Nonlinear dynamic analysis of the perovskite solar cell under blast impacts based on the modified strain gradient theory

Abstract

The perovskite solar cell (PSC) is one of the most burgeoning astronautic photovoltaic candidates with intrinsic microscales in thickness. One of the key concerns for the PSC within its operating status is nonlinear dynamic performance owing to ultra-thin structural attributes and dynamic impacts. This paper explores the size-dependent nonlinear dynamic behavior of the PSC under blast impacts. A multi-scale nonlinear dynamics framework is developed based on the modified strain gradient theory, extending the analytical nonlinear response analysis to microscale composites. Considering the von-Kármán geometric nonlinearity, the size-dependent motion equations are derived, and can be subsequently resolved by the fourth-order Runge–Kutta method to assess the transient response and blast impact resistance of the PSC. Diverse effects of the microplate theories, material length scale parameters, structural geometry, elastic foundations, damping, and blast impulses on the size-dependent nonlinear dynamic characteristics are systematically examined. Concluding remarks from the present study will be beneficial to the astronautic design and space deployment of the PSC energy harvesting devices with improved impact carrying capacity and safety.

Appendices

Appendix A

$$\begin{aligned} (\sigma_{xx} )_{l} & = (\lambda_{l} + 2\mu_{l} )\left[ {\frac{\partial u}{{\partial x}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)^{2} + \frac{1}{4}z\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - z\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{{5z^{3} }}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }}} \right] \\ & \quad + \lambda_{l} \left[ {\frac{\partial v}{{\partial y}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)^{2} + \frac{1}{4}z\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} - z\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{{5z^{3} }}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right], \\ \end{aligned}$$

(A.1)

$$\begin{aligned} (\sigma_{yy} )_{l} & = (\lambda_{l} + 2\mu_{l} )\left[ {\frac{\partial v}{{\partial y}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)^{2} + \frac{1}{4}z\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} - z\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{{5z^{3} }}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right] \\ & \quad + \lambda_{l} \left[ {\frac{\partial u}{{\partial x}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)^{2} + \frac{1}{4}z\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - z\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{{5z^{3} }}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }}} \right], \\ \end{aligned}$$

(A.2)

$$\begin{aligned} (\sigma_{zz} )_{l} & = \lambda_{l} \left[ {\frac{\partial u}{{\partial x}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)^{2} + \frac{1}{4}z\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - z\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{{5z^{3} }}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }}} \right] \\ & \quad + \lambda_{l} \left[ {\frac{\partial v}{{\partial y}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)^{2} + \frac{1}{4}z\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} - z\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{{5z^{3} }}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right], \\ \end{aligned}$$

(A.3)

$$\begin{aligned} (\sigma_{xy} )_{l} & = 2\mu_{l} \left[ {\frac{1}{2}\left( {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}} + \frac{{\partial w_{b} }}{\partial y}\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial y}\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{b} }}{\partial y}\frac{{\partial w_{s} }}{\partial x} + \frac{{\partial w_{s} }}{\partial y}\frac{{\partial w_{s} }}{\partial x}} \right)} \right. \\ & \left. {\quad + \frac{1}{4}z\frac{{\partial^{2} w_{s} }}{\partial x\partial y} - z\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{{5z^{3} }}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{\partial x\partial y}} \right], \\ \end{aligned}$$

(A.4)

$$(\sigma_{xz} )_{l} = \frac{5}{4}\mu_{l} \left( {1 - \frac{{4z^{2} }}{{h^{2} }}} \right)\frac{{\partial w_{s} }}{\partial x},$$

(A.5)

$$(\sigma_{yz} )_{l} = \frac{5}{4}\mu_{l} \left( {1 - \frac{{4z^{2} }}{{h^{2} }}} \right)\frac{{\partial w_{s} }}{\partial y},$$

(A.6)

$$\begin{aligned} (p_{x} )_{l} \; & = 2\mu _{l} l_{0}^{2} \left[ {\frac{{\partial ^{2} u}}{{\partial x^{2} }} + \frac{{\partial ^{2} v}}{{\partial x\partial y}} + \left( {\frac{{\partial w_{b} }}{{\partial y}} + \frac{{\partial w_{s} }}{{\partial y}}} \right)\left( {\frac{{\partial ^{2} w_{b} }}{{\partial x\partial y}} + \frac{{\partial ^{2} w_{s} }}{{\partial x\partial y}}} \right) + \left( {\frac{{\partial w_{b} }}{{\partial x}} + \frac{{\partial w_{s} }}{{\partial x}}} \right)\left( {\frac{{\partial ^{2} w_{b} }}{{\partial x^{2} }} + \frac{{\partial ^{2} w_{s} }}{{\partial x^{2} }}} \right)} \right. \\ & \quad \left. { + \frac{z}{4}\left( {\frac{{\partial ^{3} w_{s} }}{{\partial x\partial y^{2} }} + \frac{{\partial ^{3} w_{s} }}{{\partial x^{3} }}} \right) - z\left( {\frac{{\partial ^{3} w_{b} }}{{\partial x\partial y^{2} }} + \frac{{\partial ^{3} w_{b} }}{{\partial x^{3} }}} \right) - \frac{{5z^{3} }}{{3h^{2} }}\left( {\frac{{\partial ^{3} w_{s} }}{{\partial x^{3} }} + \frac{{\partial ^{3} w_{s} }}{{\partial x\partial y^{2} }}} \right)} \right], \\ \end{aligned}$$

(A.7)

$$\begin{aligned} (p_{y} )_{l} & = 2\mu_{l} l_{0}^{2} \left[ {\frac{{\partial^{2} u}}{\partial x\partial y} + \frac{{\partial^{2} v}}{{\partial y^{2} }} + \left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)\left( {\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) + \left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)\left( {\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{{\partial^{2} w_{s} }}{\partial x\partial y}} \right)} \right. \\ & \left. {\quad + \frac{1}{4}z\left( {\frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{3} w_{s} }}{{\partial y^{3} }}} \right) - z\left( {\frac{{\partial^{3} w_{b} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{3} w_{b} }}{{\partial y^{3} }}} \right) - \frac{5}{{3h^{2} }}z^{3} \left( {\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}}} \right)} \right], \\ \end{aligned}$$

(A.8)

$$(p_{z} )_{l} = 2\mu_{l} l_{0}^{2} \left[ { - \left( {\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + \frac{{\partial^{2} w_{b} }}{{\partial x^{2} }}} \right) + \left( {\frac{1}{4} - \frac{{5z^{2} }}{{h^{2} }}} \right)\left( {\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} + \frac{{\partial^{2} w_{s} }}{{\partial x^{2} }}} \right)} \right],$$

(A.9)

$$\begin{aligned} (\tau_{xxx}^{1} )_{l} & = 2\mu_{l} l_{1}^{2} \left[ {\frac{1}{5}\left( {2\frac{{\partial^{2} u}}{{\partial x^{2} }} - \frac{{\partial^{2} u}}{{\partial y^{2} }} - 2\frac{{\partial^{2} v}}{\partial x\partial y}} \right) - \frac{1}{5}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}\frac{{\partial w_{b} }}{\partial x} - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{b} }}{\partial x} - \frac{1}{5}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x}} \right. \\ & \quad - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} - \frac{2}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ & \quad + \frac{2}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{2}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{2}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} + \frac{2}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ & \quad \left. { + z\left( {\frac{1}{10}\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} - \frac{3}{20}\frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }} + \frac{2}{{h^{2} }}\frac{{\partial w_{s} }}{\partial x}} \right) - \frac{1}{5}z\left( {2\frac{{\partial^{3} w_{b} }}{{\partial x^{3} }} - 3\frac{{\partial^{3} w_{b} }}{{\partial x\partial y^{2} }}} \right) - \frac{{z^{3} }}{{h^{2} }}\left( {\frac{2}{3}\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} - \frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }}} \right)} \right], \\ \end{aligned}$$

(A.10)

$$\begin{aligned} (\tau_{yyy}^{1} )_{l} & = 2\mu_{l} l_{1}^{2} \left[ {\frac{1}{5}\left( {2\frac{{\partial^{2} v}}{{\partial y^{2} }} - \frac{{\partial^{2} v}}{{\partial x^{2} }} - 2\frac{{\partial^{2} u}}{\partial x\partial y}} \right) + \frac{2}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + \frac{2}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + \frac{2}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{b} }}{\partial y}} \right. \\ & \quad + \frac{2}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial y} - \frac{2}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ & \quad - \frac{1}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ & \left. {\quad + z\left( {\frac{1}{10}\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} - \frac{3}{20}\frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}} + \frac{2}{{h^{2} }}\frac{{\partial w_{s} }}{\partial y}} \right) - \frac{1}{5}z\left( {2\frac{{\partial^{3} w_{b} }}{{\partial y^{3} }} - 3\frac{{\partial^{3} w_{b} }}{{\partial x^{2} \partial y}}} \right) - \frac{{z^{3} }}{{h^{2} }}\left( {\frac{2}{3}\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} - \frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}}} \right)} \right], \\ \end{aligned}$$

(A.11)

$$(\tau_{zzz}^{1} )_{l} = 2\mu_{l} l_{1}^{2} \left[ {\frac{1}{5}\left( {\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + \frac{{\partial^{2} w_{b} }}{{\partial x^{2} }}} \right) - \left( {\frac{3}{10} - \frac{{2z^{2} }}{{h^{2} }}} \right)\left( {\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} + \frac{{\partial^{2} w_{s} }}{{\partial x^{2} }}} \right)} \right],$$

(A.12)

$$\begin{aligned} (\tau_{xxy}^{1} )_{l} & = (\tau_{xyx}^{1} )_{l} = (\tau_{yxx}^{1} )_{l} = 2\mu_{l} l_{1}^{2} \left[ {\frac{1}{15}\left( {8\frac{{\partial^{2} u}}{\partial x\partial y} + 4\frac{{\partial^{2} v}}{{\partial x^{2} }} - 3\frac{{\partial^{2} v}}{{\partial y^{2} }}} \right) - \frac{1}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}} \right. \\ & \quad - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{b} }}{\partial y} - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial y} + \frac{8}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ & \quad + \frac{4}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{4}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{4}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} + \frac{4}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ & \quad \left. {+ z\left( {\frac{1}{5}\frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}} - \frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} + \frac{2}{{3h^{2} }}\frac{{\partial w_{s} }}{\partial y}} \right) - \frac{1}{5}z\left( {4\frac{{\partial^{3} w_{b} }}{{\partial x^{2} \partial y}} - \frac{{\partial^{3} w_{b} }}{{\partial y^{3} }}} \right) - \frac{{z^{3} }}{{3h^{2} }}\left( {4\frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}} - \frac{{\partial^{3} w_{s} }}{{\partial y^{3} }}} \right)} \right], \\ \end{aligned}$$

(A.13)

$$(\tau_{xxz}^{1} )_{l} = (\tau_{xzx}^{1} )_{l} = (\tau_{zxx}^{1} )_{l} = 2\mu_{l} l_{1}^{2} \left[ { - \frac{1}{15}\left( {4\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}} \right) + \left( {\frac{1}{10} - \frac{{2z^{2} }}{{3h^{2} }}} \right)\left( {4\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right)} \right],$$

(A.14)

$$\begin{aligned}& (\tau_{xyy}^{1} )_{l} = (\tau_{yxy}^{1} )_{l} = (\tau_{yyx}^{1} )_{l}\\ &\quad = 2\mu_{l} l_{1}^{2} \left[ { - \frac{1}{15}\left( {3\frac{{\partial^{2} u}}{{\partial x^{2} }} - 4\frac{{\partial^{2} u}}{{\partial y^{2} }} - 8\frac{{\partial^{2} v}}{\partial x\partial y}} \right) + \frac{4}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + \frac{4}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right. \\ & \qquad + \frac{4}{15}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} + \frac{4}{15}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} + \frac{8}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ & \qquad - \frac{1}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ & \left. {\qquad + z\left( { - \frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} + \frac{1}{5}\frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }} + \frac{2}{{3h^{2} }}\frac{{\partial w_{s} }}{\partial x}} \right) - \frac{1}{5}z\left( {4\frac{{\partial^{3} w_{b} }}{{\partial x\partial y^{2} }} - \frac{{\partial^{3} w_{b} }}{{\partial x^{3} }}} \right) - \frac{{z^{3} }}{{3h^{2} }}\left( {4\frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }} - \frac{{\partial^{3} w_{s} }}{{\partial x^{3} }}} \right)} \right], \\ \end{aligned}$$

(A.15)

$$(\tau_{xyz}^{1} )_{l} = (\tau_{yzx}^{1} )_{l} = (\tau_{yxz}^{1} )_{l} = (\tau_{xzy}^{1} )_{l} = (\tau_{zxy}^{1} )_{l} = (\tau_{zyx}^{1} )_{l} = 2\mu_{l} l_{1}^{2} \left[ { - \left( {\frac{1}{3}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{1}{2}\frac{{\partial^{2} w_{s} }}{\partial x\partial y}} \right) - \frac{{10z^{2} }}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{\partial x\partial y}} \right],$$

