## 5 Discussions and Conclusions
### Error Analysis of Horizontal Displacement  
The influence of the control parameter \(\varepsilon_{2}\) and \(h_{2}\) on the error of horizontal displacement after 5 iterations when \(\alpha\!=\!2.0\) is then discussed.  
After five iterations, the improved homotopy analysis method is adopted to solve the Equations (4.24) and (4.25) to derive the solution to the original Equations (2.3c) and (2.4c). Thus,  
\[\begin{array}{l}U_{b}=-0.20000h_{2}\alpha^{2}-0.20000h_{2}^{2}\alpha^{2}-0. 066667h_{2}^{3}\alpha^{2}\\ -0.13333\varepsilon_{2}h_{2}\alpha^{2}-0.46667\varepsilon_{2}h_{2}^{2}\alpha^{ 2}-0.60000\varepsilon_{2}h_{2}^{3}\alpha^{2}\\ -0.3333\varepsilon_{2}h_{2}^{4}\alpha^{2}-0.066667\varepsilon_{2}h_{2}^{5} \alpha^{2}+0.010871h_{2}\alpha^{4}\\ +3.7741\times 10^{-3}h_{2}^{2}\alpha^{4}+0.017903\varepsilon_{2}h_{2}\alpha^{4}+0.027904\varepsilon_{2}h_{2}^{2}\alpha^{4}\\ +0.017361\varepsilon_{2}h_{2}^{3}\alpha^{4}+3.7741\times 10^{-3}\varepsilon_{2}h_{2} ^{4}\alpha^{4}-2.6973\times 10^{-4}h_{2}\alpha^{6}\\ -1.6042\times 10^{-3}\varepsilon_{2}h_{2}\alpha^{6}-1.1329\times 10^{-3}\varepsilon_{ 2}h_{2}^{2}\alpha^{6}\\ -2.6973\times 10^{-4}\varepsilon_{2}h_{2}^{3}\alpha^{6}+8.2194\times 10^{-5} \varepsilon_{2}h_{2}\alpha^{8}\\ +2.1630\times 10^{-5}\varepsilon_{2}h_{2}^{2}\alpha^{8}-1.8596\times 10^{-6} \varepsilon_{2}h_{2}\alpha^{10}\end{array} \tag{5.6}\]  
When \(\alpha\!=\!2.0\), Equation (5.6) becomes  
\[\begin{array}{l}U_{b}=-0.64333h_{2}-0.73961h_{2}^{2}-0.26667h_{2}^{3}\\ -0.33042\varepsilon_{2}h_{2}-1.4872\varepsilon_{2}h_{2}^{2}-2.1395\varepsilon_ {2}h_{2}^{3}\\ -1.2729\varepsilon_{2}h_{2}^{4}-0.26667\varepsilon_{2}h_{2}^{5}\end{array} \tag{5.7}\]  
When \(\alpha\!=\!2.0\), the exact solution is \(U_{b}=0.16064\). After 5 time iterations, the relative error between the results by IHAM method and the exact solution can be written as  
\[\Delta_{U}=\frac{\left|U_{b}^{[n=5]}-U_{b}\right|}{U_{b}}\times 100\% \tag{5.8}\]  
When \(h_{2}=-0.465\), Equation (5.7) becomes  
\[U_{b}=0.16603-0.0065196\varepsilon_{2} \tag{5.9}\]According to Equations (5.8) and (5.9), the relative error curve of control parameter \(\Delta_{U}-\varepsilon_{2}\) is a broken line. When the relative error is less than \(\Delta_{U}\leq 1\%\), the convergence region is \(R_{\varepsilon_{2}}=[0.58,1]\), which is the effective region of \(\varepsilon_{2}\).  
Figure 6 shows the relative errors curve \(\Delta_{U}-\varepsilon_{2}\) of Equation (2.3c) when \(\alpha=2.0\), \(h_{2}=-0.465\). From Figure 6, when \(\varepsilon_{2}=8/\pi^{2}\), the relative error is the minimum.  
When \(\varepsilon_{2}=8/\pi^{2}\), Equation (5.7) becomes  
\[\begin{array}{l}U_{b}=-0.91116h_{2}-1.9451h_{2}^{2}-2.0009h_{2}^{3}\\ -1.0318h_{2}^{4}-0.21615h_{2}^{5}\end{array} \tag{5.10}\]  
According to Equations (5.8) and (5.10), the relative error curve of control parameter \(\Delta_{U}-h_{2}\) is a quintic polynomial curve. When the relative error is less than \(\Delta_{U}\leq 1\%\), the convergence interval is \(R_{h_{2}}=[-0.75,-0.42]\), which is the effective region of \(h_{2}\).  
Figure 7 depicts the relative errors curve \(\Delta_{U}-h_{2}\) of Equation (2.3c) when \(\alpha=2.0\), \(\varepsilon_{2}=8/\pi^{2}\).