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Novel study on functionally graded anisotropic doubly curved nanoshells

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Abstract

This work aims at investigating the dispersion of elastic eaves in doubly curved nanoshells. The nonlocal strain gradient theory is adopted in conjunction with a higher-order shear deformation shell theory, to include the size-dependent effects. The nanoshells are made of functionally graded anisotropic materials, whose properties are changed exponentially through the thickness direction. Hamilton’s principle is employed to obtain the governing equations of wave motion which are solved analytically to compute the wave frequencies as well as phase velocities as a function of the wave number. The sensitivity of the wave response is analyzed for the exponential factor, small-scale parameters, geometrical conditions as well as wave number. In addition, the accuracy of modeling the nanoshells with less elastic coefficients compared to the anisotropic model is studied to disregard any kind of complex equations.

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Correspondence to Behrouz Karami.

Appendix

Appendix

$$ \begin{aligned} & {\mathcal{L}}_{l} \left\{ {C_{61} \left( {\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \beta }} \right) + D_{61} \frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + E_{61} \frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + F_{61} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + C_{62} \left( {\frac{{\partial^{2} v_{0} }}{{\partial \beta^{2} }} + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \beta }} \right)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {D_{62} \frac{{\partial^{2} \theta_{\beta } }}{{\partial \beta^{2} }} + E_{62} \frac{{\partial^{2} v_{0}^{*} }}{{\partial \beta^{2} }} + F_{62} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \beta^{2} }} + C_{64} \left( {\frac{{\partial \theta_{\beta } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{{\partial \beta^{2} }} - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \beta }} \right) + D_{64} \left( {2\frac{{\partial v_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \beta }} \right)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{64} \left( {3\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \beta }} \right) - F_{64} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + C_{65} \left( {\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \beta }} \right) + D_{65} \left( {2\frac{{\partial u_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \beta }} \right)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{65} \left( {3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \beta }} \right) - F_{65} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + C_{66} \left( {\frac{{\partial^{2} u_{0} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta }} \right) + D_{66} \left( {\frac{{\partial^{2} \theta_{\alpha } }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta }} \right)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{66} \left( {\frac{{\partial^{2} u_{0}^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta }} \right) + F_{66} \left( {\frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta }} \right) + C_{11} \left( {\frac{{\partial^{2} u_{0} }}{{\partial \alpha^{2} }} + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \alpha }} \right) + D_{11} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial \alpha^{2} }} + E_{11} \frac{{\partial^{2} u_{0}^{*} }}{{\partial \alpha^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{11} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \alpha^{2} }} + C_{12} \left( {\frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \alpha }} \right) + D_{12} \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta } + E_{12} \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta } + F_{12} \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta } + C_{14} \left( {\frac{{\partial \theta_{\beta } }}{\partial \alpha } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta }} \right)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ { - \left( {\frac{{C_{14} }}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \alpha }} \right) + D_{14} \left( {2\frac{{\partial v_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \alpha }} \right) + E_{14} \left( {3\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \alpha }} \right) - F_{14} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } + C_{15} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {C_{15} \left( {\frac{{\partial^{2} w_{0} }}{{\partial \alpha^{2} }} - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \alpha }} \right) + D_{15} \left( {2\frac{{\partial u_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right) + E_{15} \left( {3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \alpha }} \right) - F_{15} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {C_{16} \left( {\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0} }}{{\partial \alpha^{2} }}} \right) + D_{16} \left( {\frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta } }}{{\partial \alpha^{2} }}} \right) + E_{16} \left( {\frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0}^{*} }}{{\partial \alpha^{2} }}} \right) + F_{16} \left( {\frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \alpha^{2} }}} \right)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {C_{51} \left( {\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}} \right) + D_{51} \frac{{\partial \theta_{\alpha } }}{\partial \alpha } + E_{51} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + F_{51} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + C_{52} \left( {\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}} \right) + D_{52} \frac{{\partial \theta_{\beta } }}{\partial \beta } + E_{52} \frac{{\partial v_{0}^{*} }}{\partial \beta }} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {F_{52} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + C_{54} \left( {\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}} \right) + D_{54} \left( {2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}} \right) + E_{54} \left( {3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}} \right) - F_{54} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + C_{55} \theta_{\alpha } } \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {C_{55} \left( {\frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}} \right) + D_{55} \left( {2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}} \right) + E_{55} \left( {3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}} \right) - F_{55} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + C_{56} \left( {\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }} \right)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {D_{56} \left( {\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }} \right) + E_{56} \left( {\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }} \right) + F_{56} \left( {\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }} \right)} \right\} = {\mathcal{L}}_{\mu } \left\{ {I_{0} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + I_{1} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial t^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{\mu } \left\{ {I_{2} \frac{{\partial^{2} u_{0}^{*} }}{{\partial t^{2} }} + I_{3} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial t^{2} }}} \right\}, \hfill \\ & \quad \end{aligned} $$
(33)