(A.16)

$$\begin{aligned} &(\tau_{xzz}^{1} )_{l} = (\tau_{zxz}^{1} )_{l} = (\tau_{zzx}^{1} )_{l} \\ &\quad = 2\mu_{l} l_{1}^{2} \left[ { - \frac{1}{15}\left( {3\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }} + 2\frac{{\partial^{2} v}}{\partial x\partial y}} \right) - \frac{1}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{1}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right. \\ & \qquad - \frac{1}{15}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} - \frac{1}{15}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} - \frac{2}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ & \qquad - \frac{1}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ & \qquad - \left. { z\left( {\frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} + \frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }} + \frac{8}{{3h^{2} }}\frac{{\partial w_{s} }}{\partial x}} \right) + \frac{1}{5}z\left( {\frac{{\partial^{3} w_{b} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{b} }}{{\partial x\partial y^{2} }}} \right) + \frac{{z^{3} }}{{3h^{2} }}\left( {\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }}} \right)} \right], \\ \end{aligned}$$

(A.17)

$$(\tau_{yyz}^{1} )_{l} = (\tau_{yzy}^{1} )_{l} = (\tau_{zyy}^{1} )_{l} = 2\mu_{l} l_{1}^{2} \left[ { - \frac{1}{15}\left( {4\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{{\partial^{2} w_{b} }}{{\partial x^{2} }}} \right) + \left( {\frac{1}{10} - \frac{{2z^{2} }}{{3h^{2} }}} \right)\left( {4\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} - \frac{{\partial^{2} w_{s} }}{{\partial x^{2} }}} \right)} \right],$$

(A.18)

$$\begin{aligned} &(\tau_{yzz}^{1} )_{l} = (\tau_{zyz}^{1} )_{l} = (\tau_{zzy}^{1} )_{l}\\ &\quad = 2\mu_{l} l_{1}^{2} \left[ { - \frac{1}{15}\left( {2\frac{{\partial^{2} u}}{\partial x\partial y} + \frac{{\partial^{2} v}}{{\partial x^{2} }} + 3\frac{{\partial^{2} v}}{{\partial y^{2} }}} \right) - \frac{1}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}} \right. \\ & \qquad - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{b} }}{\partial y} - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial y} - \frac{2}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ & \qquad - \frac{1}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{1}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ & \qquad - \left. { z\left( {\frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}} + \frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} + \frac{8}{{3h^{2} }}\frac{{\partial w_{s} }}{\partial y}} \right) + \frac{1}{5}z\left( {\frac{{\partial^{3} w_{b} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{3} w_{b} }}{{\partial y^{3} }}} \right) + \frac{{z^{3} }}{{3h^{2} }}\left( {\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}}} \right)} \right], \\ \end{aligned}$$

(A.19)

$$(m_{xx}^{(s)} )_{l} = 2\mu_{l} l_{2}^{2} \left[ {\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \left( {\frac{3}{8} + \frac{{5z^{2} }}{{2h^{2} }}} \right)\frac{{\partial^{2} w_{s} }}{\partial x\partial y}} \right],$$

(A.20)

$$(m_{yy}^{(s)} )_{l} = 2\mu_{l} l_{2}^{2} \left[ { - \frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \left( {\frac{3}{8} + \frac{{5z^{2} }}{{2h^{2} }}} \right)\frac{{\partial^{2} w_{s} }}{\partial x\partial y}} \right],$$

(A.21)

$$(m_{xy}^{(s)} )_{l} = 2\mu_{l} l_{2}^{2} \left[ {\frac{1}{2}\left( {\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{{\partial^{2} w_{b} }}{{\partial x^{2} }}} \right) + \left( {\frac{3}{16} + \frac{{5z^{2} }}{{4h^{2} }}} \right)\left( {\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} - \frac{{\partial^{2} w_{s} }}{{\partial x^{2} }}} \right)} \right],$$

(A.22)

$$(m_{xz}^{(s)} )_{l} = 2\mu_{l} l_{2}^{2} \left[ {\frac{1}{4}\left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} - \frac{{\partial^{2} u}}{\partial x\partial y}} \right) + \frac{5z}{{2h^{2} }}\frac{{\partial w_{s} }}{\partial y}} \right],$$

(A.23)

$$(m_{yz}^{(s)} )_{l} = 2\mu_{l} l_{2}^{2} \left[ {\frac{1}{4}\left( {\frac{{\partial^{2} v}}{\partial x\partial y} - \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right) - \frac{5z}{{2h^{2} }}\frac{{\partial w_{s} }}{\partial x}} \right].$$

(A.24)

Appendix B

$$\left[ \begin{gathered} N_{xx}^{0} \hfill \\ N_{xx}^{1} \hfill \\ N_{xx}^{3} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {A_{0} } & \quad {A_{1} } & \quad {A_{3} } \\ {A_{1} } & \quad {A_{2} } & \quad {A_{4} } \\ {A_{3} } & \quad {A_{4} } & \quad {A_{6} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{\partial u}{{\partial x}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)^{2} \\ \frac{1}{4}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} \\ - \frac{5}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ \end{gathered} \right] + \left[ {\begin{array}{*{20}c} {F_{0} } & \quad {F_{1} } & \quad {F_{3} } \\ {F_{1} } & \quad {F_{2} } & \quad {F_{4} } \\ {F_{3} } & \quad {F_{4} } & \quad {F_{6} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{\partial v}{{\partial y}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)^{2} \\ \frac{1}{4}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} - \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} \\ - \frac{5}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} \\ \end{gathered} \right],$$

(B.1)

$$\left[ \begin{gathered} N_{yy}^{0} \hfill \\ N_{yy}^{1} \hfill \\ N_{yy}^{3} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {A_{0} } & \quad {A_{1} } & \quad {A_{3} } \\ {A_{1} } & \quad {A_{2} } & \quad {A_{4} } \\ {A_{3} } & \quad {A_{4} } & \quad {A_{6} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{\partial v}{{\partial y}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)^{2} \\ \frac{1}{4}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} - \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} \\ - \frac{5}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} \\ \end{gathered} \right] + \left[ {\begin{array}{*{20}c} {F_{0} } & \quad {F_{1} } & \quad {F_{3} } \\ {F_{1} } & \quad {F_{2} } & \quad {F_{4} } \\ {F_{3} } & \quad {F_{4} } & \quad {F_{6} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{\partial u}{{\partial x}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)^{2} \\ \frac{1}{4}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} \\ - \frac{5}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ \end{gathered} \right],$$

(B.2)

$$\left[ \begin{gathered} N_{xy}^{0} \hfill \\ N_{xy}^{1} \hfill \\ N_{xy}^{3} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {B_{0} } & \quad {B_{1} } & \quad {B_{3} } \\ {B_{1} } & \quad {B_{2} } & \quad {B_{4} } \\ {B_{3} } & \quad {B_{4} } & \quad {B_{6} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{1}{2}\left( {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}} + \frac{{\partial w_{b} }}{\partial y}\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial y}\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{b} }}{\partial y}\frac{{\partial w_{s} }}{\partial x} + \frac{{\partial w_{s} }}{\partial y}\frac{{\partial w_{s} }}{\partial x}} \right) \\ \frac{1}{4}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} - \frac{{\partial^{2} w_{b} }}{\partial x\partial y} \\ - \frac{5}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ \end{gathered} \right],$$

(B.3)

$$\left[ \begin{gathered} N_{xz}^{0} \hfill \\ N_{xz}^{2} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {B_{0} } & \quad {B_{2} } \\ {B_{2} } & \quad {B_{4} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{5}{8}\frac{{\partial w_{s} }}{\partial x} \\ - \frac{5}{{2h^{2} }}\frac{{\partial w_{s} }}{\partial x} \\ \end{gathered} \right],$$

(B.4)

$$\left[ \begin{gathered} N_{yz}^{0} \hfill \\ N_{yz}^{2} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {B_{0} } & \quad {B_{2} } \\ {B_{2} } & \quad {B_{4} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{5}{8}\frac{{\partial w_{s} }}{\partial y} \\ - \frac{5}{{2h^{2} }}\frac{{\partial w_{s} }}{\partial y} \\ \end{gathered} \right],$$

(B.5)

$$\left[ \begin{gathered} P_{x}^{0} \hfill \\ P_{x}^{1} \hfill \\ P_{x}^{3} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {C_{0} } & \quad {C_{1} } & \quad {C_{3} } \\ {C_{1} } & \quad {C_{2} } & \quad {C_{4} } \\ {C_{3} } & \quad {C_{4} } & \quad {C_{6} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{\partial x\partial y} + \left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)\left( {\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{{\partial^{2} w_{s} }}{\partial x\partial y}} \right) + \left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)\left( {\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{{\partial^{2} w_{s} }}{{\partial x^{2} }}} \right) \\ \frac{1}{4}\left( {\frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }} + \frac{{\partial^{3} w_{s} }}{{\partial x^{3} }}} \right) - \left( {\frac{{\partial^{3} w_{b} }}{{\partial x\partial y^{2} }} + \frac{{\partial^{3} w_{b} }}{{\partial x^{3} }}} \right) \\ - \frac{5}{{3h^{2} }}\left( {\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }}} \right) \\ \end{gathered} \right],$$

(B.6)

$$\left[ \begin{gathered} P_{y}^{0} \hfill \\ P_{y}^{1} \hfill \\ P_{y}^{3} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {C_{0} } & \quad {C_{1} } & \quad {C_{3} } \\ {C_{1} } & \quad {C_{2} } & \quad {C_{4} } \\ {C_{3} } & \quad {C_{4} } & \quad {C_{6} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{{\partial^{2} u}}{\partial x\partial y} + \frac{{\partial^{2} v}}{{\partial y^{2} }} + \left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)\left( {\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) + \left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)\left( {\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{{\partial^{2} w_{s} }}{\partial x\partial y}} \right) \\ \frac{1}{4}\left( {\frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{3} w_{s} }}{{\partial y^{3} }}} \right) - \left( {\frac{{\partial^{3} w_{b} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{3} w_{b} }}{{\partial y^{3} }}} \right) \\ - \frac{5}{{3h^{2} }}\left( {\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}}} \right) \\ \end{gathered} \right],$$

(B.7)

$$\left[ \begin{gathered} P_{z}^{0} \hfill \\ P_{z}^{2} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {C_{0} } & \quad {C_{2} } \\ {C_{2} } & \quad {C_{4} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{1}{4}\left( {\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} + \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) - \left( {\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}} \right) \\ - \frac{5}{{h^{2} }}\left( {\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} + \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) \\ \end{gathered} \right],$$

(B.8)

$$\left[ \begin{gathered} M_{xx}^{0} \hfill \\ M_{xx}^{2} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {E_{0} } & \quad {E_{2} } \\ {E_{2} } & \quad {E_{4} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{3}{8}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} + \frac{{\partial^{2} w_{b} }}{\partial x\partial y} \\ \frac{5}{{2h^{2} }}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ \end{gathered} \right],$$

(B.9)

$$\left[ \begin{gathered} M_{xy}^{0} \hfill \\ M_{xy}^{2} \hfill \\ \end{gathered} \right] = - \left[ {\begin{array}{*{20}c} {E_{0} } & \quad {E_{2} } \\ {E_{2} } & \quad {E_{4} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{3}{16}\left( {\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) + \frac{1}{2}\left( {\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}} \right) \\ \frac{5}{{4h^{2} }}\left( {\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) \\ \end{gathered} \right],$$

(B.10)

$$\left[ \begin{gathered} M_{xz}^{0} \hfill \\ M_{xz}^{1} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {E_{0} } & \quad {E_{1} } \\ {E_{1} } & \quad {E_{2} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{1}{4}\left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} - \frac{{\partial^{2} u}}{\partial x\partial y}} \right) \\ \frac{5}{{2h^{2} }}\frac{{\partial w_{s} }}{\partial y} \\ \end{gathered} \right],$$

(B.11)

$$\left[ \begin{gathered} M_{yz}^{0} \hfill \\ M_{yz}^{1} \hfill \\ \end{gathered} \right] = - \left[ {\begin{array}{*{20}c} {E_{0} } & \quad {E_{1} } \\ {E_{1} } & \quad {E_{2} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{1}{4}\left( {\frac{{\partial^{2} u}}{{\partial y^{2} }} - \frac{{\partial^{2} v}}{\partial x\partial y}} \right) \\ \frac{5}{{2h^{2} }}\frac{{\partial w_{s} }}{\partial x} \\ \end{gathered} \right],$$

(B.12)

$$\left[ \begin{gathered} T_{xxx}^{0} \hfill \\ T_{xxx}^{1} \hfill \\ T_{xxx}^{3} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {D_{0} } & \quad {D_{1} } & \quad {D_{3} } \\ {D_{1} } & \quad {D_{2} } & \quad {D_{4} } \\ {D_{3} } & \quad {D_{4} } & \quad {D_{6} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{1}{5}\left( {2\frac{{\partial^{2} u}}{{\partial x^{2} }} - \frac{{\partial^{2} u}}{{\partial y^{2} }} - 2\frac{{\partial^{2} v}}{\partial x\partial y}} \right) - \frac{1}{5}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}\frac{{\partial w_{b} }}{\partial x} - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{b} }}{\partial x} - \frac{1}{5}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} \\ - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} - \frac{2}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ + \frac{2}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{2}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{2}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} + \frac{2}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ \left( {\frac{1}{10}\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} - \frac{3}{20}\frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }} + \frac{2}{{h^{2} }}\frac{{\partial w_{s} }}{\partial x}} \right) - \frac{1}{5}\left( {2\frac{{\partial^{3} w_{b} }}{{\partial x^{3} }} - 3\frac{{\partial^{3} w_{b} }}{{\partial x\partial y^{2} }}} \right) \\ - \frac{1}{{h^{2} }}\left( {\frac{2}{3}\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} - \frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }}} \right) \\ \end{gathered} \right],$$

