## 5 Discussion about the Relative Errors  
In this section, the three iterations of \(\alpha=2.0\) are taken as an example to discuss the effect of the control parameters \(\varepsilon\) and \(\hbar\) on the relative errors.  
After three iterations, the improved homotopy analysis method is adopted to solve Equations (68) and (69) whose solution in the following form, is the approximate of the original Equations (47) and (48)  
Figure 4: Thirtieth solution to rotation angle at the free end vs. point force.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\alpha\)} & \multirow{2}{*}{
\begin{tabular}{c} **Exact** \\ **Solution** \\ \end{tabular} } & \multicolumn{2}{c}{\(\varepsilon=0\)} & \multicolumn{2}{c}{\(\varepsilon=8/\pi^{2}\)} \\  & & \multicolumn{2}{c}{**(HAM [48])**} & \multicolumn{2}{c}{**(IHAM)**} \\  & & \multicolumn{2}{c}{\(Pade[11,11]\)} & \multicolumn{2}{c}{**\(Pade[15,15]\)**} \\ \cline{2-6}  & \(\mathbf{\Theta_{b}}\) & \(\mathbf{\Theta_{b}}\) & \(\mathbf{\Delta}\) & \(\mathbf{\Theta_{b}}\) & \(\mathbf{\Delta}\) \\ \hline
7.5 & 0.86321 & 0.82935 & 3.92\% & 0.87080 & 0.879\% \\ \hline
8 & 0.87499 & 0.83850 & 4.17\% & 0.88594 & 1.25\% \\ \hline
8.5 & 0.88544 & 0.84758 & 4.27\% & 0.90079 & 1.73\% \\ \hline
9 & 0.89475 & 0.85684 & 4.24\% & 0.91557 & 2.33\% \\ \hline
9.5 & 0.90307 & 0.86644 & 4.06\% & 0.93046 & 3.03\% \\ \hline
10 & 0.91055 & 0.87648 & 3.74\% & 0.94558 & 3.85\% \\ \hline \(\infty\) & 1 & 1 & 0 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 5: _Cont._\[\theta_{b}=0.5\alpha+0.045833\mathrm{h}\alpha^{3}+0.091667\varepsilon \mathrm{h}\alpha^{3}+0.13750\mathrm{e}\mathrm{h}^{2}\alpha^{3} \tag{98}\] \[+0.045833\varepsilon\mathrm{h}^{3}\alpha^{3}+0.024400\varepsilon^{2} \mathrm{h}^{2}\alpha^{5}+0.016267\varepsilon^{2}\mathrm{h}^{3}\alpha^{5}+0.001800 \mathrm{e}^{3}\mathrm{h}^{3}\alpha^{7}\]  
When \(\alpha=2.0\), Equation (98) becomes:  
\[\begin{array}{c}\theta_{b}=1+0.36667\mathrm{h}+0.73333\mathrm{e}\mathrm{h}+1.1000\mathrm{e}\mathrm{h}^{2}+0.36667\varepsilon\mathrm{h}^{3}+0.78079 \varepsilon^{2}\mathrm{h}^{2}\\ \qquad\qquad\qquad\qquad+0.52053\varepsilon^{2}\mathrm{h}^{3}+0.23040\varepsilon ^{3}\mathrm{h}^{3}\end{array} \tag{99}\]  
When \(\alpha=2.0\), the exact solution to the ration angle at the free end is \(\Theta_{b}=0.49768\). When \(\mathrm{h}=-0.45\), Equation (99) becomes:  
\[\theta_{b}=0.83500-0.14066\varepsilon+0.11068\varepsilon^{2}-0.020995 \varepsilon^{3} \tag{100}\]  
According to Equation (100), the relative error curve with variable control parameter \(\varepsilon\) is a cubic curve. When the relative error is less than one percent, the convergence interval is \(R_{\varepsilon}=[0.5,1]\), which is the effective region of \(\varepsilon\). Figure 5 depicts the relative errors curve \(\Delta_{\Theta}-\varepsilon\) of original Equation (47) when, \(\alpha=2.0,\mathrm{h}=-0.45\).  
From Figure 5, the relative error is the minimum when \(\varepsilon=8/\pi^{2}\). When \(\varepsilon=8/\pi^{2}\), Equation (99) becomes:  
\[\theta_{b}=1+0.96108\mathrm{h}+1.4046\mathrm{h}^{2}+0.76191\mathrm{h}^{3} \tag{101}\]  
According to Equation (101), the relative error curve with variable control parameter \(\mathrm{h}\) is also cubic. When the relative error is less than one percent, the convergence interval is \(R_{t}=[-0.52,-0.4]\), which is the effective region of \(\mathrm{h}\). Figure 6 shows the relative errors curve \(\Delta_{\Theta}-\varepsilon\) of original Equation (47) when \(\alpha=20,\varepsilon=8/\pi^{2}\).  
Figure 5: Relative errors curve \(\Delta_{\Theta}-\varepsilon\) of original Equation (47) when \(\alpha=2.0,\mathrm{h}=-0.45\).  
From Figure 6, the relative error is the minimum when \(\hbar=-0.45\).