## 4 Calculation Results
### Effect of Auxiliary Parameter \(\varepsilon\) on Convergence  
It is expected that the improved homotopy analysis method derives a family of series solutions with the variable auxiliary parameter \(\varepsilon\). The efficiency of the IHAM lies in the appropriate identification for the value of \(\varepsilon\) to ensure a convergent solution promptly in a sufficiently large region. The parameters in nonlinear problems could contain physical connotations, such as the angular frequency of nonlinear vibration, the wall friction of viscous flow, etc. These parameters are usually correlated to the auxiliary parameter \(\varepsilon\). Therefore, the curve of these physical quantities can be drawn with respect to \(\varepsilon\) by regarding the auxiliary parameter \(\varepsilon\) as an independent variable. For example, below \(\Theta_{b}\) is a physical quantity  
\[\Theta_{b}=\left.\Theta(\xi,\varepsilon)\right|_{\xi-1} \tag{80}\]  
If \(\Theta_{b}\) is a function of, \(\varepsilon\), \(\Theta_{b}-\varepsilon\) can be identified. According to Theorem 1 or Theorem 2, all convergent series of \(\Theta_{b}\) given by different values of \(\varepsilon\) converge to the exact solution. If the solutions are unique, they should all converge to the same value. Therefore, there is a horizontal segment in the \(\Theta_{b}-\varepsilon\) curve to imply the converge region of \(\varepsilon\) represented by \(R_{\varepsilon}\). For simplicity, such a curve is called a \(\varepsilon\) curve, and the corresponding region \(R_{\varepsilon}\) is the effective region of \(\varepsilon\). Obviously, if the value of \(\varepsilon\) falls in the valid region, the series solution will converge without fail. Even though \(\Theta_{b}\) does not make any physical sense, the corresponding \(\varepsilon\) curve can still be attempted. The more \(\varepsilon\) curves that are drawn, the more specific value of \(\varepsilon\) will be identified. If the initial guess solution \(\theta_{0}(\xi)\), auxiliary linear operator \(\mathcal{L}\) and the auxiliary parameter \(\hbar\) are given, the effective region of \(\varepsilon\) for different physical quantities is usually almost the same. But so far this speculation has not been proved mathematically. In many cases, \(\varepsilon\) curves of physical quantities, such as the aforementioned \(\Theta_{b}-\varepsilon\) curve, can be applied to identify a suitable \(\varepsilon\) value to ensure that the series solutions \(\Theta(\xi)\) converge in the space of the whole physical context. Therefore, it can be stated that the \(\varepsilon\) curve provides an effective way to study the influence of auxiliary parameter \(\varepsilon\) on the convergence region and rate for the series solutions. By comparing the original Equation (47) with the auxiliary Equation (56), the following equation can be obtained:  
\[\cos\theta=1-\frac{\varepsilon}{2!}\theta^{2} \tag{81}\]  
Substituting \(\theta=\frac{\pi}{2}\) into the Equation (81) leads to  
\[0=1-\frac{\varepsilon}{2!}\Big{(}\frac{\pi}{2}\Big{)}^{2} \tag{82}\]  
By solving the above equations, the optimal convergence parameter in the overall regions is derived as:  
\[\varepsilon=\frac{8}{\pi^{2}} \tag{83}\]  
When \(n=1\), the \(1\)st-order approximation of \(\theta_{b}\) by IHAM is calculated by:  
\[\theta_{b}=0.5\alpha+0.045833\hbar\varepsilon\alpha^{3} \tag{84}\]  
For example, the \(1\)st-order approximation of \(\Theta_{b}\) with \(\hbar=-1,\varepsilon=8/\pi^{2}\), is  
\[\Theta_{b}=0.31831\alpha-0.023651\alpha^{3} \tag{85}\]  
In the case of \(n=1,\ \hbar=-1\), when \(\varepsilon=0,\ \varepsilon=1\), and \(\varepsilon=8/\pi^{2}\) respectively, the \(1\)st-order \(\Theta_{b}^{[n=1]}\) by IHAM and the exact solution \(\Theta_{b}\) are compared in Table 2.