(B.13)

$$\left[ \begin{gathered} T_{xyx}^{0} \hfill \\ T_{xyx}^{1} \hfill \\ T_{xyx}^{3} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {D_{0} } & \quad {D_{1} } & \quad {D_{3} } \\ {D_{1} } & \quad {D_{2} } & \quad {D_{4} } \\ {D_{3} } & \quad {D_{4} } & \quad {D_{6} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{1}{15}\left( {\frac{{8\partial^{2} u}}{\partial x\partial y} + \frac{{4\partial^{2} v}}{{\partial x^{2} }} - 3\frac{{\partial^{2} v}}{{\partial y^{2} }}} \right) - \frac{1}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{b} }}{\partial y} \\ - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial y} + \frac{8}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ + \frac{4}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{4}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{4}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} + \frac{4}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ \left( {\frac{1}{5}\frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}} - \frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} + \frac{2}{{3h^{2} }}\frac{{\partial w_{s} }}{\partial y}} \right) - \frac{1}{5}\left( {4\frac{{\partial^{3} w_{b} }}{{\partial x^{2} \partial y}} - \frac{{\partial^{3} w_{b} }}{{\partial y^{3} }}} \right) \\ - \frac{1}{{3h^{2} }}\left( {4\frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}} - \frac{{\partial^{3} w_{s} }}{{\partial y^{3} }}} \right) \\ \end{gathered} \right],$$

(B.14)

$$\left[ \begin{gathered} T_{xzx}^{0} \hfill \\ T_{xzx}^{2} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {D_{0} } & \quad {D_{2} } \\ {D_{2} } & \quad {D_{4} } \\ \end{array} } \right]\left[ \begin{gathered} - \frac{1}{15}\left( {4\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}} \right) + \frac{1}{10}\left( {4\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) \\ - \frac{2}{{3h^{2} }}\left( {4\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) \\ \end{gathered} \right],$$

(B.15)

$$\left[ \begin{gathered} T_{yyx}^{0} \hfill \\ T_{yyx}^{1} \hfill \\ T_{yyx}^{3} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {D_{0} } & \quad {D_{1} } & \quad {D_{3} } \\ {D_{1} } & \quad {D_{2} } & \quad {D_{4} } \\ {D_{3} } & \quad {D_{4} } & \quad {D_{6} } \\ \end{array} } \right]\left[ \begin{gathered} - \frac{1}{15}\left( {3\frac{{\partial^{2} u}}{{\partial x^{2} }} - 4\frac{{\partial^{2} u}}{{\partial y^{2} }} - 8\frac{{\partial^{2} v}}{\partial x\partial y}} \right) + \frac{4}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + \frac{4}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} + \frac{4}{15}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} \\ + \frac{4}{15}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} + \frac{8}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} + \frac{8}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ - \frac{1}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ \left( { - \frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} + \frac{1}{5}\frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }} + \frac{2}{{3h^{2} }}\frac{{\partial w_{s} }}{\partial x}} \right) - \frac{1}{5}\left( {4\frac{{\partial^{3} w_{b} }}{{\partial x\partial y^{2} }} - \frac{{\partial^{3} w_{b} }}{{\partial x^{3} }}} \right) \\ - \frac{1}{{3h^{2} }}\left( {4\frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }} - \frac{{\partial^{3} w_{s} }}{{\partial x^{3} }}} \right) \\ \end{gathered} \right],$$

(B.16)

$$\left[ \begin{gathered} T_{zzx}^{0} \hfill \\ T_{zzx}^{1} \hfill \\ T_{zzx}^{3} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {D_{0} } & \quad {D_{1} } & \quad {D_{3} } \\ {D_{1} } & \quad {D_{2} } & \quad {D_{4} } \\ {D_{3} } & \quad {D_{4} } & \quad {D_{6} } \\ \end{array} } \right]\left[ \begin{gathered} - \frac{1}{15}\left( {3\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }} + 2\frac{{\partial^{2} v}}{\partial x\partial y}} \right) - \frac{1}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{1}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} - \frac{1}{15}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} \\ - \frac{1}{15}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial x} - \frac{2}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ - \frac{1}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ - \left( {\frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} + \frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }} + \frac{8}{{3h^{2} }}\frac{{\partial w_{s} }}{\partial x}} \right) + \frac{1}{5}\left( {\frac{{\partial^{3} w_{b} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{b} }}{{\partial x\partial y^{2} }}} \right) \\ \frac{1}{{3h^{2} }}\left( {\frac{{\partial^{3} w_{s} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{s} }}{{\partial x\partial y^{2} }}} \right) \\ \end{gathered} \right],$$

(B.17)

$$\left[ \begin{gathered} T_{yzx}^{0} \hfill \\ T_{yzx}^{2} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {D_{0} } & \quad {D_{2} } \\ {D_{2} } & \quad {D_{4} } \\ \end{array} } \right]\left[ \begin{gathered} - \left( {\frac{1}{3}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{1}{2}\frac{{\partial^{2} w_{s} }}{\partial x\partial y}} \right) \\ - \frac{10}{{3h^{2} }}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ \end{gathered} \right],$$

(B.18)

$$\begin{aligned}\left[ \begin{gathered} T_{yyy}^{0} \hfill \\ T_{yyy}^{1} \hfill \\ T_{yyy}^{3} \hfill \\ \end{gathered} \right] = &\left[ {\begin{array}{*{20}c} {D_{0} } & \quad {D_{1} } & \quad {D_{3} } \\ {D_{1} } & \quad {D_{2} } & \quad {D_{4} } \\ {D_{3} } & \quad {D_{4} } & \quad {D_{6} } \\ \end{array} } \right]\\ &\left[ \begin{gathered} \frac{1}{5}\left( {2\frac{{\partial^{2} v}}{{\partial y^{2} }} - \frac{{\partial^{2} v}}{{\partial x^{2} }} - 2\frac{{\partial^{2} u}}{\partial x\partial y}} \right) + \frac{2}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + \frac{2}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + \frac{2}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{b} }}{\partial y} + \frac{2}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial y} \\ - \frac{2}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} - \frac{2}{5}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ - \frac{1}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ \left( {\frac{1}{10}\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} - \frac{3}{20}\frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}} + \frac{2}{{h^{2} }}\frac{{\partial w_{s} }}{\partial y}} \right) - \frac{1}{5}\left( {2\frac{{\partial^{3} w_{b} }}{{\partial y^{3} }} - 3\frac{{\partial^{3} w_{b} }}{{\partial x^{2} \partial y}}} \right) \\ - \frac{1}{{h^{2} }}\left( {\frac{2}{3}\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} - \frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}}} \right) \\ \end{gathered} \right], \end{aligned}$$

(B.19)

$$\left[ \begin{gathered} T_{yzy}^{0} \hfill \\ T_{yzy}^{2} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {D_{0} } & \quad {D_{2} } \\ {D_{2} } & \quad {D_{4} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{1}{15}\left( {\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - 4\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}} \right) - \frac{1}{10}\left( {\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - 4\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) \\ \frac{2}{{3h^{2} }}\left( {\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - 4\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) \\ \end{gathered} \right],$$

(B.20)

$$\begin{aligned} \left[ \begin{gathered} T_{zzy}^{0} \hfill \\ T_{zzy}^{1} \hfill \\ T_{zzy}^{3} \hfill \\ \end{gathered} \right] = & \left[ {\begin{array}{*{20}c} {D_{0} } & \quad {D_{1} } & \quad {D_{3} } \\ {D_{1} } & \quad {D_{2} } & \quad {D_{4} } \\ {D_{3} } & \quad {D_{4} } & \quad {D_{6} } \\ \end{array} } \right]\\ &\left[ \begin{gathered} - \frac{1}{15}\left( {2\frac{{\partial^{2} u}}{\partial x\partial y} + \frac{{\partial^{2} v}}{{\partial x^{2} }} + 3\frac{{\partial^{2} v}}{{\partial y^{2} }}} \right) - \frac{1}{5}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{1}{5}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{b} }}{\partial y} \\ - \frac{1}{5}\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}\frac{{\partial w_{s} }}{\partial y} - \frac{2}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{b} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{b} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} - \frac{2}{15}\frac{{\partial w_{s} }}{\partial x}\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ - \frac{1}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - \frac{1}{15}\frac{{\partial w_{b} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - \frac{1}{15}\frac{{\partial w_{s} }}{\partial y}\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ - \left( {\frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}} + \frac{1}{20}\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} + \frac{8}{{3h^{2} }}\frac{{\partial w_{s} }}{\partial y}} \right) + \frac{1}{5}\left( {\frac{{\partial^{3} w_{b} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{3} w_{b} }}{{\partial y^{3} }}} \right) \\ \frac{1}{{3h^{2} }}\left( {\frac{{\partial^{3} w_{s} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{s} }}{{\partial x^{2} \partial y}}} \right) \\ \end{gathered} \right], \end{aligned}$$

(B.21)

$$\left[ \begin{gathered} T_{zzz}^{0} \hfill \\ T_{zzz}^{2} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {D_{0} } & \quad {D_{2} } \\ {D_{2} } & \quad {D_{4} } \\ \end{array} } \right]\left[ \begin{gathered} \frac{1}{5}\left( {\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}} \right) - \frac{3}{10}\left( {\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} + \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) \\ \frac{2}{{h^{2} }}\left( {\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} + \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \right) \\ \end{gathered} \right],$$

(B.22)

where

$$\begin{gathered} A_{i} = \sum\limits_{l = 1}^{k} {\int_{{z_{l} }}^{{z_{l + 1} }} {(\lambda_{l} + 2\mu_{l} )z^{i} {\text{d}}z, \;\, } } i = 0,1,2,3,4,6, \;\, B_{i} = \sum\limits_{l = 1}^{k} {\int_{{z_{l} }}^{{z_{l + 1} }} {2\mu_{l} z^{i} {\text{d}}z, \;\, } } i = 0,1,2,3,4,6, \hfill \\ C_{i} = \sum\limits_{l = 1}^{k} {\int_{{z_{l} }}^{{z_{l + 1} }} {2\mu_{l} l_{0}^{2} z^{i} {\text{d}}z, \;\, } } i = 0,1,2,3,4,6, \;\, D_{i} = \sum\limits_{l = 1}^{k} {\int_{{z_{l} }}^{{z_{l + 1} }} {2\mu_{l} l_{1}^{2} z^{i} {\text{d}}z, \;\, } } i = 0,1,2,3,4,6, \hfill \\ E_{i} = \sum\limits_{l = 1}^{k} {\int_{{z_{l} }}^{{z_{l + 1} }} {2\mu_{l} l_{2}^{2} z^{i} {\text{d}}z, \;\, } } i = 0,1,2,3,4,6, \, F_{i} = \sum\limits_{l = 1}^{k} {\int_{{z_{l} }}^{{z_{l + 1} }} {\lambda_{l} z^{i} {\text{d}}z, \;\, } } i = 0,1,2,3,4,6. \hfill \\ \end{gathered}$$

(B.23)

Appendix C

$$\begin{aligned} L_{11} & = - \frac{1}{2}B_{0} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } - A_{0} \psi_{n} \phi_{m}^{(3)} + C_{0} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} + \frac{2}{3}D_{0} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} + \frac{1}{8}E_{0} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} \\ & \quad + \frac{4}{15}D_{0} \phi_{m}^{\prime } \psi_{n}^{(4)} + \frac{1}{8}E_{0} \phi_{m}^{\prime } \psi_{n}^{(4)} + C_{0} \psi_{n} \phi_{m}^{(5)} + \frac{2}{5}D_{0} \psi_{n} \phi_{m}^{(5)} , \\ \end{aligned}$$

(C.1)

$$\begin{aligned} L_{12} & = - \frac{1}{2}B_{0} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } - F_{0} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } + C_{0} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} + \frac{2}{15}D_{0} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} \\ & \quad - \frac{1}{8}E_{0} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} + C_{0} \phi_{m}^{\prime } \psi_{n}^{(4)} + \frac{2}{15}D_{0} \phi_{m}^{\prime } \psi_{n}^{(4)} - \frac{1}{8}E_{0} \phi_{m}^{\prime } \psi_{n}^{(4)} , \\ \end{aligned}$$

(C.2)

$$\begin{aligned} L_{13} & = B_{1} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } + F_{1} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } + A_{1} \psi_{n} \phi_{m}^{(3)} - 2C_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} - \frac{4}{5}D_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} \\ & \quad - C_{1} \phi_{m}^{\prime } \psi_{n}^{(4)} - \frac{2}{5}D_{1} \phi_{m}^{\prime } \psi_{n}^{(4)} - C_{1} \psi_{n} \phi_{m}^{(5)} - \frac{2}{5}D_{1} \psi_{n} \phi_{m}^{(5)} , \\ \end{aligned}$$

