## 2 Basic Equations
\begin{array}{l}\frac{\partial w}{\partial x}=\sum\limits_{m=1,3}^{\infty} \sum\limits_{n=1,3}^{\infty}\left(-\frac{2}{a}c_{n}+\frac{\delta_{m}y}{2}w_{mn }\right)\cos\frac{\delta_{m}x}{2}\sin\frac{\lambda_{n}y}{2}\\ \frac{\partial^{2}w}{\partial x^{2}}=\sum\limits_{m=1,3}^{\infty}\sum\limits _{n=1,3}^{\infty}\left\{\frac{2}{a}\Big{[}(-1)^{\frac{n-1}{2}}d_{n}+\frac{ \delta_{m}}{2}c_{n}\Big{]}-\Big{(}\frac{\delta_{m}}{2}\Big{)}^{2}w_{mn}\right\} \sin\frac{\delta_{m}x}{2}\sin\frac{\lambda_{n}y}{2}\\ \frac{\partial^{3}w}{\partial x^{3}}=\sum\limits_{m=1,3}^{\infty}\sum\limits _{n=1,3}^{\infty}\left\{\frac{2}{a}\Big{[}-e_{n}+(-1)^{\frac{n-1}{2}}\frac{ \delta_{m}}{2}d_{n}+\Big{(}\frac{\delta_{m}}{2}\Big{)}^{2}c_{n}\Big{]}-\Big{(} \frac{\delta_{m}}{2}\Big{)}^{3}w_{mm}\right\}\cos\frac{\delta_{m}x}{2}\sin \frac{\lambda_{n}y}{2}\\ \frac{\partial^{4}w}{\partial x^{4}}=\sum\limits_{m=1,3}^{\infty}\sum\limits _{n=1,3}^{\infty}\left\{\frac{2}{a}\Bigg{[}\ \ \ (-1)^{\frac{n-1}{2}}f_{n}+\frac{\delta_{m}}{2}e_{n}\right.\\ \left.-(-1)^{\frac{n-1}{2}}\Big{(}\frac{\delta_{m}}{2}\Big{)}^{2}d_{n}-\Big{(} \frac{\delta_{m}}{2}\Big{)}^{3}c_{n}\ \Bigg{]}+\Big{(}\frac{\delta_{m}}{2}\Big{)}^{4}w_{mm}\right\}\sin \frac{\delta_{m}x}{2}\sin\frac{\lambda_{n}y}{2}\\ \end{array}  \tag{8}\]  
\[ \begin{array}{l}\frac{\partial w}{\partial y}=\sum\limits_{m=1,3}^{\infty} \sum\limits_{n=1,3}^{\infty}\left(-\frac{2}{b}g_{m}+\frac{\lambda_{n}}{2}w_{mm }\right)\sin\frac{\delta_{m}x}{2}\cos\frac{\lambda_{n}y}{2}\\ \frac{\partial^{2}w}{\partial y^{2}}=\sum\limits_{m=1,3}^{\infty}\sum\limits _{n=1,3}^{\infty}\left\{\frac{2}{b}\Big{[}(-1)^{\frac{n-1}{2}}h_{m}+\frac{ \lambda_{n}}{2}g_{m}\Big{]}-\Big{(}\frac{\lambda_{n}}{2}\Big{)}^{2}w_{mm} \right\}\sin\frac{\delta_{m}x}{2}\sin\frac{\lambda_{n}y}{2}\\ \frac{\partial^{3}w}{\partial y^{3}}=\sum\limits_{m=1,3}^{\infty}\sum\limits_{n=1,3}^{\infty}\left\{\frac{2}{b}\Big{[}-i_{m}+(-1)^{\frac{n-1}{2}}\frac{ \lambda_{n}}{2}h_{m}+\Big{(}\frac{\lambda_{n}}{2}\Big{)}^{2}g_{m}\Big{]}-\Big{(} \frac{\lambda_{n}}{2}\Big{)}^{3}w_{mm}\right\}\sin\frac{\delta_{m}x}{2}\cos \frac{\lambda_{n}y}{2}\\ \frac{\partial^{4}w}{\partial y^{4}}=\sum\limits_{m=1,3}^{\infty}\sum\limits_{n=1,3}^{\infty}\left\{\frac{2}{b}\Bigg{[}\ \ \ (-1)^{\frac{n-1}{2}}j_{m}+\frac{\lambda_{n}}{2}i_{m}\right.\\ \left.-(-1)^{\frac{n-1}{2}}\Big{(}\frac{\lambda_{n}}{2}\Big{)}^{2}h_{m}-\Big{(} \frac{\lambda_{n}}{2}\Big{)}^{3}g_{m}\ \Bigg{]}+\Big{(}\frac{\lambda_{n}}{2}\Big{)}^{4}w_{mm}\right\}\sin \frac{\delta_{m}x}{2}\sin\frac{\lambda_{n}y}{2}\\ \end{array}  \tag{9}\]  
\[\begin{array}{l}\frac{\partial^{2}w}{\partial x\partial y}=\sum\limits_{m=1,3}^{ \infty}\sum\limits_{n=1,3}^{\infty}\left(-\frac{2}{b}\frac{\delta_{m}}{2}g_{m}+ \frac{\delta_{m}}{2}\frac{\lambda_{n}}{2}w_{mm}\right)\cos\frac{\delta_{m}x}{2} \cos\frac{\lambda_{n}y}{2}\\ \frac{\partial^{2}w}{\partial x\partial y^{2}}=\sum\limits_{m=1,3}^{\infty}\sum \limits_{n=1,3}^{\infty}\left\{-\frac{2}{a}l_{n}+\frac{2}{b}\frac{\delta_{m}}{2} \Big{[}(-1)^{\frac{n-1}{2}}h_{m}+\frac{\lambda_{n}}{2}g_{m}\Big{]}-\frac{ \delta_{m}}{2}\Big{(}\frac{\lambda_{n}}{2}\Big{)}^{2}w_{mm}\right\}\cos\frac{ \delta_{m}x}{2}\sin\frac{\lambda_{n}y}{2}\\ \frac{\partial^{3}w}{\partial x^{2}\partial y}=\sum\limits_{m=1,3}^{\infty}\sum \limits_{n=1,3}^{\infty}\left\{-\frac{2}{b}k_{m}+\frac{2}{a}\frac{\lambda_{n}}{2} \Big{[}(-1)^{\frac{n-1}{2}}d_{n}+\frac{\delta_{m}}{2}c_{n}\Big{]}-\Big{(} \frac{\delta_{m}}{2}\Big{)}^{2}\frac{\lambda_{n}}{2}w_{mm}\right\}\sin\frac{ \delta_{m}x}{2}\cos\frac{\lambda_{n}y}{2}\\ \frac{\partial^{4}w}{\partial x^{2}\partial y^{2}}=\sum\limits_{m=1,3}^{\infty} \sum\limits_{n=1,3}^{\infty}\left\{ \begin{array}{l}\ \frac{4}{ab}(-1)^{\frac{m-1}{2}}\bigg{[}(-1)^{\frac{n-1}{2}}\frac{ \partial^{2}w}{\partial x\partial y}\bigg{|}_{x=a,y=b}+\frac{\lambda_{n}}{2} \frac{\partial w}{\partial x}\bigg{|}_{x=a,y=b}\end{array} \right]\\ -\frac{2}{a}\Big{[}(-1)^{\frac{n-1}{2}}\Big{[}(\frac{\lambda_{n}}{2}\Big{)}^{2}d_{ n}-\frac{2}{b}\frac{\delta_{m}}{2}l_{n}\Big{]}\]where  
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