## 3 Operator Reformulation of the Problem
 \tag{30}\]  
Since \(\mathbb{F}\in\mathcal{D}(\mathbb{L})\), i.e., \(\varphi_{0}(L)-\varphi_{2}^{\prime}(L)=0\), and \(\mathbb{G}\in\mathcal{D}(\mathbb{L})\), i.e., \(g_{3}(0)=0\), we simplify \(I_{1}\):  
\[I_{1}=\int_{0}^{L}K(\xi)\Big{(}\varphi_{0}(\xi)-\varphi_{2}^{\prime}(\xi)\Big{)} \overline{\widetilde{g}}_{3}(\xi)d\xi. \tag{31a}\]Evaluating the second integral from Equation (29), we get  
\[ \begin{split} I_{2}&=\int_{0}^{L}S\rho(\xi)\bigg{[}\frac{1 }{S\rho(\xi)}\bigg{(}\Big{(}EI(\xi)\varphi_{0}^{\prime}(\xi)\Big{)}^{\prime}-K( \xi)\varphi_{0}(\xi)\bigg{)}+\frac{K(\xi)}{S\rho(\xi)}\varphi_{2}^{\prime}(\xi) \bigg{]}\overline{g}_{1}(\xi)d\xi\\ &=EI(L)\varphi_{0}^{\prime}(L)\overline{g}_{1}(L)-\int_{0}^{L}EI( \xi)\varphi_{0}^{\prime}(\xi)\overline{g^{\prime}}_{1}(\xi)d\xi-\int_{0}^{L}K( \xi)\Big{[}\varphi_{0}(\xi)-\varphi_{2}^{\prime}(\xi)\Big{]}\overline{g}_{1}( \xi)d\xi.\end{split} \]  
Since \(\mathbb{F}\in\mathcal{D}(\mathbb{L})\), i.e., \(EI(L)\varphi_{0}^{\prime}(L)=\theta\varphi_{4}\), we obtain for \(I_{2}\)  
\[I_{2}=\theta\varphi_{4}\overline{g}_{1}(L)-\int_{0}^{L}EI(\xi)\varphi_{0}^{ \prime}(\xi)\overline{g^{\prime}}_{1}(\xi)d\xi-\int_{0}^{L}K(\xi)\Big{[} \varphi_{0}(\xi)-\varphi_{2}^{\prime}(\xi)\Big{]}\overline{g}_{1}(\xi)d\xi. \tag{31b}\]  
For the integrals \(I_{3},\ I_{4}\) and \(I_{5}\), we have  
\[I_{3}=\int_{0}^{L}EI(\xi)\varphi_{1}^{\prime}(\xi)\overline{g^{\prime}}_{0}( \xi)d\xi, \tag{31c}\]  
\[I_{4}=\int_{0}^{L}K(\xi)\Big{(}\varphi_{1}(\xi)-\varphi_{3}^{\prime}(\xi) \Big{)}\Big{(}\overline{g_{0}}(\xi)-\overline{g^{\prime}_{2}}(\xi)\Big{)}d\xi, \tag{31d}\]  
\[I_{5}=\bigg{(}-\frac{\theta}{C}\varphi_{1}(L)-\frac{1}{CR}\varphi_{4}\bigg{)} \overline{g^{4}}. \tag{31e}\]  
Collecting Equations (31a)-(31e), we get from Equation (29)  
\[ \begin{split} 2i\Big{(}\mathbb{L}\mathbb{F},\mathbb{G}\Big{)}_{ \mathcal{H}}&=\int_{0}^{L}K(\xi)\Big{(}\varphi_{0}(\xi)-\varphi_ {2}^{\prime}(\xi)\Big{)}\overline{g^{\prime}_{3}}(\xi)d\xi-\int_{0}^{L}K(\xi) \Big{(}\varphi_{0}(\xi)-\varphi_{2}^{\prime}(\xi)\Big{)}\overline{g}_{1}(\xi) d\xi-\\ &\int_{0}^{L}EI(\xi)\varphi_{0}^{\prime}(\xi)\overline{g}_{1}(\xi) d\xi+\theta\varphi_{4}\overline{g_{1}}(L)+\int_{0}^{L}EI(\xi)\varphi_{1}^{\prime}( \xi)\overline{g^{\prime}_{0}}(\xi)d\xi+\\ &\int_{0}^{L}K(\xi)\Big{(}\varphi_{1}(\xi)-\varphi_{3}^{\prime}( \xi)\Big{)}\Big{(}\overline{g_{0}}(\xi)-\overline{g^{\prime}_{2}}(\xi)\Big{)} d\xi-\frac{\theta}{C}\varphi_{1}(L)\overline{g_{4}}-\frac{1}{CR} \varphi_{4}\overline{g_{4}}.\end{split}  \tag{32}\]  
Thus, rearranging the terms in Equation (32), we obtain  
\[ \begin{split} 2i\Big{(}\mathbb{L}\mathbb{F},\mathbb{G} \Big{)}_{\mathcal{H}}=&-\int_{0}^{L}K(\xi)\Big{(}\varphi_{0}( \xi)-\varphi_{2}^{\prime}(\xi)\Big{)}\Big{(}\overline{g_{1}}(\xi)-\overline {g^{\prime}_{3}}(\xi)\Big{)}d\xi+\\ &\int_{0}^{L}K(\xi)\Big{(}\varphi_{1}(\xi)-\varphi_{3}^{\prime}( \xi)\Big{)}\Big{(}\overline{g_{0}}(\xi)-\overline{g^{\prime}_{2}}(\xi)\Big{)} d\xi+\\ &\int_{0}^{L}EI(\xi)\Big{(}\varphi_{1}^{\prime}(\xi)\overline{g^{ \prime}_{0}}(\xi)-\varphi_{0}^{\prime}(\xi)\overline{g^{\prime}_{1}}(\xi)\Big{)} d\xi-\\ &\frac{\theta}{C}\varphi_{1}(L)\overline{g_{4}}-\frac{1}{CR} \varphi_{4}\overline{g_{4}}+\theta\varphi_{4}\overline{g_{1}}.\end{split}  \tag{33}\]  
Now, we use the definition of \(\mathbb{L}^{*}\) to evaluate \(\Big{(}\mathbb{F},\mathbb{L}^{*}\mathbb{G}\Big{)}_{\mathcal{H}}\) and have  
\[\Big{(}\mathbb{L}^{*}\mathbb{G}\Big{)}(x)=-i\left[ \begin{array}{c}g_{1}(x)\\ \frac{1}{S\rho(x)}\bigg{(}\Big{(}EI(x)g^{\prime}_{0}(x)\Big{)}^{\prime}-K(x)g_{ 0}(x)\bigg{)}+\frac{K(x)}{S\rho(x)}g^{\prime}_{2}(x)\\ g_{3}(x)\\ -\frac{1}{\rho(x)}\Big{(}K(x)g_{0}(x)\Big{)}^{\prime}+\frac{1}{\rho(x)}\Big{(}K( x)g^{\prime}_{2}(x)\Big{)}^{\prime}\\ \Theta g_{1}(L)-\frac{1}{CR}\overline{g_{4}}\end{array} \right]=\left[ \begin{array}{c} \widetilde{g_{0}}(x)\\ \widetilde{g_{1}}(x)\\ \widetilde{g_{2}}(x)\\ \widetilde{g_{3}}(x)\\ \widetilde{g_{4}}\end{array} \right]. \tag{34}\]Evaluating an inner product induced by the norm in Equation (19) yields  
\[