(C.3)

$$\begin{aligned} L_{14} & = - \frac{1}{2}B_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime 2} - F_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime 2} - A_{0} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{\prime \prime } + 5C_{0} \phi_{m}^{\prime } \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } + \frac{16}{{15}}D_{0} \phi_{m}^{\prime } \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } \\ & \quad - \frac{1}{2}B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } + 2C_{0} \psi_{n} \phi_{m}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{5}{3}D_{0} \psi_{n} \phi_{m}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + 2C_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime \prime 2} + \frac{8}{15}D_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime \prime 2} \\ & \quad + C_{0} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{(3)} + \frac{2}{15}D_{0} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{(3)} + 3C_{0} \psi_{n}^{2} \phi_{m}^{\prime \prime } \phi_{m}^{(3)} + \frac{6}{5}D_{0} \psi_{n}^{2} \phi_{m}^{\prime \prime } \phi_{m}^{(3)} + \frac{1}{3}D_{0} \phi_{m} \psi_{n} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} \\ & \quad + 2C_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime } \psi_{n}^{(3)} + \frac{4}{5}D_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime } \psi_{n}^{(3)} + C_{0} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(4)} + \frac{2}{5}D_{0} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(4)} + \frac{4}{15}D_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{(4)} , \\ \end{aligned}$$

(C.4)

$$\begin{aligned} L_{15} & = - \frac{1}{4}B_{1} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } + \frac{{5B_{3} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} + \frac{{2D_{1} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } }}{{h^{2} }} - \frac{1}{4}F_{1} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } + \frac{{5F_{3} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} - \frac{1}{4}A_{1} \psi_{n} \phi_{m}^{(3)} \\ & \quad + \frac{{5A_{3} \psi_{n} \phi_{m}^{(3)} }}{{3h^{2} }} + \frac{{2D_{1} \psi_{n} \phi_{m}^{(3)} }}{{h^{2} }} + \frac{1}{2}C_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} - \frac{{10C_{3} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} }}{{3h^{2} }} + \frac{1}{5}D_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} - \frac{{4D_{3} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} }}{{3h^{2} }} \\ & \quad + \frac{1}{4}C_{1} \phi_{m}^{\prime } \psi_{n}^{(4)} - \frac{{5C_{3} \phi_{m}^{\prime } \psi_{n}^{(4)} }}{{3h^{2} }} + \frac{1}{10}D_{1} \phi_{m}^{\prime } \psi_{n}^{(4)} - \frac{{2D_{3} \phi_{m}^{\prime } \psi_{n}^{(4)} }}{{3h^{2} }} + \frac{1}{4}C_{1} \psi_{n} \phi_{m}^{(5)} - \frac{{5C_{3} \psi_{n} \phi_{m}^{(5)} }}{{3h^{2} }} \\ & \quad + \frac{1}{10}D_{1} \psi_{n} \phi_{m}^{(5)} - \frac{{2D_{3} \psi_{n} \phi_{m}^{(5)} }}{{3h^{2} }}, \\ \end{aligned}$$

(C.5)

$$\begin{aligned} L_{16} & = - B_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime 2} - 2F_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime 2} - 2A_{0} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{\prime \prime } + 10C_{0} \phi_{m}^{\prime } \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } + \frac{32}{{15}}D_{0} \phi_{m}^{\prime } \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } \\ & \quad - B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } + 4C_{0} \psi_{n} \phi_{m}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{10}{3}D_{0} \psi_{n} \phi_{m}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + 4C_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime \prime 2} + \frac{16}{{15}}D_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime \prime 2} \\ & \quad + 2C_{0} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{(3)} + \frac{4}{15}D_{0} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{(3)} + 6C_{0} \psi_{n}^{2} \phi_{m}^{\prime \prime } \phi_{m}^{(3)} + \frac{12}{5}D_{0} \psi_{n}^{2} \phi_{m}^{\prime \prime } \phi_{m}^{(3)} + \frac{2}{3}D_{0} \phi_{m} \psi_{n} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} \\ & \quad + 4C_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime } \psi_{n}^{(3)} + \frac{8}{5}D_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime } \psi_{n}^{(3)} + 2C_{0} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(4)} + \frac{4}{5}D_{0} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(4)} + \frac{8}{15}D_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{(4)} , \\ \end{aligned}$$

(C.6)

$$\begin{aligned} L_{17} & = - \frac{1}{2}B_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime 2} - F_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime 2} - A_{0} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{\prime \prime } + 5C_{0} \phi_{m}^{\prime } \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } + \frac{16}{{15}}D_{0} \phi_{m}^{\prime } \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } \\ & \quad - \frac{1}{2}B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } + 2C_{0} \psi_{n} \phi_{m}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{5}{3}D_{0} \psi_{n} \phi_{m}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + 2C_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime \prime 2} + \frac{8}{15}D_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime \prime 2} \\ & \quad + C_{0} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{(3)} + \frac{2}{15}D_{0} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{(3)} + 3C_{0} \psi_{n}^{2} \phi_{m}^{\prime \prime } \phi_{m}^{(3)} + \frac{6}{5}D_{0} \psi_{n}^{2} \phi_{m}^{\prime \prime } \phi_{m}^{(3)} + \frac{1}{3}D_{0} \phi_{m} \psi_{n} \psi_{n}^{\prime \prime } \phi_{m}^{(3)} \\ & \quad + 2C_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime } \psi_{n}^{(3)} + \frac{4}{5}D_{0} \phi_{m} \phi_{m}^{\prime } \psi_{n}^{\prime } \psi_{n}^{(3)} + C_{0} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(4)} + \frac{2}{5}D_{0} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(4)} + \frac{4}{15}D_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{(4)} , \\ \end{aligned}$$

(C.7)

$$L_{111} = I_{0} \psi_{n} \phi_{m}^{\prime } , \, L_{112} = - I_{1} \psi_{n} \phi_{m}^{\prime } , \, L_{113} = I_{2} \psi_{n} \phi_{m}^{\prime } ,$$

(C.8)

$$\begin{aligned} L_{21} & = - \frac{1}{2}B_{0} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } - F_{0} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } + C_{0} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} + \frac{2}{15}D_{0} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} - \frac{1}{8}E_{0} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} + C_{0} \psi_{n}^{\prime } \phi_{m}^{(4)} \\ & \quad + \frac{2}{15}D_{0} \psi_{n}^{\prime } \phi_{m}^{(4)} - \frac{1}{8}E_{0} \psi_{n}^{\prime } \phi_{m}^{(4)} , \\ \end{aligned}$$

(C.9)

$$\begin{aligned} L_{22} & = - \frac{1}{2}B_{0} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } - A_{0} \phi_{m} \psi_{n}^{(3)} + C_{0} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} + \frac{2}{3}D_{0} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} + \frac{1}{8}E_{0} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} + \frac{4}{15}D_{0} \psi_{n}^{\prime } \phi_{m}^{(4)} \\ & + \frac{1}{8}E_{0} \psi_{n}^{\prime } \phi_{m}^{(4)} + C_{0} \phi_{m} \psi_{n}^{(5)} + \frac{2}{5}D_{0} \phi_{m} \psi_{n}^{(5)} , \\ \end{aligned}$$

(C.10)

$$\begin{aligned} L_{23} & = B_{1} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } + F_{1} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } + A_{1} \phi_{m} \psi_{n}^{(3)} - 2C_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} - \frac{4}{5}D_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} - C_{1} \psi_{n}^{\prime } \phi_{m}^{(4)} - \frac{2}{5}D_{1} \psi_{n}^{\prime } \phi_{m}^{(4)} \\ & - C_{1} \phi_{m} \psi_{n}^{(5)} - \frac{2}{5}D_{1} \phi_{m} \psi_{n}^{(5)} , \\ \end{aligned}$$

(C.11)

$$\begin{aligned} L_{24} & = - \frac{1}{2}B_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime } - F_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime } - \frac{1}{2}B_{0} \phi_{m} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } + 2C_{0} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{\prime \prime 2} + \frac{8}{15}D_{0} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{\prime \prime 2} \\ & \quad - A_{0} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{\prime \prime } + 5C_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime } \psi_{n}^{\prime \prime } + \frac{16}{{15}}D_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime } \psi_{n}^{\prime \prime } + 2C_{0} \phi_{m} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{5}{3}D_{0} \phi_{m} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } \\ & \quad + 2C_{0} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime } \phi_{m}^{(3)} + \frac{4}{5}D_{0} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime } \phi_{m}^{(3)} + C_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{(3)} + \frac{2}{15}D_{0} \psi_{n} \phi_{m}^{\prime 2} 2\psi_{n}^{(3)} + \frac{1}{3}D_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} \\ & \quad + 3C_{0} \phi_{m}^{2} \psi_{n}^{\prime \prime } \psi_{n}^{(3)} + \frac{6}{5}D_{0} \phi_{m}^{2} \psi_{n}^{\prime \prime } \psi_{n}^{(3)} + \frac{4}{15}D_{0} \phi_{m} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{(4)} + C_{0} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(4)} + \frac{2}{5}D_{0} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(4)} , \\ \end{aligned}$$

(C.12)

$$\begin{aligned} L_{25} & = - \frac{1}{4}B_{1} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } + \frac{{5B_{3} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } }}{{3h^{2} }} + \frac{{2D_{1} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } }}{{h^{2} }} - \frac{1}{4}F_{1} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } + \frac{{5F_{3} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } }}{{3h^{2} }} - \frac{1}{4}A_{1} \phi_{m} \psi_{n}^{(3)} \\ & \quad + \frac{{5A_{3} \phi_{m} \psi_{n}^{(3)} }}{{3h^{2} }} + \frac{{2D_{1} \phi_{m} \psi_{n}^{(3)} }}{{h^{2} }} + \frac{1}{2}C_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} - \frac{{10C_{3} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} }}{{3h^{2} }} + \frac{1}{5}D_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} - \frac{{4D_{3} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} }}{{3h^{2} }} \\ & \quad + \frac{1}{4}C_{1} \psi_{n}^{\prime } \phi_{m}^{(4)} - \frac{{5C_{3} \psi_{n}^{\prime } \phi_{m}^{(4)} }}{{3h^{2} }} + \frac{1}{10}D_{1} \psi_{n}^{\prime } \phi_{m}^{(4)} - \frac{{2D_{3} \psi_{n}^{\prime } \phi_{m}^{(4)} }}{{3h^{2} }} + \frac{1}{4}C_{1} \phi_{m} \psi_{n}^{(5)} - \frac{{5C_{3} \phi_{m} \psi_{n}^{(5)} }}{{3h^{2} }} \\ & \quad + \frac{1}{10}D_{1} \phi_{m} \psi_{n}^{(5)} - \frac{{2D_{3} \phi_{m} \psi_{n}^{(5)} }}{{3h^{2} }}, \\ \end{aligned}$$

(C.13)

$$\begin{aligned} L_{26} & = - B_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime } - 2F_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime } - B_{0} \phi_{m} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } + 4C_{0} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{\prime \prime 2} + \frac{16}{{15}}D_{0} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{\prime \prime 2} \\ & \quad - 2A_{0} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{\prime \prime } + 10C_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime } \psi_{n}^{\prime \prime } + \frac{32}{{15}}D_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime } \psi_{n}^{\prime \prime } + 4C_{0} \phi_{m} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{10}{3}D_{0} \phi_{m} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } \\ & \quad + 4C_{0} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime } \phi_{m}^{(3)} + \frac{8}{5}D_{0} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime } \phi_{m}^{(3)} + 2C_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{(3)} + \frac{4}{15}D_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{(3)} + \frac{2}{3}D_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} \\ & \quad + 6C_{0} \phi_{m}^{2} \psi_{n}^{\prime \prime } \psi_{n}^{(3)} + \frac{12}{5}D_{0} \phi_{m}^{2} \psi_{n}^{\prime \prime } \psi_{n}^{(3)} + \frac{8}{15}D_{0} \phi_{m} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{(4)} + 2C_{0} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(4)} + \frac{4}{5}D_{0} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(4)} , \\ \end{aligned}$$

(C.14)

$$\begin{aligned} L_{27} & = - \frac{1}{2}B_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime } - F_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime } - \frac{1}{2}B_{0} \phi_{m} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } + 2C_{0} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{\prime 2} + \frac{8}{15}D_{0} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{\prime 2} \\ & \quad - A_{0} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{\prime \prime } + 5C_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime } \psi_{n}^{\prime \prime } + \frac{16}{{15}}D_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime } \psi_{n}^{\prime \prime } + 2C_{0} \phi_{m} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{5}{3}D_{0} \phi_{m} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } \\ & \quad + 2C_{0} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime } \phi_{m}^{(3)} + \frac{4}{5}D_{0} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime } \phi_{m}^{(3)} + C_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{(3)} + \frac{2}{15}D_{0} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{(3)} + \frac{1}{3}D_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{(3)} \\ & \quad + 3C_{0} \phi_{m}^{2} \psi_{n}^{\prime \prime } \psi_{n}^{(3)} + \frac{6}{5}D_{0} \phi_{m}^{2} \psi_{n}^{\prime \prime } \psi_{n}^{(3)} + \frac{4}{15}D_{0} \phi_{m} \psi_{n} \psi_{n}^{\prime } \phi_{m}^{(4)} + C_{0} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(4)} + \frac{2}{5}D_{0} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(4)} , \\ \end{aligned}$$

(C.15)

$$L_{211} = I_{0} \phi_{m} \psi_{n}^{\prime } , \, L_{212} = - I_{1} \phi_{m} \psi_{n}^{\prime } , \, L_{213} = I_{2} \phi_{m} \psi_{n}^{\prime } ,$$

