## 4 Horizontal Displacement of Cantilever Beam by IHAM  
### Zero Order Deformation Equation  
The nonlinear boundary value Equations (2.3c) and (2.4c) are then solved by IHAM to give explicit expression for \(U_{b}\). To start with, \(X(\xi)\) is expressed by the power series of \(\xi\), so that  
\[X(\xi)=\sum_{k=1}^{+\infty}c_{k}\xi^{k} \tag{4.1}\]  
where \(c_{k}\) is a constant coefficient. Linear Equation (2.7c) is specified as initial guess. Accordingly, we choose  
\[X_{0}(\xi)=\xi \tag{4.2}\]  
The equivalent nonlinear equation of Equation (2.3c) is  
\[X^{\prime}-1+\frac{\varepsilon_{2}}{2!}\theta^{2}=0 \tag{4.3}\]  
subject to boundary condition of  
\[X(0)=0 \tag{4.4}\]  
where \(\varepsilon_{2}\) is the correction coefficient of the convergent region, and \(\varepsilon_{2}\in[0,1]\). When \(\varepsilon_{2}\)\(=0\), Equation (4.3) becomes the linear Equation (2.5c), so that  
\[X^{\prime}-1=0 \tag{4.5}\]  
When \(\varepsilon_{2}\)\(=1\), Equation (4.3) becomes  
\[X^{\prime}-1+\frac{1}{2!}\theta^{2}=0 \tag{4.6}\]  
Approximation expression of the second-order [PERSON] expansion of cosine function can be written as  
\[\cos\theta\approx 1-\frac{1}{2!}\theta^{2} \tag{4.7}\]  
Figure 2: Comparison of vertical displacement at the free end \(V_{b}\) with \(n=30\). According to the original Equation (2.3c) and the equivalent Equation (4.3), we define a nonlinear operator as  
\[ \begin{array}{c}\mathrm{N}_{u}[\mu(\xi;q),\psi(\xi,q),q,\varepsilon_{2}]=\frac{ \mathrm{d}\mu(\xi;q)}{\mathrm{d}\xi}-\cos[q\cdot\psi(\xi,q)]\\ +\frac{\varepsilon_{2}}{2!}\big{(}1-q^{2}\big{)}[\psi(\xi,q)]^{2}\end{array}  \tag{4.8}\]  
where \(q\in[0,1]\) is an embedding parameter; \(\mu(\xi;q)\), \(\psi(\xi;q)\) are the functions dependent on \(\xi\) and \(q\); \(\varepsilon_{2}\) is a control parameter for the convergent region, and \(\varepsilon_{2}\in[0,1]\). When \(q=0\), there is  
\[\mathrm{N}_{u}[\mu(\xi;0),\psi(\xi,0),0,\varepsilon_{2}]=\frac{\mathrm{d}\mu( \xi;0)}{\mathrm{d}\xi}-1+\frac{\varepsilon_{2}}{2!}[\psi(\xi,0)]^{2} \tag{4.9}\]  
If \(q=1\), it yields  
\[\mathrm{N}_{u}[\mu(\xi;1),\psi(\xi,1),1,\varepsilon_{2}]=\frac{\mathrm{d}\mu( \xi;1)}{\mathrm{d}\xi}-\sin[\psi(\xi,1)] \tag{4.10}\]  
Specifying  
\[\mathrm{L}_{u}[\mu(\xi;q)]=\frac{\partial\mu(\xi;q)}{\partial\xi} \tag{4.11}\]  
as an auxiliary linear operator, and denoting \(h_{2}\in[-1,0)\) as a nonzero auxiliary parameter (convergence-control parameter), a homotopy is constructed as  
\[\mathcal{H}_{u}[\mu(\xi;q),q]:=(1-q)\mathrm{L}_{u}[\mu(\xi;q)-X_{0}(\xi)]\]  
\[-qh_{2}\mathrm{N}_{u}[\mu(\xi;q),\psi(\xi,q),q,\varepsilon_{2}] \tag{4.12}\]  
Accordingly, when \(q=0\), we have  
\[\mathcal{H}_{u}[\mu(\xi;0),0]:=\mathrm{L}_{u}[\mu(\xi;0)-X_{0}(\xi)] \tag{4.13}\]  
If \(q=1\), it becomes  
\[ \begin{array}{c}\mathcal{H}_{u}[\mu(\xi;1),1]:=-h_{2}\mathrm{N}_{u}[\mu(\xi ;1),1,\varepsilon_{2}]\\ =-h_{2}\Big{\{}\frac{\mathrm{d}\mu(\xi;1)}{\mathrm{d}\xi}-\cos[\psi(\xi,1)] \Big{\}}\end{array}  \tag{4.14}\]  
Thus, by enforcing  
\[\mathcal{H}_{u}[\mu(\xi;q),q]=0 \tag{4.15}\]  
we have a family of equations  
\[(1-q)\mathrm{L}_{u}[\mu(\xi;q)-X_{0}(\xi)]=qh_{2}\mathrm{N}_{u}[\mu(\xi;q), \psi(\xi,q),q,\varepsilon_{2}] \tag{4.16}\]  
subject to boundary conditions  
\[\mu(0;q)=0 \tag{4.17}\]  
When \(q=0\), we have  
\[\mu(\xi;0)=X_{0}(\xi),\ \psi(\xi;0)=\theta_{0}(\xi) \tag{4.18}\]  
When \(q=1\), Equations (4.16) and (4.17) are equivalent to the original Equations (2.3c) and (2.4c), provided that  
\[\mu(\xi;1)=X(\xi),\ \psi(\xi;1)=\theta(\xi) \tag{4.19}\]  
Thus, as \(q\) increases from \(0\) to \(1\), \(\mu(\xi;q)\) evolves from the known initial guess solution \(X_{0}(\xi)\) to the exact solution \(X(\xi)\) of Equations (2.3c) and (2.4c).