$$ \begin{aligned} & {\mathcal{L}}_{l} \left\{ {C_{21} \left( {\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \beta }} \right) + D_{21} \frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + E_{21} \frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + F_{21} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + C_{22} \left( {\frac{{\partial^{2} v_{0} }}{{\partial \beta^{2} }} + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \beta }} \right)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {D_{22} \frac{{\partial^{2} \theta_{\beta } }}{{\partial \beta^{2} }} + E_{22} \frac{{\partial^{2} v_{0}^{*} }}{{\partial \beta^{2} }} + F_{22} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \beta^{2} }} + C_{24} \left( {\frac{{\partial \theta_{\beta } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{{\partial \beta^{2} }} - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \beta }} \right) + D_{24} \left( {2\frac{{\partial v_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \beta }} \right)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{24} \left( {3\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \beta }} \right) - F_{24} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + C_{25} \left( {\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \beta }} \right) + D_{25} \left( {2\frac{{\partial u_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \beta }} \right)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{25} \left( {3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \beta }} \right) - F_{25} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + C_{26} \left( {\frac{{\partial^{2} u_{0} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta }} \right) + D_{26} \left( {\frac{{\partial^{2} \theta_{\alpha } }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta }} \right)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{26} \left( {\frac{{\partial^{2} u_{0}^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta }} \right) + F_{26} \left( {\frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta }} \right) + C_{61} \left( {\frac{{\partial^{2} u_{0} }}{{\partial \alpha^{2} }} + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \alpha }} \right) + D_{61} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial \alpha^{2} }} + E_{61} \frac{{\partial^{2} u_{0}^{*} }}{{\partial \alpha^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{61} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \alpha^{2} }} + C_{62} \left( {\frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \alpha }} \right) + D_{62} \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta } + E_{62} \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta } + F_{62} \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta } + C_{64} \left( {\frac{{\partial \theta_{\beta } }}{\partial \alpha } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta }} \right)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ { - \left( {\frac{{C_{64} }}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \alpha }} \right) + D_{64} \left( {2\frac{{\partial v_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \alpha }} \right) + E_{64} \left( {3\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \alpha }} \right) - F_{64} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } + C_{15} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {C_{65} \left( {\frac{{\partial^{2} w_{0} }}{{\partial \alpha^{2} }} - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \alpha }} \right) + D_{65} \left( {2\frac{{\partial u_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right) + E_{65} \left( {3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \alpha }} \right) - F_{65} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {C_{66} \left( {\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0} }}{{\partial \alpha^{2} }}} \right) + D_{66} \left( {\frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta } }}{{\partial \alpha^{2} }}} \right) + E_{66} \left( {\frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0}^{*} }}{{\partial \alpha^{2} }}} \right) + F_{66} \left( {\frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \alpha^{2} }}} \right)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {C_{41} \left( {\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}} \right) + D_{41} \frac{{\partial \theta_{\alpha } }}{\partial \alpha } + E_{41} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + F_{41} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + C_{42} \left( {\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}} \right) + D_{42} \frac{{\partial \theta_{\beta } }}{\partial \beta } + E_{42} \frac{{\partial v_{0}^{*} }}{\partial \beta }} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {F_{42} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + C_{44} \left( {\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}} \right) + D_{44} \left( {2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}} \right) + E_{44} \left( {3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}} \right) - F_{44} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + C_{45} \theta_{\alpha } } \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {C_{45} \left( {\frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}} \right) + D_{45} \left( {2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}} \right) + E_{45} \left( {3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}} \right) - F_{45} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + C_{46} \left( {\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }} \right)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {D_{46} \left( {\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }} \right) + E_{46} \left( {\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }} \right) + F_{46} \left( {\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }} \right)} \right\} = {\mathcal{L}}_{\mu } \left\{ {I_{0} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + I_{1} \frac{{\partial^{2} \theta_{\beta } }}{{\partial t^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{\mu } \left\{ {I_{2} \frac{{\partial^{2} v_{0}^{*} }}{{\partial t^{2} }} + I_{3} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial t^{2} }}} \right\}, \hfill \\ & \quad \end{aligned} $$
(34)
$$ \begin{aligned} & {\mathcal{L}}_{l} \left\{ {C_{41} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg) + D_{41} \frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + E_{41} \frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + F_{41} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + C_{42} \Bigg(\frac{{\partial^{2} v_{0} }}{{\partial \beta^{2} }} + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {D_{42} \frac{{\partial^{2} \theta_{\beta } }}{{\partial \beta^{2} }} + E_{42} \frac{{\partial^{2} v_{0}^{*} }}{{\partial \beta^{2} }} + F_{42} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \beta^{2} }} + C_{44} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{{\partial \beta^{2} }} - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \beta }\Bigg) + D_{44} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{44} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \beta }\Bigg) - F_{44} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + C_{45} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \beta }\Bigg) + D_{45} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{45} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \beta }\Bigg) - F_{45} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + C_{46} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta }\Bigg) + D_{46} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{46} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta }\Bigg) + F_{46} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta }\Bigg) + C_{51} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \alpha^{2} }} + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + D_{51} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial \alpha^{2} }} + E_{51} \frac{{\partial^{2} u_{0}^{*} }}{{\partial \alpha^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{51} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \alpha^{2} }} + C_{52} \Bigg(\frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + D_{52} \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta } + E_{52} \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta } + F_{52} \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta } + C_{54} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \alpha } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ { - \Bigg(\frac{{C_{54} }}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + D_{54} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + E_{54} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) - F_{54} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } + C_{55} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {C_{55} \Bigg(\frac{{\partial^{2} w_{0} }}{{\partial \alpha^{2} }} - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \alpha }\Bigg) + D_{55} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \alpha }\Bigg) + E_{55} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \alpha }\Bigg) - F_{55} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {C_{56} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0} }}{{\partial \alpha^{2} }}\Bigg) + D_{56} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta } }}{{\partial \alpha^{2} }}\Bigg) + E_{56} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0}^{*} }}{{\partial \alpha^{2} }}\Bigg) + F_{56} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \alpha^{2} }}\Bigg)} \right\} \hfill \\ & \quad - \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {C_{11} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + D_{11} \frac{{\partial \theta_{\alpha } }}{\partial \alpha } + E_{11} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + F_{11} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + C_{12} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + D_{12} \frac{{\partial \theta_{\beta } }}{\partial \beta } + E_{12} \frac{{\partial v_{0}^{*} }}{\partial \beta }} \right\} \hfill \\ & \quad - \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {F_{12} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + C_{14} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg) + D_{14} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + E_{14} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - F_{14} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + C_{15} \theta_{\alpha } } \right\} \hfill \\ & \quad - \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {C_{15} \Bigg(\frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + D_{15} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg) + E_{15} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - F_{15} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + C_{16} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad - \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {D_{16} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + E_{16} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) + F_{16} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} - \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {C_{21} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + D_{21} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad - \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {E_{21} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + F_{21} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + C_{22} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + D_{22} \frac{{\partial \theta_{\beta } }}{\partial \beta } + E_{22} \frac{{\partial v_{0}^{*} }}{\partial \beta } + F_{22} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + C_{24} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg)} \right\} \hfill \\ & \quad - \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {D_{24} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + E_{24} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - F_{24} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + C_{25} \Bigg(\theta_{\alpha } + \frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + D_{25} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg)} \right\} \hfill \\ & \quad - \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {E_{25} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - F_{25} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + C_{26} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + D_{26} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + E_{26} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad - \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {F_{26} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} = {\mathcal{L}}_{\mu } \left\{ {I_{0} \frac{{\partial^{2} w_{0} }}{{\partial t^{2} }}} \right\}, \hfill \\ & \quad \end{aligned} $$
(35)
$$ \begin{aligned} & {\mathcal{L}}_{l} \left\{ {D_{61} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg) + E_{61} \frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + F_{61} \frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + G_{61} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + D_{62} \Bigg(\frac{{\partial^{2} v_{0} }}{{\partial \beta^{2} }} + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{62} \frac{{\partial^{2} \theta_{\beta } }}{{\partial \beta^{2} }} + F_{62} \frac{{\partial^{2} v_{0}^{*} }}{{\partial \beta^{2} }} + G_{62} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \beta^{2} }} + D_{64} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{{\partial \beta^{2} }} - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \beta }\Bigg) + E_{64} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{64} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \beta }\Bigg) - G_{64} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + D_{65} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \beta }\Bigg) + E_{65} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{65} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \beta }\Bigg) - G_{65} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + D_{66} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta }\Bigg) + E_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{66} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta }\Bigg) + G_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta }\Bigg) + D_{11} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \alpha^{2} }} + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + E_{11} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial \alpha^{2} }} + F_{11} \frac{{\partial^{2} u_{0}^{*} }}{{\partial \alpha^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {G_{11} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \alpha^{2} }} + D_{12} \Bigg(\frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + E_{12} \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta } + F_{12} \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta } + G_{12} \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta } + D_{14} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \alpha } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ { - \Bigg(\frac{{D_{14} }}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + E_{14} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + F_{14} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) - G_{14} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } + D_{15} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {D_{15} \Bigg(\frac{{\partial^{2} w_{0} }}{{\partial \alpha^{2} }} - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \alpha }\Bigg) + E_{15} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \alpha }\Bigg) + F_{15} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \alpha }\Bigg) - G_{15} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {D_{16} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0} }}{{\partial \alpha^{2} }}\Bigg) + E_{16} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta } }}{{\partial \alpha^{2} }}\Bigg) + F_{16} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0}^{*} }}{{\partial \alpha^{2} }}\Bigg) + G_{16} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \alpha^{2} }}\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {D_{51} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + E_{51} \frac{{\partial \theta_{\alpha } }}{\partial \alpha } + F_{51} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + G_{51} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + D_{52} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + E_{52} \frac{{\partial \theta_{\beta } }}{\partial \beta } + F_{52} \frac{{\partial v_{0}^{*} }}{\partial \beta }} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {G_{52} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + D_{54} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg) + E_{54} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + F_{54} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - G_{54} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + D_{55} \theta_{\alpha } } \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {D_{55} \Bigg(\frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + E_{55} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg) + F_{55} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - G_{55} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + D_{56} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {E_{56} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + F_{56} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) + G_{56} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} - {\mathcal{L}}_{l} \left\{ {C_{51} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + D_{51} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad - {\mathcal{L}}_{l} \left\{ {E_{51} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + F_{51} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + C_{52} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + D_{52} \frac{{\partial \theta_{\beta } }}{\partial \beta } + E_{52} \frac{{\partial v_{0}^{*} }}{\partial \beta } + F_{52} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + C_{54} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg)} \right\} \hfill \\ & \quad - {\mathcal{L}}_{l} \left\{ {D_{54} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + E_{54} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - F_{54} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + C_{55} \Bigg(\theta_{\alpha } + \frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + D_{55} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg)} \right\} \hfill \\ & \quad - {\mathcal{L}}_{l} \left\{ {E_{55} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - F_{55} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + C_{56} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + D_{56} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + E_{56} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad - {\mathcal{L}}_{l} \left\{ {F_{56} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} = {\mathcal{L}}_{\mu } \left\{ {I_{1} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + I_{2} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial t^{2} }} + I_{3} \frac{{\partial^{2} u_{0}^{*} }}{{\partial t^{2} }} + I_{4} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial t^{2} }}} \right\}, \hfill \\ & \quad \end{aligned} $$
(36)
$$ \begin{aligned} & {\mathcal{L}}_{l} \left\{ {D_{21} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg) + E_{21} \frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + F_{21} \frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + G_{21} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + D_{22} \Bigg(\frac{{\partial^{2} v_{0} }}{{\partial \beta^{2} }} + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{22} \frac{{\partial^{2} \theta_{\beta } }}{{\partial \beta^{2} }} + F_{22} \frac{{\partial^{2} v_{0}^{*} }}{{\partial \beta^{2} }} + G_{22} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \beta^{2} }} + D_{24} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{{\partial \beta^{2} }} - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \beta }\Bigg) + E_{24} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{24} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \beta }\Bigg) - G_{24} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + D_{25} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \beta }\Bigg) + E_{25} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{25} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \beta }\Bigg) - G_{25} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + D_{26} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta }\Bigg) + E_{26} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{26} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta }\Bigg) + G_{26} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta }\Bigg) + D_{61} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \alpha^{2} }} + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + E_{61} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial \alpha^{2} }} + F_{61} \frac{{\partial^{2} u_{0}^{*} }}{{\partial \alpha^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {G_{61} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \alpha^{2} }} + D_{62} \Bigg(\frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + E_{62} \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta } + F_{62} \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta } + G_{62} \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta } + D_{64} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \alpha } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ { - \Bigg(\frac{{D_{64} }}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + E_{64} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + F_{64} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) - G_{64} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } + D_{65} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {D_{65} \Bigg(\frac{{\partial^{2} w_{0} }}{{\partial \alpha^{2} }} - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \alpha }\Bigg) + E_{65} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \alpha }\Bigg) + F_{65} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \alpha }\Bigg) - G_{65} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {D_{66} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0} }}{{\partial \alpha^{2} }}\Bigg) + E_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta } }}{{\partial \alpha^{2} }}\Bigg) + F_{66} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0}^{*} }}{{\partial \alpha^{2} }}\Bigg) + G_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \alpha^{2} }}\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {D_{41} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + E_{41} \frac{{\partial \theta_{\alpha } }}{\partial \alpha } + F_{41} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + G_{41} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + D_{42} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + E_{42} \frac{{\partial \theta_{\beta } }}{\partial \beta } + F_{42} \frac{{\partial v_{0}^{*} }}{\partial \beta }} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {G_{42} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + D_{44} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg) + E_{44} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + F_{44} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - G_{44} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + D_{45} \theta_{\alpha } } \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {D_{45} \Bigg(\frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + E_{45} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg) + F_{45} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - G_{45} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + D_{46} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {E_{46} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + F_{46} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) + G_{46} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} - {\mathcal{L}}_{l} \left\{ {C_{41} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + D_{41} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad - {\mathcal{L}}_{l} \left\{ {E_{41} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + F_{41} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + C_{42} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + D_{42} \frac{{\partial \theta_{\beta } }}{\partial \beta } + E_{42} \frac{{\partial v_{0}^{*} }}{\partial \beta } + F_{42} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + C_{44} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg)} \right\} \hfill \\ & \quad - {\mathcal{L}}_{l} \left\{ {D_{44} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + E_{44} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - F_{44} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + C_{45} \Bigg(\theta_{\alpha } + \frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + D_{45} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg)} \right\} \hfill \\ & \quad - {\mathcal{L}}_{l} \left\{ {E_{45} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - F_{45} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + C_{46} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + D_{46} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + E_{46} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad - {\mathcal{L}}_{l} \left\{ {F_{46} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} = {\mathcal{L}}_{\mu } \left\{ {I_{1} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + I_{2} \frac{{\partial^{2} \theta_{\beta } }}{{\partial t^{2} }} + I_{3} \frac{{\partial^{2} v_{0}^{*} }}{{\partial t^{2} }} + I_{4} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial t^{2} }}} \right\}, \hfill \\ & \quad \end{aligned} $$
(37)
$$ \begin{aligned} & {\mathcal{L}}_{l} \left\{ {E_{61} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg) + F_{61} \frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + G_{61} \frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + H_{61} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + E_{62} \Bigg(\frac{{\partial^{2} v_{0} }}{{\partial \beta^{2} }} + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{62} \frac{{\partial^{2} \theta_{\beta } }}{{\partial \beta^{2} }} + G_{62} \frac{{\partial^{2} v_{0}^{*} }}{{\partial \beta^{2} }} + H_{62} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \beta^{2} }} + E_{64} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{{\partial \beta^{2} }} - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \beta }\Bigg) + F_{64} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {G_{64} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \beta }\Bigg) - H_{64} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + E_{65} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \beta }\Bigg) + F_{65} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {G_{65} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \beta }\Bigg) - H_{65} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + E_{66} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta }\Bigg) + F_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {G_{66} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta }\Bigg) + H_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta }\Bigg) + E_{11} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \alpha^{2} }} + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + F_{11} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial \alpha^{2} }} + G_{11} \frac{{\partial^{2} u_{0}^{*} }}{{\partial \alpha^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {H_{11} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \alpha^{2} }} + E_{12} \Bigg(\frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + F_{12} \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta } + G_{12} \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta } + H_{12} \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta } + E_{14} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \alpha } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ { - \frac{{E_{14} }}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + F_{14} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + G_{14} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) - H_{14} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } + E_{15} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{15} \Bigg(\frac{{\partial^{2} w_{0} }}{{\partial \alpha^{2} }} - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \alpha }\Bigg) + F_{15} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \alpha }\Bigg) + G_{15} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \alpha }\Bigg) - H_{15} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{16} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0} }}{{\partial \alpha^{2} }}\Bigg) + F_{16} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta } }}{{\partial \alpha^{2} }}\Bigg) + G_{16} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0}^{*} }}{{\partial \alpha^{2} }}\Bigg) + H_{16} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \alpha^{2} }}\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {E_{51} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + F_{51} \frac{{\partial \theta_{\alpha } }}{\partial \alpha } + G_{51} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + H_{51} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + E_{52} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + F_{52} \frac{{\partial \theta_{\beta } }}{\partial \beta } + G_{52} \frac{{\partial v_{0}^{*} }}{\partial \beta }} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {H_{52} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + E_{54} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg) + F_{54} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + G_{54} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - H_{54} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + E_{55} \theta_{\alpha } } \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {E_{55} \Bigg(\frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + F_{55} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg) + G_{55} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - H_{55} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + E_{56} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {F_{56} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + G_{56} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) + H_{56} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} - 2{\mathcal{L}}_{l} \left\{ {D_{51} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + E_{51} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad - 2{\mathcal{L}}_{l} \left\{ {F_{51} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + G_{51} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + D_{52} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + E_{52} \frac{{\partial \theta_{\beta } }}{\partial \beta } + F_{52} \frac{{\partial v_{0}^{*} }}{\partial \beta } + G_{52} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + D_{54} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg)} \right\} \hfill \\ & \quad - 2{\mathcal{L}}_{l} \left\{ {E_{54} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + F_{54} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - G_{54} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + D_{55} \Bigg(\theta_{\alpha } + \frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + E_{55} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg)} \right\} \hfill \\ & \quad - 2{\mathcal{L}}_{l} \left\{ {F_{55} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - G_{55} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + D_{56} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + E_{56} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + F_{56} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad - 2{\mathcal{L}}_{l} \left\{ {G_{56} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} = {\mathcal{L}}_{\mu } \left\{ {I_{2} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + I_{3} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial t^{2} }} + I_{4} \frac{{\partial^{2} u_{0}^{*} }}{{\partial t^{2} }} + I_{5} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial t^{2} }}} \right\}, \hfill \\ & \quad \end{aligned} $$
(38)
$$ \begin{aligned} & {\mathcal{L}}_{l} \left\{ {E_{21} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg) + F_{21} \frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + G_{21} \frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + H_{21} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + E_{22} \Bigg(\frac{{\partial^{2} v_{0} }}{{\partial \beta^{2} }} + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{22} \frac{{\partial^{2} \theta_{\beta } }}{{\partial \beta^{2} }} + G_{22} \frac{{\partial^{2} v_{0}^{*} }}{{\partial \beta^{2} }} + H_{22} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \beta^{2} }} + E_{24} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{{\partial \beta^{2} }} - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \beta }\Bigg) + F_{24} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {G_{24} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \beta }\Bigg) - H_{24} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + E_{25} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \beta }\Bigg) + F_{25} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{25} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \beta }\Bigg) - H_{25} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + E_{26} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta }\Bigg) + F_{26} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {G_{26} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta }\Bigg) + H_{26} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta }\Bigg) + E_{61} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \alpha^{2} }} + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + F_{61} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial \alpha^{2} }} + G_{61} \frac{{\partial^{2} u_{0}^{*} }}{{\partial \alpha^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {H_{61} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \alpha^{2} }} + E_{62} \Bigg(\frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + F_{62} \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta } + G_{62} \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta } + H_{62} \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta } + E_{64} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \alpha } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ { - \Bigg(\frac{{E_{64} }}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + F_{64} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + G_{64} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) - H_{64} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } + E_{65} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{65} \Bigg(\frac{{\partial^{2} w_{0} }}{{\partial \alpha^{2} }} - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \alpha }\Bigg) + F_{65} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \alpha }\Bigg) + G_{65} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \alpha }\Bigg) - H_{65} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {E_{66} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0} }}{{\partial \alpha^{2} }}\Bigg) + F_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta } }}{{\partial \alpha^{2} }}\Bigg) + G_{66} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0}^{*} }}{{\partial \alpha^{2} }}\Bigg) + H_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \alpha^{2} }}\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {E_{41} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + F_{41} \frac{{\partial \theta_{\alpha } }}{\partial \alpha } + G_{41} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + H_{41} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + E_{42} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + F_{42} \frac{{\partial \theta_{\beta } }}{\partial \beta } + G_{42} \frac{{\partial v_{0}^{*} }}{\partial \beta }} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {H_{42} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + E_{44} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg) + F_{44} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + G_{44} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - H_{44} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + E_{45} \theta_{\alpha } } \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {E_{45} \Bigg(\frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + F_{45} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg) + G_{45} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - H_{45} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + E_{46} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {F_{46} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + G_{46} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) + H_{46} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} - 2{\mathcal{L}}_{l} \left\{ {D_{41} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + E_{41} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad - 2{\mathcal{L}}_{l} \left\{ {F_{41} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + G_{41} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + D_{42} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + E_{42} \frac{{\partial \theta_{\beta } }}{\partial \beta } + F_{42} \frac{{\partial v_{0}^{*} }}{\partial \beta } + G_{42} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + D_{44} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg)} \right\} \hfill \\ & \quad - 2{\mathcal{L}}_{l} \left\{ {E_{44} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + F_{44} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - G_{44} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + D_{45} \Bigg(\theta_{\alpha } + \frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + E_{45} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg)} \right\} \hfill \\ & \quad - 2{\mathcal{L}}_{l} \left\{ {F_{45} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - G_{45} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + D_{46} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + E_{46} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + F_{46} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad - 2{\mathcal{L}}_{l} \left\{ {G_{46} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} = {\mathcal{L}}_{\mu } \left\{ {I_{2} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + I_{3} \frac{{\partial^{2} \theta_{\beta } }}{{\partial t^{2} }} + I_{4} \frac{{\partial^{2} v_{0}^{*} }}{{\partial t^{2} }} + I_{5} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial t^{2} }}} \right\}, \hfill \\ & \quad \end{aligned} $$