(C.16)

$$\begin{aligned} L_{31} & = - B_{1} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - F_{1} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - A_{1} \psi_{n} \phi_{m}^{(4)} + 2C_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} + \frac{4}{5}D_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} + C_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} \\ & \quad + \frac{2}{5}D_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} + C_{1} \psi_{n} \phi_{m}^{(6)} + \frac{2}{5}D_{1} \psi_{n} \phi_{m}^{(6)} , \\ \end{aligned}$$

(C.17)

$$\begin{aligned} L_{32} & = - B_{1} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - F_{1} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + C_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} + \frac{2}{5}D_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} - A_{1} \phi_{m} \psi_{n}^{(4)} + 2C_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} \\ & \quad + \frac{4}{5}D_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} + C_{1} \phi_{m} \psi_{n}^{(6)} + \frac{2}{5}D_{1} \phi_{m} \psi_{n}^{(6)} , \\ \end{aligned}$$

(C.18)

$$\begin{aligned} L_{33} & = k_{w} \phi_{m} \psi_{n} - k_{p} \psi_{n} \phi_{m}^{\prime \prime } - k_{p} \phi_{m} \psi_{n}^{\prime \prime } + 2B_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + 2C_{0} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{8}{15}D_{0} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + E_{0} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } \\ & \quad + 2F_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + A_{2} \psi_{n} \phi_{m}^{(4)} + C_{0} \psi_{n} \phi_{m}^{(4)} + \frac{4}{15}D_{0} \psi_{n} \phi_{m}^{(4)} + \frac{1}{2}E_{0} \psi_{n} \phi_{m}^{(4)} - 3C_{2} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} \\ & \quad - \frac{6}{5}D_{2} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} + A_{2} \phi_{m} \psi_{n}^{(4)} + C_{0} \phi_{m} \psi_{n}^{(4)} + \frac{4}{15}D_{0} \phi_{m} \psi_{n}^{(4)} + \frac{1}{2}E_{0} \phi_{m} \psi_{n}^{(4)} - 3C_{2} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} \\ & \quad - \frac{6}{5}D_{2} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} - C_{2} \psi_{n} \phi_{m}^{(6)} - \frac{2}{5}D_{2} \psi_{n} \phi_{m}^{(6)} - C_{2} \phi_{m} \psi_{n}^{(6)} - \frac{2}{5}D_{2} \phi_{m} \psi_{n}^{(6)} , \\ \end{aligned}$$

(C.19)

$$L_{34} = - B_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - A_{0} \psi_{n}^{2} \phi_{m}^{\prime \prime 2} - F_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } ,$$

(C.20)

$$L_{35} = - B_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - F_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - A_{0} \phi_{m}^{2} \psi_{n}^{\prime \prime 2} ,$$

(C.21)

$$\begin{aligned} L_{{36}} & = B_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} - 2F_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} - B_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} - F_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} + 7C_{1} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime 2}} + \frac{8}{5}D_{1} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime 2}} \\ & \quad - B_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} - F_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} - B_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} + 2F_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} + 4C_{1} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} \\ & \quad + \frac{{11}}{5}D_{1} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} + 7C_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} + \frac{8}{5}D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} + 4C_{1} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} + \frac{{11}}{5}D_{1} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} \\ & \quad - A_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(3)}} + 8C_{1} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} + 2D_{1} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} + 4C_{1} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} + \frac{{14}}{5}D_{1} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} \\ & \quad + 3C_{1} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} + \frac{6}{5}D_{1} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} - A_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} + 8C_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} + 2D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} \\ & \quad + 4C_{1} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} + \frac{{14}}{5}D_{1} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} + 3C_{1} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} + \frac{6}{5}D_{1} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} + C_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} \\ & \quad + \frac{2}{5}D_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} + 4C_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + \frac{8}{5}D_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + \frac{3}{5}D_{1} \phi _{m} \psi _{n} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + C_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} \\ & \quad + \frac{2}{5}D_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} + \frac{3}{5}D_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + 4C_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + \frac{8}{5}D_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + C_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} \\ & \quad + \frac{2}{5}D_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} + C_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} + \frac{2}{5}D_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} , \\ \end{aligned}$$

(C.22)

$$\begin{aligned} L_{37} & = - B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - \frac{1}{2}A_{0} \psi_{n}^{3} \phi_{m}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{1}{2}F_{0} \phi_{m}^{2} \psi_{n} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{1}{2}F_{0} \phi_{m} \psi_{n}^{2} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } \\ & \quad - \frac{1}{2}A_{0} \phi_{m}^{3} \psi_{n}^{\prime 2} \psi_{n}^{\prime \prime } , \\ \end{aligned}$$

(C.23)

$$\begin{aligned} L_{38} & = k_{w} \phi_{m} \psi_{n} - k_{p} \psi_{n} \phi_{m}^{\prime \prime } - \frac{1}{2}B_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{10B_{4} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} - \frac{1}{2}C_{0} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{10C_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{h^{2} }} \\ & \quad - \frac{4}{5}D_{0} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{28D_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} + \frac{3}{8}E_{0} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{5E_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{2h^{2} }} - \frac{1}{2}F_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{10F_{4} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} \\ & \quad - \frac{1}{4}A_{2} \psi_{n} \phi_{m}^{(4)} + \frac{{5A_{4} \psi_{n} \phi_{m}^{(4)} }}{{3h^{2} }} - \frac{1}{4}C_{0} \psi_{n} \phi_{m}^{(4)} + \frac{{5C_{2} \psi_{n} \phi_{m}^{(4)} }}{{h^{2} }} - \frac{2}{5}D_{0} \psi_{n} \phi_{m}^{(4)} + \frac{{14D_{2} \psi_{n} \phi_{m}^{(4)} }}{{3h^{2} }} \\ & \quad + \frac{3}{16}E_{0} \psi_{n} \phi_{m}^{(4)} + \frac{{5E_{2} \psi_{n} \phi_{m}^{(4)} }}{{4h^{2} }} + \frac{3}{4}C_{2} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} - \frac{{5C_{4} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} }}{{h^{2} }} + \frac{3}{10}D_{2} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} - \frac{{2D_{4} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} }}{{h^{2} }} \\ & \quad - \frac{1}{4}A_{2} \phi_{m} \psi_{n}^{(4)} + \frac{{5A_{4} \phi_{m} \psi_{n}^{(4)} }}{{3h^{2} }} - \frac{1}{4}C_{0} \phi_{m} \psi_{n}^{(4)} + \frac{{5C_{2} \phi_{m} \psi_{n}^{(4)} }}{{h^{2} }} - \frac{2}{5}D_{0} \phi_{m} \psi_{n}^{(4)} + \frac{{14D_{2} \phi_{m} \psi_{n}^{(4)} }}{{3h^{2} }} \\ & \quad + \frac{3}{16}E_{0} \phi_{m} \psi_{n}^{(4)} + \frac{{5E_{2} \phi_{m} \psi_{n}^{(4)} }}{{4h^{2} }} + \frac{3}{4}C_{2} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} - \frac{{5C_{4} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} }}{{h^{2} }} + \frac{3}{10}D_{2} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} - \frac{{2D_{4} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} }}{{h^{2} }} \\ & \quad + \frac{1}{4}C_{2} \psi_{n} \phi_{m}^{(6)} - \frac{{5C_{4} \psi_{n} \phi_{m}^{(6)} }}{{3h^{2} }} + \frac{1}{10}D_{2} \psi_{n} \phi_{m}^{(6)} - \frac{{2D_{4} \psi_{n} \phi_{m}^{(6)} }}{{3h^{2} }} + \frac{1}{4}C_{2} \phi_{m} \psi_{n}^{(6)} - \frac{{5C_{4} \phi_{m} \psi_{n}^{(6)} }}{{3h^{2} }} \\ & \quad + \frac{1}{10}D_{2} \phi_{m} \psi_{n}^{(6)} - \frac{{2D_{4} \phi_{m} \psi_{n}^{(6)} }}{{3h^{2} }} - k_{p} \phi_{m} \psi_{n}^{\prime \prime } , \\ \end{aligned}$$

(C.24)

$$L_{39} = - B_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - A_{0} \psi_{n}^{2} \phi_{m}^{\prime \prime 2} - F_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } ,$$

(C.25)

$$L_{390} = - B_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - F_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - A_{0} \phi_{m}^{2} \psi_{n}^{\prime \prime 2},$$

(C.26)

$$\begin{aligned} L_{391} & = - \frac{1}{2}B_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} + \frac{{10B_{3} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} }}{{3h^{2} }} - 4F_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - 2B_{1} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } - 2F_{1} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } \\ & \quad - \frac{5}{4}A_{1} \psi_{n}^{2} \phi_{m}^{\prime \prime 2} + \frac{{5A_{3} \psi_{n}^{2} \phi_{m}^{\prime \prime 2} }}{{3h^{2} }} + 14C_{1} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime 2} + \frac{16}{5}D_{1} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime 2} - 2B_{1} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } \\ & \quad - 2F_{1} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } - 2B_{1} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{3}{2}F_{1} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{10F_{3} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} + 8C_{1} \psi_{n} \phi_{m}^{\prime \prime 2} \psi_{n}^{\prime \prime } \\ & \quad + \frac{22}{5}D_{1} \psi_{n} \phi_{m}^{\prime \prime 2} \psi_{n}^{\prime \prime } - \frac{5}{4}A_{1} \phi_{m}^{2} \psi_{n}^{\prime \prime 2} + \frac{{5A_{3} \phi_{m}^{2} \psi_{n}^{\prime \prime 2} }}{{3h^{2} }} + 14C_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime 2} + \frac{16}{5}D_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime 2} + 8C_{1} \phi_{m} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime 2} \\ & \quad + \frac{22}{5}D_{1} \phi_{m} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime 2} - 2A_{1} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(3)} + 16C_{1} \phi_{m}^{\prime } \psi_{n}^{\prime 2} \phi_{m}^{(3)} + 4D_{1} \phi_{m}^{\prime } \psi_{n}^{\prime 2} \phi_{m}^{(3)} + 8C_{1} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } \phi_{m}^{(3)} \\ & \quad + \frac{28}{5}D_{1} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } \phi_{m}^{(3)} + 6C_{1} \psi_{n}^{2} \phi_{m}^{(3)2} + \frac{12}{5}D_{1} \psi_{n}^{2} \phi_{m}^{(3)2} - 2A_{1} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(3)} + 16C_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime } \psi_{n}^{(3)} \\ & \quad + 4D_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime } \psi_{n}^{(3)} + 8C_{1} \phi_{m} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{(3)} + \frac{28}{5}D_{1} \phi_{m} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{(3)} + 6C_{1} \phi_{m}^{2} \psi_{n}^{(3)2} + \frac{12}{5}D_{1} \phi_{m}^{2} \psi_{n}^{(3)2} \\ & \quad + 2C_{1} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{(4)} + \frac{4}{5}D_{1} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{(4)} + 8C_{1} \psi_{n}^{2} \phi_{m}^{\prime \prime } \phi_{m}^{(4)} + \frac{16}{5}D_{1} \psi_{n}^{2} \phi_{m}^{\prime \prime } \phi_{m}^{(4)} + \frac{6}{5}D_{1} \phi_{m} \psi_{n} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} \\ & \quad + 2C_{1} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{(4)} + \frac{4}{5}D_{1} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{(4)} + \frac{6}{5}D_{1} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} + 8C_{1} \phi_{m}^{2} \psi_{n}^{\prime \prime } \psi_{n}^{(4)} + \frac{16}{5}D_{1} \phi_{m}^{2} \psi_{n}^{\prime \prime } \psi_{n}^{(4)} \\ & \quad + 2C_{1} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(5)} + \frac{4}{5}D_{1} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(5)} + 2C_{1} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(5)} + \frac{4}{5}D_{1} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(5)} , \\ \end{aligned}$$

(C.27)

$$\begin{aligned} L_{392} & = - 3B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - \frac{3}{2}A_{0} \psi_{n}^{3} \phi_{m}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{3}{2}F_{0} \phi_{m}^{2} \psi_{n} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{3}{2}F_{0} \phi_{m} \psi_{n}^{2} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } \\ & \quad - \frac{3}{2}A_{0} \phi_{m}^{3} \psi_{n}^{\prime 2} \psi_{n}^{\prime \prime } , \\ \end{aligned}$$

