq 0.3\). When \(\varepsilon=8/\pi^{2}\), the convergence region of \(\Theta_{b}\) is \(0<\alpha\leq 1.3\). It can be stated that the improved homotopy analysis method expands the convergence region by 333% for this specific case. In the case of \(n=1,\bar{\textsc{h}}=-1\), the 1 st-order solution \(\Theta_{b}^{[n-1]}\) to rotation angle at the free end versus the point force by the improved homotopy analysis method, and the exact solution \(\Theta_{b}\) are compared in Figure 2, from which the optimal control parameter of the convergence region for the improved homotopy analysis method is identified to be \(\varepsilon=8/\pi^{2}\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(\alpha\)} & \multicolumn{2}{c}{Exact} & \multirow{2}{*}{\(\varepsilon=0\) (HAM [48])} & \multirow{2}{*}{\(\varepsilon=8/\pi^{2}\) (IHAM)} & \multirow{2}{*}{\(\varepsilon=0\) (IHAM)} \\ \cline{2-2} \cline{5-8}  & \(\Theta_{b}\) & & & & & \\ \hline
1.0 & 0.29371 & 0.31831 & 8.38\% & 0.29466 & 0.324\% & 0.28913 & 1.56\% \\ \hline
1.1 & 0.31839 & 0.35014 & 9.97\% & 0.31866 & 0.0857\% & 0.31130 & 2.22\% \\ \hline
1.2 & 0.34206 & 0.38197 & 11.7\% & 0.34110 & 0.279\% & 0.33155 & 3.07\% \\ \hline
1.3 & 0.36473 & 0.41380 & 13.5\% & 0.36184 & 0.791\% & 0.34970 & 4.12\% \\ \hline
1.4 & 0.38641 & 0.44563 & 15.3\% & 0.38074 & 1.47\% & 0.36557 & 5.39\% \\ \hline
1.5 & 0.40714 & 0.47746 & 17.3\% & 0.39764 & 2.33\% & 0.37899 & 6.92\% \\ \hline
1.6 & 0.42685 & 0.50930 & 19.2\% & 0.41242 & 3.40\% & 0.38978 & 8.71\% \\ \hline
1.7 & 0.44587 & 0.54113 & 21.4\% & 0.42493 & 4.70\% & 0.39777 & 10.8\% \\ \hline
1.8 & 0.46394 & 0.57296 & 23.5\% & 0.43502 & 6.23\% & 0.40279 & 13.2\% \\ \hline
1.9 & 0.48120 & 0.60479 & 25.7\% & 0.44257 & 8.03\% & 0.40465 & 15.9\% \\ \hline
2 & 0.49768 & 0.63662 & 27.9\% & 0.44741 & 10.1\% & 0.40319 & 19.0\% \\ \hline \(\infty\) & 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 2: Rotation angle at free end vs. point force for different \(\varepsilon\). From Table 2, the series solutions to rotation angle converge to the exact solution and the relative error is less than 1% for \(0<\alpha\leq 0.3\) when \(\varepsilon\) increases from 0 to 1. Thus, when \(0<\alpha\leq 0.3\), the effective region of \(\varepsilon\) is specified to be \(R_{s}=[0,1]\).