(39)
$$ \begin{aligned} &{\mathcal{L}}_{l} \left\{ {F_{61} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg) + G_{61} \frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + H_{61} \frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + J_{61} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + F_{62} \Bigg(\frac{{\partial^{2} v_{0} }}{{\partial \beta^{2} }} + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {G_{62} \frac{{\partial^{2} \theta_{\beta } }}{{\partial \beta^{2} }} + H_{62} \frac{{\partial^{2} v_{0}^{*} }}{{\partial \beta^{2} }} + J_{62} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \beta^{2} }} + F_{64} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{{\partial \beta^{2} }} - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \beta }\Bigg) + G_{64} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {H_{64} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \beta }\Bigg) - J_{64} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + F_{65} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \beta }\Bigg) + G_{65} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {H_{65} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \beta }\Bigg) - J_{65} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + F_{66} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta }\Bigg) + G_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {H_{66} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta }\Bigg) + J_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta }\Bigg) + F_{11} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \alpha^{2} }} + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + G_{11} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial \alpha^{2} }} + H_{11} \frac{{\partial^{2} u_{0}^{*} }}{{\partial \alpha^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {J_{11} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \alpha^{2} }} + F_{12} \Bigg(\frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + G_{12} \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta } + H_{12} \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta } + J_{12} \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta } + F_{14} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \alpha } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ { - \Bigg(\frac{{F_{14} }}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + G_{14} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + H_{14} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) - J_{14} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } + F_{15} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{15} \Bigg(\frac{{\partial^{2} w_{0} }}{{\partial \alpha^{2} }} - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \alpha }\Bigg) + G_{15} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \alpha }\Bigg) + H_{15} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \alpha }\Bigg) - J_{15} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{16} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0} }}{{\partial \alpha^{2} }}\Bigg) + G_{16} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta } }}{{\partial \alpha^{2} }}\Bigg) + H_{16} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0}^{*} }}{{\partial \alpha^{2} }}\Bigg) + J_{16} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \alpha^{2} }}\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {F_{51} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + G_{51} \frac{{\partial \theta_{\alpha } }}{\partial \alpha } + H_{51} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + J_{51} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + F_{52} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + G_{52} \frac{{\partial \theta_{\beta } }}{\partial \beta } + H_{52} \frac{{\partial v_{0}^{*} }}{\partial \beta }} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {J_{52} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + F_{54} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg) + G_{54} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + H_{54} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - J_{54} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + F_{55} \theta_{\alpha } } \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {F_{55} \Bigg(\frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + G_{55} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg) + H_{55} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - J_{55} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + F_{56} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\alpha } }}\left\{ {G_{56} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + H_{56} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) + J_{56} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} - 3{\mathcal{L}}_{l} \left\{ {E_{51} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + F_{51} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad - 3{\mathcal{L}}_{l} \left\{ {G_{51} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + H_{51} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + E_{52} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + F_{52} \frac{{\partial \theta_{\beta } }}{\partial \beta } + G_{52} \frac{{\partial v_{0}^{*} }}{\partial \beta } + H_{52} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + E_{54} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg)} \right\} \hfill \\ & \quad - 3{\mathcal{L}}_{l} \left\{ {F_{54} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + G_{54} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - H_{54} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + E_{55} \Bigg(\theta_{\alpha } + \frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + F_{55} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg)} \right\} \hfill \\ & \quad - 3{\mathcal{L}}_{l} \left\{ {G_{55} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - H_{55} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + E_{56} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + F_{56} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + G_{56} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad - 3{\mathcal{L}}_{l} \left\{ {H_{56} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} = {\mathcal{L}}_{\mu } \left\{ {I_{3} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + I_{4} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial t^{2} }} + I_{5} \frac{{\partial^{2} u_{0}^{*} }}{{\partial t^{2} }} + I_{6} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial t^{2} }}} \right\}, \hfill \\ & \quad \end{aligned} $$
(40)