(C.28)

$$\begin{aligned} L_{393} & = - \frac{3}{2}B_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} + \frac{{10B_{3} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} }}{{3h^{2} }} - 2F_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - B_{1} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } - F_{1} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } \\ & \quad - \frac{5}{4}A_{1} \psi_{n}^{2} \phi_{m}^{\prime \prime 2} + \frac{{5A_{3} \psi_{n}^{2} \phi_{m}^{\prime \prime 2} }}{{3h^{2} }} + 7C_{1} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime 2} + \frac{8}{5}D_{1} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime 2} - B_{1} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } - F_{1} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } \\ & \quad - B_{1} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - \frac{1}{2}F_{1} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{10F_{3} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} + 4C_{1} \psi_{n} \phi_{m}^{\prime \prime 2} \psi_{n}^{\prime \prime } + \frac{11}{5}D_{1} \psi_{n} \phi_{m}^{\prime \prime 2} \psi_{n}^{\prime \prime } \\ & \quad - \frac{5}{4}A_{1} \phi_{m}^{2} \psi_{n}^{\prime \prime 2} + \frac{{5A_{3} \phi_{m}^{2} \psi_{n}^{\prime \prime 2} }}{{3h^{2} }} + 7C_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime 2} + \frac{8}{5}D_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime 2} + 4C_{1} \phi_{m} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime 2} + \frac{11}{5}D_{1} \phi_{m} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime 2} \\ & \quad - A_{1} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(3)} + 8C_{1} \phi_{m}^{\prime } \psi_{n}^{\prime 2} \phi_{m}^{(3)} + 2D_{1} \phi_{m}^{\prime } \psi_{n}^{\prime 2} \phi_{m}^{(3)} + 4C_{1} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } \phi_{m}^{(3)} + \frac{14}{5}D_{1} \psi_{n} \phi_{m}^{\prime } \psi_{n}^{\prime \prime } \phi_{m}^{(3)} \\ & \quad + 3C_{1} \psi_{n}^{2} \phi_{m}^{(3)2} + \frac{6}{5}D_{1} \psi_{n}^{2} \phi_{m}^{(3)2} - A_{1} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(3)} + 8C_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime } \psi_{n}^{(3)} + 2D_{1} \phi_{m}^{\prime 2} \psi_{n}^{\prime } \psi_{n}^{(3)} \\ & \quad + 4C_{1} \phi_{m} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{(3)} + \frac{14}{5}D_{1} \phi_{m} \psi_{n}^{\prime } \phi_{m}^{\prime \prime } \psi_{n}^{(3)} + 3C_{1} \phi_{m}^{2} \psi_{n}^{(3)2} + \frac{6}{5}D_{1} \phi_{m}^{2} \psi_{n}^{(3)2} + C_{1} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{(4)} \\ & \quad c + \frac{2}{5}D_{1} \phi_{m} \psi_{n}^{\prime 2} \phi_{m}^{(4)} + 4C_{1} \psi_{n}^{2} \phi_{m}^{\prime \prime } \phi_{m}^{(4)} + \frac{8}{5}D_{1} \psi_{n}^{2} \phi_{m}^{\prime \prime } \phi_{m}^{(4)} + \frac{3}{5}D_{1} \phi_{m} \psi_{n} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} + C_{1} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{(4)} \\ & \quad + \frac{2}{5}D_{1} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{(4)} + \frac{3}{5}D_{1} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} + 4C_{1} \phi_{m}^{2} \psi_{n}^{\prime \prime } \psi_{n}^{(4)} + \frac{8}{5}D_{1} \phi_{m}^{2} \psi_{n}^{\prime \prime } \psi_{n}^{(4)} \\ & \quad + C_{1} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(5)} + \frac{2}{5}D_{1} \psi_{n}^{2} \phi_{m}^{\prime } \phi_{m}^{(5)} + C_{1} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(5)} + \frac{2}{5}D_{1} \phi_{m}^{2} \psi_{n}^{\prime } \psi_{n}^{(5)} , \\ \end{aligned}$$

(C.29)

$$\begin{aligned} L_{394} = - 3B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - \frac{3}{2}A_{0} \psi_{n}^{3} \phi_{m}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{3}{2}F_{0} \phi_{m}^{2} \psi_{n} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{3}{2}F_{0} \phi_{m} \psi_{n}^{2} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } - \frac{3}{2}A_{0} \phi_{m}^{3} \psi_{n}^{\prime 2} \psi_{n}^{\prime \prime } , \\ \end{aligned}$$

(C.30)

$$\begin{aligned} L_{395} & { = } - B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - \frac{1}{2}A_{0} \psi_{n}^{3} \phi_{m}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{1}{2}F_{0} \phi_{m}^{2} \psi_{n} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{1}{2}F_{0} \phi_{m} \psi_{n}^{2} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } \\ & \quad - \frac{1}{2}A_{0} \phi_{m}^{3} \psi_{n}^{\prime 2} \psi_{n}^{\prime \prime } , \\ \end{aligned}$$

(C.31)

$$\begin{gathered} L_{311} = C_{d} I_{0} \phi_{m} \psi_{n} ,L_{312} = C_{d} I_{0} \phi_{m} \psi_{n} ,L_{313} = I_{1} \psi_{n} \phi_{m}^{\prime \prime } ,L_{314} = I_{1} \phi_{m} \psi_{n}^{\prime \prime } , \hfill \\ L_{315} = I_{0} \phi_{m} \psi_{n} - I_{3} \psi_{n} \phi_{m}^{\prime \prime } - I_{3} \phi_{m} \psi_{n}^{\prime \prime } ,L_{316} = I_{0} \phi_{m} \psi_{n} + I_{5} \psi_{n} \phi_{m}^{\prime \prime } + I_{5} \phi_{m} \psi_{n}^{\prime \prime } , \hfill \\ \end{gathered}$$

(C.32)

$$\begin{aligned} L_{41} & = \frac{1}{4}B_{1} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - \frac{{5B_{3} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} - \frac{{2D_{1} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{h^{2} }} + \frac{1}{4}F_{1} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - \frac{{5F_{3} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} + \frac{1}{4}A_{1} \psi_{n} \phi_{m}^{(4)} \\ & \quad - \frac{{5A_{3} \psi_{n} \phi_{m}^{(4)} }}{{3h^{2} }} - \frac{{2D_{1} \psi_{n} \phi_{m}^{(4)} }}{{h^{2} }} - \frac{1}{2}C_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} + \frac{{10C_{3} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} }}{{3h^{2} }} - \frac{1}{5}D_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} + \frac{{4D_{3} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} }}{{3h^{2} }} \\ & \quad - \frac{1}{4}C_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} + \frac{{5C_{3} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} }}{{3h^{2} }} - \frac{1}{10}D_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} + \frac{{2D_{3} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} }}{{3h^{2} }} - \frac{1}{4}C_{1} \psi_{n} \phi_{m}^{(6)} + \frac{{5C_{3} \psi_{n} \phi_{m}^{(6)} }}{{3h^{2} }} \\ & \quad - \frac{1}{10}D_{1} \psi_{n} \phi_{m}^{(6)} + \frac{{2D_{3} \psi_{n} \phi_{m}^{(6)} }}{{3h^{2} }}, \\ \end{aligned}$$

(C.33)

$$\begin{aligned} L_{42} & = \frac{1}{4}B_{1} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - \frac{{5B_{3} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} - \frac{{2D_{1} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{h^{2} }} + \frac{1}{4}F_{1} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - \frac{{5F_{3} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} - \frac{1}{4}C_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} \\ & \quad + \frac{{5C_{3} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} }}{{3h^{2} }} - \frac{1}{10}D_{1} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} + \frac{{2D_{3} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} }}{{3h^{2} }} + \frac{1}{4}A_{1} \phi_{m} \psi_{n}^{(4)} - \frac{{5A_{3} \phi_{m} \psi_{n}^{(4)} }}{{3h^{2} }} - \frac{{2D_{1} \phi_{m} \psi_{n}^{(4)} }}{{h^{2} }} \\ & \quad - \frac{1}{2}C_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} + \frac{{10C_{3} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} }}{{3h^{2} }} - \frac{1}{5}D_{1} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} + \frac{{4D_{3} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} }}{{3h^{2} }} - \frac{1}{4}C_{1} \phi_{m} \psi_{n}^{(6)} + \frac{{5C_{3} \phi_{m} \psi_{n}^{(6)} }}{{3h^{2} }} \\ & \quad - \frac{1}{10}D_{1} \phi_{m} \psi_{n}^{(6)} + \frac{{2D_{3} \phi_{m} \psi_{n}^{(6)} }}{{3h^{2} }}, \\ \end{aligned}$$

(C.34)

$$\begin{aligned} L_{43} & = k_{w} \phi_{m} \psi_{n} - k_{p} \psi_{n} \phi_{m}^{\prime \prime } - k_{p} \phi_{m} \psi_{n}^{\prime \prime } - \frac{1}{2}B_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{10B_{4} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} - \frac{1}{2}C_{0} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{10C_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{h^{2} }} \\ & \quad - \frac{4}{5}D_{0} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{28D_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} + \frac{3}{8}E_{0} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{5E_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{2h^{2} }} - \frac{1}{2}F_{2} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } + \frac{{10F_{4} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } }}{{3h^{2} }} \\ & \quad - \frac{1}{4}A_{2} \psi_{n} \phi_{m}^{(4)} + \frac{{5A_{4} \psi_{n} \phi_{m}^{(4)} }}{{3h^{2} }} - \frac{1}{4}C_{0} \psi_{n} \phi_{m}^{(4)} + \frac{{5C_{2} \psi_{n} \phi_{m}^{(4)} }}{{h^{2} }} - \frac{2}{5}D_{0} \psi_{n} \phi_{m}^{(4)} + \frac{{14D_{2} \psi_{n} \phi_{m}^{(4)} }}{{3h^{2} }} \\ & \quad + \frac{3}{16}E_{0} \psi_{n} \phi_{m}^{(4)} + \frac{{5E_{2} \psi_{n} \phi_{m}^{(4)} }}{{4h^{2} }} + \frac{3}{4}C_{2} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} - \frac{{5C_{4} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} }}{{h^{2} }} + \frac{3}{10}D_{2} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} - \frac{{2D_{4} \psi_{n}^{\prime \prime } \phi_{m}^{(4)} }}{{h^{2} }} \\ & \quad - \frac{1}{4}A_{2} \phi_{m} \psi_{n}^{(4)} + \frac{{5A_{4} \phi_{m} \psi_{n}^{(4)} }}{{3h^{2} }} - \frac{1}{4}C_{0} \phi_{m} \psi_{n}^{(4)} + \frac{{5C_{2} \phi_{m} \psi_{n}^{(4)} }}{{h^{2} }} - \frac{2}{5}D_{0} \phi_{m} \psi_{n}^{(4)} + \frac{{14D_{2} \phi_{m} \psi_{n}^{(4)} }}{{3h^{2} }} \\ & \quad + \frac{3}{16}E_{0} \phi_{m} \psi_{n}^{(4)} + \frac{{5E_{2} \phi_{m} \psi_{n}^{(4)} }}{{4h^{2} }} + \frac{3}{4}C_{2} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} - \frac{{5C_{4} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} }}{{h^{2} }} + \frac{3}{10}D_{2} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} - \frac{{2D_{4} \phi_{m}^{\prime \prime } \psi_{n}^{(4)} }}{{h^{2} }} \\ & \quad + \frac{1}{4}C_{2} \psi_{n} \phi_{m}^{(6)} - \frac{{5C_{4} \psi_{n} \phi_{m}^{(6)} }}{{3h^{2} }} + \frac{1}{10}D_{2} \psi_{n} \phi_{m}^{(6)} - \frac{{2D_{4} \psi_{n} \phi_{m}^{(6)} }}{{3h^{2} }} + \frac{1}{4}C_{2} \phi_{m} \psi_{n}^{(6)} - \frac{{5C_{4} \phi_{m} \psi_{n}^{(6)} }}{{3h^{2} }} \\ & \quad + \frac{1}{10}D_{2} \phi_{m} \psi_{n}^{(6)} - \frac{{2D_{4} \phi_{m} \psi_{n}^{(6)} }}{{3h^{2} }}, \\ \end{aligned}$$

(C.35)

$$L_{44} = - B_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - A_{0} \psi_{n}^{2} \phi_{m}^{\prime \prime 2} - F_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } ,$$

(C.36)

$$L_{45} { = } - B_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - F_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - A_{0} \phi_{m}^{2} \psi_{n}^{\prime \prime 2} ,$$

