## Application of Perturbation Method
### Perturbation Solution on \(P_{m}\)  
Next, the load is selected as the perturbation parameter but not the central deflection, such that  
\[P_{m}=\frac{P}{16}=\frac{1}{16}\frac{qa^{4}}{E_{0}t^{4}}. \tag{59}\]  
Thus Equation (39) is changed as  
\[\frac{d^{2}}{d\eta^{2}}\bigg{[}(1-\eta)\frac{dW}{d\eta}\bigg{]}=\frac{S}{4K} \frac{dW}{d\eta}-\frac{P_{m}}{K} \tag{60}\]while Equation (40) and boundary conditions keep unchanged. \(W\) and \(S\) may be expanded with respect to \(P_{m}\)  
\[\left\{ \begin{array}{l}W=\overline{W}_{1}(\eta)P_{m}+\overline{W}_{3}(\eta)P_{ m}^{3}+\overline{W}_{5}(\eta)P_{m}^{5}+\cdots\\ S=\overline{S}_{2}(\eta)P_{m}^{2}+\overline{S}_{4}(\eta)P_{m}^{4}+\overline{S}_ {6}(\eta)P_{m}^{6}+\cdots\end{array} \right., \tag{61}\]  
where \(\overline{W}_{i}(i=1,3,5,\ldots)\) and \(\overline{S}_{i}(i=2,4,6,\ldots)\) are unknown functions of \(\eta\), i.e., \(W=W(P_{m},\eta)\) and \(S=S(P_{m},\eta)\). Similarly, the even exponential terms with respect to \(P_{m}\) in the expansion of \(W\), and the odd terms in \(S\) are not considered because they will be eliminated during the perturbation. By comparing the coefficients of \(P_{m}^{1}\) in Equation (60), we obtain the differential equation for \(\overline{W}_{1}\)  
\[\frac{d^{2}}{d\eta^{2}}\left[(1-\eta)\frac{d\overline{W}_{1}}{d\eta}\right]=- \frac{1}{K} \tag{62}\]  
which may be solved under the boundary conditions  
\[\left\{ \begin{array}{l}\overline{W}_{1}=0,\;\lambda_{1}\frac{d^{2}\overline{ W}_{1}}{d\eta^{2}}-\frac{d\overline{W}_{1}}{d\eta}=0,\;\mbox{at}\;\eta=0\\ \frac{d\overline{W}_{1}}{d\eta}\
eq\infty,\;\mbox{at}\;\eta=1\end{array} \right.. \tag{63}\]  
Thus, we obtain  
\[\overline{W}_{1}=\frac{\eta^{2}+2\lambda_{1}\eta}{4K}. \tag{64}\]  
By comparing the coefficients of \(P_{m}^{2}\) in Equation (40), we obtain the differential equation used for \(\overline{S}_{2}\)  
\[\frac{d^{2}}{d\eta^{2}}\left[(1-\eta)\overline{S}_{2}\right]+\frac{V}{2} \left(\frac{d\overline{W}_{1}}{d\eta}\right)^{2}=0 \tag{65}\]  
which may be solved under the boundary conditions  
\[\left\{ \begin{array}{l}\lambda_{2}\frac{d\overline{S}_{2}}{d\eta}-\overline {S}_{2}=0,\;\mbox{at}\;\eta=0\\ \overline{S}_{2}\
eq\infty,\;\mbox{at}\;\eta=1\end{array} \right.. \tag{66}\]  
Using the determined \(\overline{W}_{1}\), we obtain  
\[\overline{S}_{2}=\frac{V}{96K^{2}}[\eta^{3}+(4\lambda_{1}+1)\eta^{2}+(6{ \lambda_{1}}^{2}+4\lambda_{1}+1)\eta+\lambda_{2}(6{\lambda_{1}}^{2}+4\lambda_ {1}+1)]. \tag{67}\]  
Similarly, by comparing the coefficients of \(P_{m}^{3}\) in Equation (60), we have the differential equation for \(\overline{W}_{3}\)  
\[\frac{d^{2}}{d\eta^{2}}\left[(1-\eta)\frac{d\overline{W}_{3}}{d\eta}\right]= \frac{\overline{S}_{2}}{4K}\frac{d\overline{W}_{1}}{d\eta} \tag{68}\]  
which may be solved by the boundary conditions as follows,  
\[\left\{ \begin{array}{l}\overline{W}_{3}=0,\;\lambda_{1}\frac{d^{2}\overline{ W}_{3}}{d\eta^{2}}-\frac{d\overline{W}_{3}}{d\eta}=0,\;\mbox{at}\;\eta=0\\ \frac{d\overline{W}_{3}}{d\eta}\
eq\infty,\;\mbox{at}\;\eta=1\end{array} \right.. \tag{69}\]  
Using the determined \(\overline{W}_{1}\) and \(\overline{S}_{2}\), we obtain  
\[\overline{W}_{3}=-\frac{V}{276480K^{4}}\left\{ \begin{array}{l}2\eta^{6}+6(3 \lambda_{1}+1)\eta^{5}+15(5{\lambda_{1}}^{2}+4\lambda_{1}+1)\eta^{4}\\ +20[6{\lambda_{1}}^{3}+9{\lambda_{1}}^{2}+5\lambda_{1}+1+\lambda_{2}(6{\lambda _{1}}^{2}+4\lambda_{1}+1)]\eta^{3}\\ +30[6{\lambda_{1}}^{3}+9{\lambda_{1}}^{2}+5\lambda_{1}+1\\ +\lambda_{2}(18{\lambda_{1}}^{3}+18{\lambda_{1}}^{2}+7\lambda_{1}+1)](\eta^{2 }+2\lambda_{1}\eta)\end{array} \right\}. \tag{70}\]Similarly, by comparing the coefficients of \(P_{m}^{4}\) in Equation (40), we have the differential equation for \(\overline{S}_{4}\)  
\[\frac{d^{2}}{d\eta^{2}}\big{[}(1-\eta)\overline{S}_{4}\big{]}+V\frac{d\overline{ W}_{1}}{d\eta}\frac{d\overline{W}_{3}}{d\eta}=0 \tag{71}\]  
which may be solved following the boundary conditions, as follows,  
\[\left\{