$$ \begin{aligned} & {\mathcal{L}}_{l} \left\{ {F_{21} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg) + G_{21} \frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + H_{21} \frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + J_{21} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + F_{22} \Bigg(\frac{{\partial^{2} v_{0} }}{{\partial \beta^{2} }} + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {G_{22} \frac{{\partial^{2} \theta_{\beta } }}{{\partial \beta^{2} }} + H_{22} \frac{{\partial^{2} v_{0}^{*} }}{{\partial \beta^{2} }} + J_{22} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \beta^{2} }} + F_{24} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{{\partial \beta^{2} }} - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \beta }\Bigg) + G_{24} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {H_{24} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \beta }\Bigg) - J_{24} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + F_{25} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \beta }\Bigg) + G_{25} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {H_{25} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \beta }\Bigg) - J_{25} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + F_{26} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta }\Bigg) + G_{26} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {H_{26} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta }\Bigg) + J_{26} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \beta^{2} }} + \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta }\Bigg) + F_{61} \Bigg(\frac{{\partial^{2} u_{0} }}{{\partial \alpha^{2} }} + \frac{1}{{R_{\alpha } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + G_{61} \frac{{\partial^{2} \theta_{\alpha } }}{{\partial \alpha^{2} }} + H_{61} \frac{{\partial^{2} u_{0}^{*} }}{{\partial \alpha^{2} }}} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {J_{61} \frac{{\partial^{2} \theta_{\alpha }^{*} }}{{\partial \alpha^{2} }} + F_{62} \Bigg(\frac{{\partial^{2} v_{0} }}{\partial \alpha \partial \beta } + \frac{1}{{R_{\beta } }}\frac{{\partial w_{0} }}{\partial \alpha }\Bigg) + G_{62} \frac{{\partial^{2} \theta_{\beta } }}{\partial \alpha \partial \beta } + H_{62} \frac{{\partial^{2} v_{0}^{*} }}{\partial \alpha \partial \beta } + J_{62} \frac{{\partial^{2} \theta_{\beta }^{*} }}{\partial \alpha \partial \beta } + F_{64} \Bigg(\frac{{\partial \theta_{\beta } }}{\partial \alpha } + \frac{{\partial^{2} w_{0} }}{\partial \alpha \partial \beta }\Bigg)} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ { - \Bigg(\frac{{F_{64} }}{{R_{\beta } }}\frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + G_{64} \Bigg(2\frac{{\partial v_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + H_{64} \Bigg(3\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } - \frac{1}{{R_{\beta } }}\frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) - J_{64} \frac{1}{{R_{\beta } }}\frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha } + F_{65} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{65} \Bigg(\frac{{\partial^{2} w_{0} }}{{\partial \alpha^{2} }} - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0} }}{\partial \alpha }\Bigg) + G_{65} \Bigg(2\frac{{\partial u_{0}^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha } }}{\partial \alpha }\Bigg) + H_{65} \Bigg(3\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } - \frac{1}{{R_{\alpha } }}\frac{{\partial u_{0}^{*} }}{\partial \alpha }\Bigg) - J_{65} \frac{1}{{R_{\alpha } }}\frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha }} \right\} \hfill \\ & \quad + {\mathcal{L}}_{l} \left\{ {F_{66} \Bigg(\frac{{\partial^{2} u_{0} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0} }}{{\partial \alpha^{2} }}\Bigg) + G_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha } }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta } }}{{\partial \alpha^{2} }}\Bigg) + H_{66} \Bigg(\frac{{\partial^{2} u_{0}^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} v_{0}^{*} }}{{\partial \alpha^{2} }}\Bigg) + J_{66} \Bigg(\frac{{\partial^{2} \theta_{\alpha }^{*} }}{\partial \alpha \partial \beta } + \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial \alpha^{2} }}\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {F_{41} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + G_{41} \frac{{\partial \theta_{\alpha } }}{\partial \alpha } + H_{41} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + J_{41} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + F_{42} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + G_{42} \frac{{\partial \theta_{\beta } }}{\partial \beta } + H_{42} \frac{{\partial v_{0}^{*} }}{\partial \beta }} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {J_{42} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + F_{44} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg) + G_{44} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + H_{44} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - J_{44} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + F_{45} \theta_{\alpha } } \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {F_{45} \Bigg(\frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + G_{45} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg) + H_{45} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - J_{45} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + F_{46} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad + \frac{{{\mathcal{L}}_{l} }}{{R_{\beta } }}\left\{ {G_{46} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + H_{46} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg) + J_{46} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} - 3{\mathcal{L}}_{l} \left\{ {E_{41} \Bigg(\frac{{\partial u_{0} }}{\partial \alpha } + \frac{{w_{0} }}{{R_{\alpha } }}\Bigg) + F_{41} \frac{{\partial \theta_{\alpha } }}{\partial \alpha }} \right\} \hfill \\ & \quad - 3{\mathcal{L}}_{l} \left\{ {G_{41} \frac{{\partial u_{0}^{*} }}{\partial \alpha } + H_{41} \frac{{\partial \theta_{\alpha }^{*} }}{\partial \alpha } + E_{42} \Bigg(\frac{{\partial v_{0} }}{\partial \beta } + \frac{{w_{0} }}{{R_{\beta } }}\Bigg) + F_{42} \frac{{\partial \theta_{\beta } }}{\partial \beta } + G_{42} \frac{{\partial v_{0}^{*} }}{\partial \beta } + H_{42} \frac{{\partial \theta_{\beta }^{*} }}{\partial \beta } + E_{44} \Bigg(\theta_{\beta } + \frac{{\partial w_{0} }}{\partial \beta } - \frac{{v_{0} }}{{R_{\beta } }}\Bigg)} \right\} \hfill \\ & \quad - 3{\mathcal{L}}_{l} \left\{ {F_{44} \Bigg(2v_{0}^{*} - \frac{{\theta_{\beta } }}{{R_{\beta } }}\Bigg) + G_{44} \Bigg(3\theta_{\beta }^{*} - \frac{{v_{0}^{*} }}{{R_{\beta } }}\Bigg) - H_{44} \frac{{\theta_{\beta }^{*} }}{{R_{\beta } }} + E_{45} \Bigg(\theta_{\alpha } + \frac{{\partial w_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{R_{\alpha } }}\Bigg) + F_{45} \Bigg(2u_{0}^{*} - \frac{{\theta_{\alpha } }}{{R_{\alpha } }}\Bigg)} \right\} \hfill \\ & \quad - 3{\mathcal{L}}_{l} \left\{ {G_{45} \Bigg(3\theta_{\varepsilon }^{*} - \frac{{u_{0}^{*} }}{{R_{\alpha } }}\Bigg) - H_{45} \frac{{\theta_{\alpha }^{*} }}{{R_{\alpha } }} + E_{46} \Bigg(\frac{{\partial u_{0} }}{\partial \beta } + \frac{{\partial v_{0} }}{\partial \alpha }\Bigg) + F_{46} \Bigg(\frac{{\partial \theta_{\alpha } }}{\partial \beta } + \frac{{\partial \theta_{\beta } }}{\partial \alpha }\Bigg) + G_{46} \Bigg(\frac{{\partial u_{0}^{*} }}{\partial \beta } + \frac{{\partial v_{0}^{*} }}{\partial \alpha }\Bigg)} \right\} \hfill \\ & \quad - 3{\mathcal{L}}_{l} \left\{ {H_{46} \Bigg(\frac{{\partial \theta_{\alpha }^{*} }}{\partial \beta } + \frac{{\partial \theta_{\beta }^{*} }}{\partial \alpha }\Bigg)} \right\} = {\mathcal{L}}_{\mu } \left\{ {I_{3} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + I_{4} \frac{{\partial^{2} \theta_{\beta } }}{{\partial t^{2} }} + I_{5} \frac{{\partial^{2} v_{0}^{*} }}{{\partial t^{2} }} + I_{6} \frac{{\partial^{2} \theta_{\beta }^{*} }}{{\partial t^{2} }}} \right\}, \hfill \\ & \quad \end{aligned} $$
(41)

where

$$ (C_{ij} ,D_{ij} ,E_{ij} ,F_{ij} ,G_{ij} ,H_{ij} ,J_{ij} ) = \int\limits_{{ - {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}^{{{\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} {C_{ij} } (1,z,z^{2} ,z^{3} ,z^{4} ,z^{5} ,z^{6} )dz\,\,(i,j = 1,2,3,4,5,6) $$
(42)

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Karami, B., Janghorban, M. & Tounsi, A. Novel study on functionally graded anisotropic doubly curved nanoshells. Eur. Phys. J. Plus 135, 103 (2020). https://doi.org/10.1140/epjp/s13360-019-00079-y

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