## 3 Solution Methodology  
### Boundary Discontinues Fourier Series Method  
We proposed a boundary discontinuous Fourier series method as the first proposed analytical solution. The functions expressing a displacement solution to the boundary value problem when mixed-type boundary conditions were included into the mathematical formulation initially. These functions, which are adapted for FSDT-based cross-ply panels, are as follows:  
\[\{u_{1},\theta_{1}\} = \sum_{m=0}^{\infty}\sum_{n=1}^{\infty}\{\mathrm{U}_{\mathrm{mn}}, X_{\mathrm{mn}}\}\cos(\alpha\xi_{1})\sin(\beta\xi_{2}),\quad 0<\xi_{1}<a;0<\xi_{2}<b \tag{22}\] \[\{u_{2},\theta_{2}\} = \sum_{m=1}^{\infty}\sum_{n=0}^{\infty}\{\mathrm{V}_{\mathrm{mn}}, Y_{\mathrm{mn}}\}\sin(\alpha\xi_{1})\cos(\beta\xi_{2}),\quad 0<\xi_{1}<a;0<\xi_{2}<b\] (23) \[\{u_{3}\} = \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\{W_{\mathrm{mn}}\}\sin( \alpha\xi_{1})\sin(\beta\xi_{2}),\quad 0\leq x_{1}\leq a;0\leq x_{2}\leq b \tag{24}\]  
The unknown coefficients (\(U_{\mathrm{mn}}\), \(V_{\mathrm{mn}}\), \(W_{\mathrm{mn}}\), \(X_{\mathrm{mn}}\), \(Y_{\mathrm{mn}}\)) regarding the displacements are represented as \(5\,mn+2\,m+2\,n\). The assumed solution functions in Equations (22)-(24) do not entirely ensure that the prescribed boundary conditions. Thus, whenever the relevant boundary conditions are not satisfied, the assumed solution function or its derivative at the boundaries is forced to satisfy it. The derivatives of the assumed solution functions are achieved by expanding them in a double Fourier series in the form discussed by [PERSON] [46], [PERSON] [47] and [PERSON] [21,22]. Boundary Fourier coefficients arising from the discontinuities of the assumed functions or their derivatives at the boundaries are handled by utilizing the Lebesgue integration theory. This theory was considered the procedure for the differentiation of these functions. The function \(u_{1}\) given by Equation (22) and its first partial derivative, \(u_{1,1}\), given by Equation (25) and obtained by term-wise differentiation, are not satisfied at the edges, \(\xi_{1}=0\), a, and thus violate the boundary constraints and complementary boundary constraints, respectively, at these edges. Therefore, \(u_{1}\) is forced to vanish at these edges, while for further differentiation, \(u_{1,11}\) is expanded in a double Fourier series in the form advised by [PERSON] [21,22], thereby satisfying the complementary boundary constraints (inequality). The partial derivatives are obtained as follows:  
\[u_{1,1}=-\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\alpha U_{mn}\sin(\alpha\xi_{1} )\sin(\beta\xi_{2}) \tag{25}\]  
\[u_{1,11}=\sum_{m=1}^{\infty}\frac{1}{2}\overline{a}_{n}\sin(\beta\xi_{2})+\sum _{m=1}^{\infty}\sum_{n=1}^{\infty}\Bigl{[}-\alpha^{2}U_{mn}+\gamma_{m} \overline{a}_{n}+\psi_{m}\overline{b}_{n}\Bigr{]}\cos(\alpha\xi_{1})\sin(\beta \xi_{2}), \tag{26}\]  
\[u_{1,22}=-\frac{1}{2}\sum_{n=1}^{\infty}\beta^{2}U_{0n}\sin(\beta\xi_{2})-\sum _{m=1}^{\infty}\sum_{n=1}^{\infty}\beta^{2}U_{mn}\cos(\alpha\xi_{1})\sin( \beta\xi_{2}) \tag{27}\]  
In the same way as \(u_{1},\ u_{3,1}\) is forced to vanish at the edges, \(\xi_{1}=0\) and a. Thus, the derivatives of \(u_{3,1}\) and \(u_{3,11}\) are obtained as shown in Equations (28) and (29) by expanding them in a double Fourier series as previously discussed. \[u_{3,1}=\sum_{m=1}^{\infty}\frac{1}{2}\overline{c}_{n}\sin(\beta\xi_{2})+\sum _{m=1}^{\infty}\sum_{n=1}^{\infty}\Bigl{[}\alpha W_{mn}+\gamma_{m}\overline{c} _{n}+\psi_{m}\overline{d}_{n}\Bigr{]}\cos(\alpha\tilde{\xi}_{1})\sin(\beta \xi_{2}) \tag{28}\]  
\[u_{3,11}=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}-\alpha\Bigl{[}\alpha W_{mn}+ \gamma_{m}\overline{c}_{n}+\psi_{m}\overline{d}_{n}\Bigr{]}\sin(\alpha\tilde{ \xi}_{1})\sin(\beta\xi_{2}) \tag{29}\]in which  
\[(\gamma_{n},\delta_{n})= \begin{cases}(0,1),&n=\text{odd}\\ (1,0),&n=\text{even}\end{cases}  \tag{30}\]  
Other derivatives of the assumed solution functions can be obtained through term-wise differentiation. The details of the proposed method are available in [PERSON] [21; 22].