(C.37)

$$\begin{aligned} L_{{46}} & = \frac{9}{4}B_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} - \frac{{5B_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} }}{{3h^{2} }} - \frac{{8D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} }}{{3h^{2} }} + \frac{1}{2}F_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} - \frac{{10F_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} }}{{3h^{2} }} + \frac{1}{4}B_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} \\ & \quad - \frac{{5B_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} }}{{3h^{2} }} - \frac{{2D_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} }}{{h^{2} }} + \frac{1}{4}F_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} - \frac{{5F_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} }}{{3h^{2} }} + \frac{5}{4}A_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime 2}} - \frac{{5A_{3} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime 2}} }}{{3h^{2} }} \\ & \quad - \frac{{2D_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime 2}} }}{{h^{2} }} - \frac{7}{4}C_{1} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} + \frac{{35C_{3} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{2}{5}D_{1} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} + \frac{{8D_{3} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} }}{{3h^{2} }} + \frac{1}{4}B_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} \\ & \quad - \frac{{5B_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} - \frac{{2D_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} }}{{h^{2} }} + \frac{1}{4}F_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} - \frac{{5F_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} + \frac{1}{4}B_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} \\ & \quad - \frac{{5B_{3} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} - \frac{{4D_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} + 2F_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} - C_{1} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} + \frac{{20C_{3} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} \\ & \quad - \frac{{11}}{{20}}D_{1} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} + \frac{{11D_{3} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} + \frac{5}{4}A_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime 2}} - \frac{{5A_{3} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{{2D_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime 2}} }}{{h^{2} }} - \frac{7}{4}C_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} \\ & \quad + \frac{{35C_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{2}{5}D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} + \frac{{8D_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} - C_{1} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} + \frac{{20C_{3} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{{11}}{{20}}D_{1} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} \\ & \quad + \frac{{11D_{3} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} + \frac{1}{4}A_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(3)}} - \frac{{5A_{3} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(3)}} }}{{3h^{2} }} - \frac{{2D_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(3)}} }}{{h^{2} }} - 2C_{1} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} \\ & \quad + \frac{{40C_{3} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - \frac{1}{2}D_{1} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} + \frac{{10D_{3} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - C_{1} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} + \frac{{20C_{3} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} }}{{3h^{2} }} \\ & \quad - \frac{7}{{10}}D_{1} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} + \frac{{14D_{3} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - \frac{3}{4}C_{1} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} + \frac{{5C_{3} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} }}{{h^{2} }} - \frac{3}{{10}}D_{1} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} \\ & \quad + \frac{{2D_{3} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} }}{{h^{2} }} + \frac{1}{4}A_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} - \frac{{5A_{3} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{3h^{2} }} - \frac{{2D_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{h^{2} }} - 2C_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} \\ & \quad + \frac{{40C_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{3h^{2} }} - \frac{1}{2}D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} + \frac{{10D_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{3h^{2} }} - C_{1} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} + \frac{{20C_{3} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} }}{{3h^{2} }} \\ & \quad - \frac{7}{{10}}D_{1} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} + \frac{{14D_{3} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} }}{{3h^{2} }} - \frac{3}{4}C_{1} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} + \frac{{5C_{3} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} }}{{h^{2} }} - \frac{3}{{10}}D_{1} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} \\ & \quad + \frac{{2D_{3} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} }}{{h^{2} }} - \frac{1}{4}C_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} + \frac{{5C_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} }}{{3h^{2} }} - \frac{1}{{10}}D_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} + \frac{{2D_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} }}{{3h^{2} }} \\ & \quad - C_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + \frac{{20C_{3} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{3h^{2} }} - \frac{2}{5}D_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + \frac{{8D_{3} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{3h^{2} }} - \frac{3}{{20}}D_{1} \phi _{m} \psi _{n} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} \\ & \quad + \frac{{D_{3} \phi _{m} \psi _{n} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{h^{2} }} - \frac{1}{4}C_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} + \frac{{5C_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} }}{{3h^{2} }} - \frac{1}{{10}}D_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} + \frac{{2D_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} }}{{3h^{2} }} \\ & \quad - \frac{3}{{20}}D_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + \frac{{D_{3} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{h^{2} }} - C_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + \frac{{20C_{3} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{3h^{2} }} - \frac{2}{5}D_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} \\ & \quad + \frac{{8D_{3} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{3h^{2} }} - \frac{1}{4}C_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} + \frac{{5C_{3} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} }}{{3h^{2} }} - \frac{1}{{10}}D_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} + \frac{{2D_{3} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} }}{{3h^{2} }} \\ & \quad - \frac{1}{4}C_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} + \frac{{5C_{3} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} }}{{3h^{2} }} - \frac{1}{{10}}D_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} + \frac{{2D_{3} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} }}{{3h^{2} }}, \\ \end{aligned}$$

(C.38)

$$\begin{aligned} L_{47} = - B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - \frac{1}{2}A_{0} \psi_{n}^{3} \phi_{m}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{1}{2}F_{0} \phi_{m}^{2} \psi_{n} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{1}{2}F_{0} \phi_{m} \psi_{n}^{2} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } - \frac{1}{2}A_{0} \phi_{m}^{3} \psi_{n}^{\prime 2} \psi_{n}^{\prime \prime } , \\ \end{aligned}$$

(C.39)

$$\begin{aligned} L_{{48}} & = k_{w} \phi _{m} \psi _{n} - k_{p} \psi _{n} \phi _{m} ^{{\prime \prime }} - \frac{{25}}{{32}}B_{0} \psi _{n} \phi _{m} ^{{\prime \prime }} + \frac{{25B_{2} \psi _{n} \phi _{m} ^{{\prime \prime }} }}{{4h^{2} }} - \frac{{25B_{4} \psi _{n} \phi _{m} ^{{\prime \prime }} }}{{2h^{4} }} - \frac{{80D_{2} \psi _{n} \phi _{m} ^{{\prime \prime }} }}{{3h^{4} }} - \frac{{25E_{2} \psi _{n} \phi _{m} ^{{\prime \prime }} }}{{2h^{4} }} \\ & \quad - k_{p} \phi _{m} \psi _{n} ^{{\prime \prime }} - \frac{{25}}{{32}}B_{0} \phi _{m} \psi _{n} ^{{\prime \prime }} + \frac{{25B_{2} \phi _{m} \psi _{n} ^{{\prime \prime }} }}{{4h^{2} }} - \frac{{25B_{4} \phi _{m} \psi _{n} ^{{\prime \prime }} }}{{2h^{4} }} - \frac{{80D_{2} \phi _{m} \psi _{n} ^{{\prime \prime }} }}{{3h^{4} }} - \frac{{25E_{2} \phi _{m} \psi _{n} ^{{\prime \prime }} }}{{2h^{4} }} + \frac{1}{8}B_{2} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} \\ & \quad - \frac{{5B_{4} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} + \frac{{50B_{6} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{9h^{4} }} + \frac{1}{8}C_{0} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} - \frac{{5C_{2} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{h^{2} }} + \frac{{50C_{4} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{h^{4} }} + \frac{6}{5}D_{0} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} \\ & \quad - \frac{{18D_{2} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{h^{2} }} + \frac{{200D_{4} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{4} }} + \frac{9}{{64}}E_{0} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} + \frac{{15E_{2} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{8h^{2} }} + \frac{{25E_{4} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{4h^{4} }} + \frac{1}{8}F_{2} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} \\ & \quad - \frac{{5F_{4} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} + \frac{{50F_{6} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{9h^{4} }} + \frac{1}{{16}}A_{2} \psi _{n} \phi _{m} ^{{(4)}} - \frac{{5A_{4} \psi _{n} \phi _{m} ^{{(4)}} }}{{6h^{2} }} + \frac{{25A_{6} \psi _{n} \phi _{m} ^{{(4)}} }}{{9h^{4} }} + \frac{1}{{16}}C_{0} \psi _{n} \phi _{m} ^{{(4)}} \\ & \quad - \frac{{5C_{2} \psi _{n} \phi _{m} ^{{(4)}} }}{{2h^{2} }} + \frac{{25C_{4} \psi _{n} \phi _{m} ^{{(4)}} }}{{h^{4} }} + \frac{3}{5}D_{0} \psi _{n} \phi _{m} ^{{(4)}} - \frac{{9D_{2} \psi _{n} \phi _{m} ^{{(4)}} }}{{h^{2} }} + \frac{{100D_{4} \psi _{n} \phi _{m} ^{{(4)}} }}{{3h^{4} }} + \frac{9}{{128}}E_{0} \psi _{n} \phi _{m} ^{{(4)}} \\ & \quad + \frac{{15E_{2} \psi _{n} \phi _{m} ^{{(4)}} }}{{16h^{2} }} + \frac{{25E_{4} \psi _{n} \phi _{m} ^{{(4)}} }}{{8h^{4} }} - \frac{3}{{16}}C_{2} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + \frac{{5C_{4} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{2h^{2} }} - \frac{{25C_{6} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{3h^{4} }} - \frac{3}{{40}}D_{2} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} \\ & \quad + \frac{{D_{4} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{h^{2} }} - \frac{{10D_{6} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{3h^{4} }} + \frac{1}{{16}}A_{2} \phi _{m} \psi _{n} ^{{(4)}} - \frac{{5A_{4} \phi _{m} \psi _{n} ^{{(4)}} }}{{6h^{2} }} + \frac{{25A_{6} \phi _{m} \psi _{n} ^{{(4)}} }}{{9h^{4} }} + \frac{1}{{16}}C_{0} \phi _{m} \psi _{n} ^{{(4)}} \\ & \quad - \frac{{5C_{2} \phi _{m} \psi _{n} ^{{(4)}} }}{{2h^{2} }} + \frac{{25C_{4} \phi _{m} \psi _{n} ^{{(4)}} }}{{h^{4} }} + \frac{3}{5}D_{0} \phi _{m} \psi _{n} ^{{(4)}} - \frac{{9D_{2} \phi _{m} \psi _{n} ^{{(4)}} }}{{h^{2} }} + \frac{{100D_{4} \phi _{m} \psi _{n} ^{{(4)}} }}{{3h^{4} }} + \frac{9}{{128}}E_{0} \phi _{m} \psi _{n} ^{{(4)}} \\ & \quad + \frac{{15E_{2} \phi _{m} \psi _{n} ^{{(4)}} }}{{16h^{2} }} + \frac{{25E_{4} \phi _{m} \psi _{n} ^{{(4)}} }}{{8h^{4} }} - \frac{3}{{16}}C_{2} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + \frac{{5C_{4} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{2h^{2} }} - \frac{{25C_{6} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{3h^{4} }} - \frac{3}{{40}}D_{2} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} \\ & \quad + \frac{{D_{4} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{h^{2} }} - \frac{{10D_{6} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{3h^{4} }} - \frac{1}{{16}}C_{2} \psi _{n} \phi _{m} ^{{(6)}} + \frac{{5C_{4} \psi _{n} \phi _{m} ^{{(6)}} }}{{6h^{2} }} - \frac{{25C_{6} \psi _{n} \phi _{m} ^{{(6)}} }}{{9h^{4} }} - \frac{1}{{40}}D_{2} \psi _{n} \phi _{m} ^{{(6)}} \\ & \quad + \frac{{D_{4} \psi _{n} \phi _{m} ^{{(6)}} }}{{3h^{2} }} - \frac{{10D_{6} \psi _{n} \phi _{m} ^{{(6)}} }}{{9h^{4} }} - \frac{1}{{16}}C_{2} \phi _{m} \psi _{n} ^{{(6)}} + \frac{{5C_{4} \phi _{m} \psi _{n} ^{{(6)}} }}{{6h^{2} }} - \frac{{25C_{6} \phi _{m} \psi _{n} ^{{(6)}} }}{{9h^{4} }} - \frac{1}{{40}}D_{2} \phi _{m} \psi _{n} ^{{(6)}} \\ & \quad + \frac{{D_{4} \phi _{m} \psi _{n} ^{{(6)}} }}{{3h^{2} }} - \frac{{10D_{6} \phi _{m} \psi _{n} ^{{(6)}} }}{{9h^{4} }}, \\ \end{aligned}$$

(C.40)

$$L_{49} { = } - B_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - A_{0} \psi_{n}^{2} \phi_{m}^{\prime \prime 2} - F_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } ,$$

(C.41)

$$L_{490} = - B_{0} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - F_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime \prime } \psi_{n}^{\prime \prime } - A_{0} \phi_{m}^{2} \psi_{n}^{\prime \prime 2} ,$$

(C.42)

$$\begin{aligned} L_{{491}} & = 2B_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} - \frac{{16D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} }}{{3h^{2} }} + F_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} - \frac{{20F_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} }}{{3h^{2} }} + \frac{1}{2}B_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} - \frac{{10B_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} }}{{3h^{2} }} \\ & \quad - \frac{{4D_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} }}{{h^{2} }} + \frac{1}{2}F_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} - \frac{{10F_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} }}{{3h^{2} }} + \frac{5}{4}A_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime 2}} - \frac{{5A_{3} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{{4D_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime 2}} }}{{h^{2} }} \\ & \quad - \frac{7}{2}C_{1} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} + \frac{{70C_{3} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{4}{5}D_{1} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} + \frac{{16D_{3} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} }}{{3h^{2} }} + \frac{1}{2}B_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} - \frac{{10B_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} \\ & \quad - \frac{{4D_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} }}{{h^{2} }} + \frac{1}{2}F_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} - \frac{{10F_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} + \frac{1}{2}B_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} - \frac{{10B_{3} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} \\ & \quad - \frac{{8D_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} + \frac{3}{2}F_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} + \frac{{10F_{3} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} - 2C_{1} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} + \frac{{40C_{3} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} \\ & \quad - \frac{{11}}{{10}}D_{1} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} + \frac{{22D_{3} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} + \frac{5}{4}A_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime 2}} - \frac{{5A_{3} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{{4D_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime 2}} }}{{h^{2} }} - \frac{7}{2}C_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} \\ & \quad + \frac{{70C_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{4}{5}D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} + \frac{{16D_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} - 2C_{1} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} + \frac{{40C_{3} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} \\ & \quad - \frac{{11}}{{10}}D_{1} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} + \frac{{22D_{3} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} + \frac{1}{2}A_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(3)}} - \frac{{10A_{3} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(3)}} }}{{3h^{2} }} - \frac{{4D_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(3)}} }}{{h^{2} }} \\ & \quad - 4C_{1} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} + \frac{{80C_{3} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - D_{1} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} + \frac{{20D_{3} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - 2C_{1} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} \\ & \quad + \frac{{40C_{3} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - \frac{7}{5}D_{1} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} + \frac{{28D_{3} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - \frac{3}{2}C_{1} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} + \frac{{10C_{3} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} }}{{h^{2} }} \\ & \quad - \frac{3}{5}D_{1} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} + \frac{{4D_{3} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} }}{{h^{2} }} + \frac{1}{2}A_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} - \frac{{10A_{3} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{3h^{2} }} - \frac{{4D_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{h^{2} }} \\ & \quad - 4C_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} + \frac{{80C_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{3h^{2} }} - D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} + \frac{{20D_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{3h^{2} }} - 2C_{1} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} \\ & \quad + \frac{{40C_{3} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} }}{{3h^{2} }} - \frac{7}{5}D_{1} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} + \frac{{28D_{3} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} }}{{3h^{2} }} - \frac{3}{2}C_{1} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} + \frac{{10C_{3} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} }}{{h^{2} }} \\ & \quad - \frac{3}{5}D_{1} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} + \frac{{4D_{3} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} }}{{h^{2} }} - \frac{1}{2}C_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} + \frac{{10C_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} }}{{3h^{2} }} - \frac{1}{5}D_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} \\ & \quad + \frac{{4D_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} }}{{3h^{2} }} - 2C_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + \frac{{40C_{3} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{3h^{2} }} - \frac{4}{5} D_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + \frac{{16D_{3} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{3h^{2} }} \\ & \quad - \frac{3}{{10}}D_{1} \phi _{m} \psi _{n} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + \frac{{2D_{3} \phi _{m} \psi _{n} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{h^{2} }} - \frac{1}{2}C_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} + \frac{{10C_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} }}{{3h^{2} }} - \frac{1}{5}D_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} \\ & \quad + \frac{{4D_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} }}{{3h^{2} }} - \frac{3}{{10}}D_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + \frac{{2D_{3} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{h^{2} }} - 2C_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + \frac{{40C_{3} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{3h^{2} }} \\ & \quad - \frac{4}{5}D_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + \frac{{16D_{3} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{3h^{2} }} - \frac{1}{2}C_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} + \frac{{10C_{3} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} }}{{3h^{2} }} - \frac{1}{5}D_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} \\ & \quad + \frac{{4D_{3} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} }}{{3h^{2} }} - \frac{1}{2}C_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} + \frac{{10C_{3} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} }}{{3h^{2} }} - \frac{1}{5}D_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} + \frac{{4D_{3} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} }}{{3h^{2} }}, \\ \end{aligned}$$

