## 3 Numerical Solution
By substituting Equation (56) into Equation (50), we get  
\[w(x,t)=\psi(x)\cdot q(t) \tag{43}\]5  
We now introduce the independent time variables as follows [55]:  
\[w(x,t)=\psi(x)\cdot q(t) \tag{43}\]6  
Therefore, the derivatives with respect to the time it takes the following form due to the chain rule:  
\[w(x,t)=\psi(x)\cdot q(t) \tag{43}\]7\[w(x,t)=\psi(x)\cdot q(t) \tag{43}\]8  
where  
\[w(x,t)=\psi(x)\cdot q(t) \tag{43}\]9  
Following the MTS method, the solution of Equation (57) can be expressed as follows [8]:  
\[\psi(x)=(cos\beta x-cosh\beta x)-\frac{cos\beta l+cosh\beta l}{sin\beta l+sinh \beta l}(sin\beta x-sinh\beta x),\quad\beta l=1.8751 \tag{44}\]0  
Substituting Equation (62) into Equation (57) combined with Equations (58)-(60), and by equating the coefficients of similar power of \(\epsilon\) to zero yields  
\[\psi(x)=(cos\beta x-cosh\beta x)-\frac{cos\beta l+cosh\beta l}{sin\beta l+sinh \beta l}(sin\beta x-sinh\beta x),\quad\beta l=1.8751 \tag{44}\]1  
\[\psi(x)=(cos\beta x-cosh\beta x)-\frac{cos\beta l+cosh\beta l}{sin\beta l+sinh \beta l}(sin\beta x-sinh\beta x),\quad\beta l=1.8751 \tag{44}\]2  
\[\psi(x)=(cos\beta x-cosh\beta x)-\frac{cos\beta l+cosh\beta l}{sin\beta l+sinh \beta l}(sin\beta x-sinh\beta x),\quad\beta l=1.8751 \tag{44}\]3  
whose general solution can be defined as follows:  
\[\psi(x)=(cos\beta x-cosh\beta x)-\frac{cos\beta l+cosh\beta l}{sin\beta l+sinh \beta l}(sin\beta x-sinh\beta x),\quad\beta l=1.8751 \tag{44}\]4  
\(A(T_{1},T_{2})\) being an unknown complex function. It can be determined by eliminating the secular terms from \(q_{2}\) and \(q_{3}\). At the same time, \(\overline{A}(T_{1},T_{2})\) is the complex conjugate of \(A\). The substitution of Equation (66) into Equation (64) yields  
\[\psi(x)=(cos\beta x-cosh\beta x)-\frac{cos\beta l+cosh\beta l}{sin\beta l+sinh \beta l}(sin\beta x-sinh\beta x),\quad\beta l=1.8751 \tag{44}\]5  
\(cc\) is the complex conjugate of the preceding terms. The secular terms are simplified from \(q_{2}\) by equating the coefficients of \(e^{\pm i\omega_{0}T_{0}}\) to zero as follows:  
\[\psi(x)=(cos\beta x-cosh\beta x)-\frac{cos\beta l+cosh\beta l}{sin\beta l+sinh \beta l}(sin\beta x-sinh\beta x),\quad\beta l=1.8751 \tag{44}\]6  
Therefore, it is  
\[\psi(x)=(cos\beta x-cosh\beta x)-\frac{cos\beta l+cosh\beta l}{sin\beta l+sinh \beta l}(sin\beta x-sinh\beta x),\quad\beta l=1.8751 \tag{44}\]7  
Based on Equation (68), the particular solution of Equation (67) is defined as follows:  
\[\psi(x)=(cos\beta x-cosh\beta x)-\frac{cos\beta l+cosh\beta l}{sin\beta l+sinh \beta l}(sin\beta x-sinh\beta x),\quad\beta l=1.8751 \tag{44}\]8  
Bu substituting Equations (66) and (70) into Equation (65) yields  
\[ \begin{array}{rl}\big{(}D_{0}^{2}+\omega_{0}^{2}\big{)}q_{3}=&-\Bigg{[}\hat {\eta}_{1}i\omega_{0}+\frac{2\hat{\eta}_{2}^{2}}{3\omega_{0}^{2}}+\hat{\eta}_{ 3}\Bigg{]}A^{3}e^{3i\omega_{0}T_{0}}\\ &+\Bigg{[}A^{2}\overline{A}\left(\frac{10\hat{\eta}_{2}^{2}-9\omega_{0}^{2} \hat{\eta}_{3}^{2}}{3\omega_{0}^{2}}\right)-2i\omega_{0}(D_{2}A+\mu_{0}A\\ &+\frac{1}{2}\hat{\eta}_{1}A^{2}\overline{A}\Bigg{)}\Bigg{]}e^{i\omega_{0}T_{0 }}+cc\end{array}  \tag{71}\]The secular terms are simplified from \(q_{3}\) by equating to zero the coefficients of \(e^{\pm i\omega_{0}T_{0}}\) in Equation (71), as follows:  
\[-2i\omega_{0}\bigg{(}D_{2}A+\mu_{0}A+\frac{1}{2}\hat{\mu}_{1}A^{2}\overline{A} \bigg{)}+A^{2}\overline{A}\Bigg{(}\frac{10\hat{\mu}_{2}^{2}-9\omega_{0}^{2} \hat{\mu}_{3}}{3\omega_{0}^{2}}\bigg{)}=0 \tag{72}\]  
where \(A(T_{2})\) is defined in polar form as follows [8]:  
\[A=\frac{1}{2}ae^{i\beta} \tag{73}\]  
\(a\) and \(\beta\) being real functions of \(T_{2}\).