## 3 Vertical Displacement of Cantilever Beam by IHAM
### Vertical Displacement at Free end of Cantilever Beam
#### 3.3.5 Homotopy-Pade Approximation  
By application of the homotopy-Pade acceleration technique [27], the convergence region can be greatly enlarged. For example, when \(n=30\), \(\varepsilon_{1}=0\), \(h_{1}=-0.3\), the \(Pade[11,11]\) homotopy-Pade approximation of \(V_{b}\) in Equation (3.44) by HAM can be written as  
\[V_{b}=\frac{\alpha}{3}\frac{f(\alpha)}{g(\alpha)} \tag{3.49a}\] \[\begin{array}{c}f(\alpha)=1+0.31708\alpha^{2}+6.1434\times 10^{-2} \alpha^{4}+6.0404\times 10^{-3}\alpha^{6}\\ +4.1539\times 10^{-4}\alpha^{8}+5.5063\times 10^{-6}\alpha^{10}\end{array}\] (3.49b) \[\begin{array}{c}g(\alpha)=1+0.42567\alpha^{2}+8.7988\times 10^{-2} \alpha^{4}+1.0914\times 10^{-2}\alpha^{6}\\ +8.0640\times 10^{-4}\alpha^{8}+2.8143\times 10^{-6}\alpha^{10}\end{array} \tag{3.49c}\]  
When the relative error \(\Delta\leq 1\%\), the convergent regions of the \(Pade[11,11]\) homotopy-Pade approximation by HAM are  
\[\alpha\in[0,5],\ V_{b}\in[0,0.71],\ 0\leq\delta_{v}\leq 0.71L \tag{3.50}\]  
When \(n=30\), \(\varepsilon_{1}=24(\pi-2)/\pi^{3}\), \(h_{1}=-0.3\), the \(Pade[15,15]\) homotopy-Pade approximation of \(V_{b}\) in Equation (3.47) by IHAM are  
\[V_{b}=\frac{\alpha}{3}\frac{f(\alpha)}{g(\alpha)} \tag{3.51a}\] \[\begin{array}{c}f(\alpha)=1+0.33026\alpha^{2}+7.1207\times 10^{-2} \alpha^{4}+8.9035\times 10^{-3}\alpha^{6}+9.2792\times 10^{-4}\alpha^{8}\\ +5.3226\times 10^{-5}\alpha^{10}+2.5321\times 10^{-6}\alpha^{12}+1.9082\times 10^{-8} \alpha^{14}\\ +1.043885\alpha^{2}+9.9028\times 10^{-2}\alpha^{4}+1.4763\times 10^{-2}\alpha^{6}+1.5584 \times 10^{-3}\alpha^{8}\\ +1.1572\times 10^{-4}\alpha^{10}+5.5647\times 10^{-6}\alpha^{12}+1.3297\times 10^{-7} \alpha^{14}\end{array} \tag{3.51b}\]When the relative error \(\Delta\leq 1\%\), the convergent regions of the \(Pade[15,15]\) homotopy-Pade approximation by IHAM are  
\[\alpha\in[0,8.1],\ V_{b}\in[0,0.79],\ 0\leq\delta_{v}\leq 0.79L \tag{3.52}\]  
Table 5 shows that numerical solutions \(V_{b}^{[P]}\) of homotopy-Pade approximation by IHAM compares with exact solutions \(V_{b}\). The relative error in Table 5 is calculated by  
\[\Delta=\frac{\left|V_{b}^{[P]}-V_{b}\right|}{V_{b}}\times 100\% \tag{3.53}\]  
From Equations (3.50) and (3.52), the improved homotopy analysis method extends the convergence region from \(\alpha\in[0,5]\) to \(\alpha\in[0,8.1]\) for the homotopy-Pade approximate solution compared with the homotopy analysis method [32], which is a 62% increase.  
Figure 2 compares the 30 th-order iteration solution and the exact solution of the vertical displacement. From Figure 2, it can be observed that  
* the difference between the linear solution and the exact solution is remarkable;the 30 th-order improved homotopy analysis solution for \(\varepsilon_{1}=24(\pi-2)/\pi^{3}\) is much closer to the exact solution than does the 30 th-order homotopy analysis solution for \(\varepsilon_{1}\)\(=0\);
* the convergence range and rate have been greatly improved by the expression by homotopy-Pade approximation;
* the Pade approximation solution \(Pade[15,15]\) by IHAM is more approximating to the exact solution than does the Pade approximation solution \(Pade[11,11]\) by HAM.