(C.43)

$$\begin{aligned} L_{492} & = - 3B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - \frac{3}{2}A_{0} \psi_{n}^{3} \phi_{m}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{3}{2}F_{0} \phi_{m}^{2} \psi_{n} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{3}{2}F_{0} \phi_{m} \psi_{n}^{2} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } \\ & \quad - \frac{3}{2}A_{0} \phi_{m}^{3} \psi_{n}^{\prime 2} \psi_{n}^{\prime \prime } , \\ \end{aligned}$$

(C.44)

$$\begin{aligned} L_{{493}} & = - \frac{1}{4}B_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} + \frac{{5B_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} }}{{3h^{2} }} - \frac{{8D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} }}{{3h^{2} }} + \frac{1}{2}F_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} - \frac{{10F_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime 2}} }}{{3h^{2} }} + \frac{1}{4}B_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} \\ & \quad - \frac{{5B_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} }}{{3h^{2} }} - \frac{{2D_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} }}{{h^{2} }} + \frac{1}{4}F_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} - \frac{{5F_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime }} }}{{3h^{2} }} - \frac{{2D_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime 2}} }}{{h^{2} }} - \frac{7}{4}C_{1} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} \\ & \quad + \frac{{35C_{3} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{2}{5}D_{1} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} + \frac{{8D_{3} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{\prime \prime 2}} }}{{3h^{2} }} + \frac{1}{4}B_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} - \frac{{5B_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} - \frac{{2D_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} }}{{h^{2} }} \\ & \quad + \frac{1}{4}F_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} - \frac{{5F_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} + \frac{1}{4}B_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} - \frac{{5B_{3} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} - \frac{{4D_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} \\ & \quad - \frac{1}{2}F_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} + \frac{{10F_{3} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} - C_{1} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} + \frac{{20C_{3} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} - \frac{{11}}{{20}}D_{1} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} \\ & \quad + \frac{{11D_{3} \psi _{n} \phi _{m} ^{{\prime \prime 2}} \psi _{n} ^{{\prime \prime }} }}{{3h^{2} }} - \frac{{2D_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime 2}} }}{{h^{2} }} - \frac{7}{4}C_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} + \frac{{35C_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{2}{5}D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} + \frac{{8D_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} \\ & \quad - C_{1} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} + \frac{{20C_{3} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} - \frac{{11}}{{20}}D_{1} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} + \frac{{11D_{3} \phi _{m} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{\prime \prime 2}} }}{{3h^{2} }} + \frac{1}{4}A_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(3)}} \\ & \quad - \frac{{5A_{3} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(3)}} }}{{3h^{2} }} - \frac{{2D_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(3)}} }}{{h^{2} }} - 2C_{1} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} + \frac{{40C_{3} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - \frac{1}{2}D_{1} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} \\ & \quad + \frac{{10D_{3} \phi _{m} ^{\prime } \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - C_{1} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} + \frac{{20C_{3} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - \frac{7}{{10}}D_{1} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} \\ & \quad + \frac{{14D_{3} \psi _{n} \phi _{m} ^{\prime } \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(3)}} }}{{3h^{2} }} - \frac{3}{4}C_{1} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} + \frac{{5C_{3} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} }}{{h^{2} }} - \frac{3}{{10}}D_{1} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} + \frac{{2D_{3} \psi _{n} ^{2} \phi _{m} ^{{(3)2}} }}{{h^{2} }} \\ & \quad + \frac{1}{4}A_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} - \frac{{5A_{3} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{3h^{2} }} - \frac{{2D_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{h^{2} }} - 2C_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} + \frac{{40C_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{3h^{2} }} \\ & \quad - \frac{1}{2}D_{1} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} + \frac{{10D_{3} \phi _{m} ^{{\prime 2}} \psi _{n} ^{\prime } \psi _{n} ^{{(3)}} }}{{3h^{2} }} - C_{1} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} + \frac{{20C_{3} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} }}{{3h^{2} }} \\ & \quad - \frac{7}{{10}}D_{1} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} + \frac{{14D_{3} \phi _{m} \psi _{n} ^{\prime } \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(3)}} }}{{3h^{2} }} - \frac{3}{4}C_{1} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} + \frac{{5C_{3} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} }}{{h^{2} }} - \frac{3}{{10}}D_{1} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} \\ & \quad + \frac{{2D_{3} \phi _{m} ^{2} \psi _{n} ^{{(3)2}} }}{{h^{2} }} - \frac{1}{4}C_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} + \frac{{5C_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} }}{{3h^{2} }} - \frac{1}{{10}}D_{1} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} + \frac{{2D_{3} \phi _{m} \psi _{n} ^{{\prime 2}} \phi _{m} ^{{(4)}} }}{{3h^{2} }} \\ & \quad - C_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + \frac{{20C_{3} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{3h^{2} }} - \frac{2}{5}D_{1} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} + \frac{{8D_{3} \psi _{n} ^{2} \phi _{m} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{3h^{2} }} - \frac{3}{{20}}D_{1} \phi _{m} \psi _{n} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} \\ & \quad + \frac{{D_{3} \phi _{m} \psi _{n} \psi _{n} ^{{\prime \prime }} \phi _{m} ^{{(4)}} }}{{h^{2} }} - \frac{1}{4}C_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} + \frac{{5C_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} }}{{3h^{2} }} - \frac{1}{{10}}D_{1} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} + \frac{{2D_{3} \psi _{n} \phi _{m} ^{{\prime 2}} \psi _{n} ^{{(4)}} }}{{3h^{2} }} \\ & \quad - \frac{3}{{20}}D_{1} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + \frac{{D_{3} \phi _{m} \psi _{n} \phi _{m} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{h^{2} }} - C_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} + \frac{{20C_{3} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{3h^{2} }} - \frac{2}{5}D_{1} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} \\ & \quad + \frac{{8D_{3} \phi _{m} ^{2} \psi _{n} ^{{\prime \prime }} \psi _{n} ^{{(4)}} }}{{3h^{2} }} - \frac{1}{4}C_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} + \frac{{5C_{3} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} }}{{3h^{2} }} - \frac{1}{{10}}D_{1} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} + \frac{{2D_{3} \psi _{n} ^{2} \phi _{m} ^{\prime } \phi _{m} ^{{(5)}} }}{{3h^{2} }} \\ & \quad - \frac{1}{4}C_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} + \frac{{5C_{3} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} }}{{3h^{2} }} - \frac{1}{{10}}D_{1} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} + \frac{{2D_{3} \phi _{m} ^{2} \psi _{n} ^{\prime } \psi _{n} ^{{(5)}} }}{{3h^{2} }}, \\ \end{aligned}$$

(C.45)

$$\begin{aligned} L_{494} & = - 3B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - \frac{3}{2}A_{0} \psi_{n}^{3} \phi_{m}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{3}{2}F_{0} \phi_{m}^{2} \psi_{n} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{3}{2}F_{0} \phi_{m} \psi_{n}^{2} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } \\ & \quad - \frac{3}{2}A_{0} \phi_{m}^{3} \psi_{n}^{\prime 2} \psi_{n}^{\prime \prime } , \\ \end{aligned}$$

(C.46)

$$\begin{aligned} L_{495} & = - B_{0} \phi_{m} \psi_{n} \phi_{m}^{\prime 2} \psi_{n}^{\prime 2} - \frac{1}{2}A_{0} \psi_{n}^{3} \phi_{m}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{1}{2}F_{0} \phi_{m}^{2} \psi_{n} \psi_{n}^{\prime 2} \phi_{m}^{\prime \prime } - \frac{1}{2}F_{0} \phi_{m} \psi_{n}^{2} \phi_{m}^{\prime 2} \psi_{n}^{\prime \prime } \\ & \quad - \frac{1}{2}A_{0} \phi_{m}^{3} \psi_{n}^{\prime 2} \psi_{n}^{\prime \prime } , \\ \end{aligned}$$

(C.47)

$$\begin{gathered} L_{411} = C_{d} I_{0} \phi_{m} \psi_{n} ,\;\, L_{412} = C_{d} I_{0} \phi_{m} \psi_{n} , \;\, L_{413} = - I_{2} \psi_{n} \phi_{m}^{\prime \prime } , \;\, L_{414} = - I_{2} \phi_{m} \psi_{n}^{\prime \prime } , \hfill \\ L_{415} = I_{0} \phi_{m} \psi_{n} + I_{5} \psi_{n} \phi_{m}^{\prime \prime } + I_{5} \phi_{m} \psi_{n}^{\prime \prime } , \;\, L_{416} = I_{0} \phi_{m} \psi_{n} - I_{4} \psi_{n} \phi_{m}^{\prime \prime } - I_{4} \phi_{m} \psi_{n}^{\prime \prime } . \hfill \\ \end{gathered}$$

(C.48)

Appendix D

$$Li_{11} = \int_{{S_{a} }} {L_{11} \phi_{m}^{\prime } \psi_{n} {\text{d}}S_{a} } , \, Li_{12} = \int_{{S_{a} }} {L_{12} \phi_{m}^{\prime } \psi_{n} {\text{d}}S_{a} } , \, Li_{13} = \int_{{S_{a} }} {L_{13} \phi_{m}^{\prime } \psi_{n} {\text{d}}S_{a} } , \, Li_{14} = \int_{{S_{a} }} {L_{14} \phi_{m}^{\prime } \psi_{n} {\text{d}}S_{a} } ,$$

(D.1)

$$Li_{15} = \int_{{S_{a} }} {L_{15} \phi_{m}^{\prime } \psi_{n} {\text{d}}S_{a} } , \, Li_{16} = \int_{{S_{a} }} {L_{16} \phi_{m}^{\prime } \psi_{n} {\text{d}}S_{a} } , \, Li_{17} = \int_{{S_{a} }} {L_{17} \phi_{m}^{\prime } \psi_{n} {\text{d}}S_{a} } , \, Li_{111} = \int_{{S_{a} }} {L_{111} \phi_{m}^{\prime } \psi_{n} {\text{d}}S_{a} } ,$$

(D.2)

$$Li_{112} = \int_{{S_{a} }} {L_{112} \phi_{m}^{\prime } \psi_{n} {\text{d}}S_{a} } ,Li_{113} = \int_{{S_{a} }} {L_{113} \phi_{m}^{\prime } \psi_{n} {\text{d}}S_{a} } ,$$

(D.3)

$$Li_{21} = \int_{{S_{a} }} {L_{21} \phi_{m} \psi_{n}^{\prime } {\text{d}}S_{a} } , \, Li_{22} = \int_{{S_{a} }} {L_{22} \phi_{m} \psi_{n}^{\prime } {\text{d}}S_{a} } , \, Li_{23} = \int_{{S_{a} }} {L_{23} \phi_{m} \psi_{n}^{\prime } {\text{d}}S_{a} } , \, Li_{24} = \int_{{S_{a} }} {L_{24} \phi_{m} \psi_{n}^{\prime } {\text{d}}S_{a} } ,$$

(D.4)

$$Li_{25} = \int_{{S_{a} }} {L_{25} \phi_{m} \psi_{n}^{\prime } {\text{d}}S_{a} } , \, Li_{26} = \int_{{S_{a} }} {L_{26} \phi_{m} \psi_{n}^{\prime } {\text{d}}S_{a} } , \, Li_{27} = \int_{{S_{a} }} {L_{27} \phi_{m} \psi_{n}^{\prime } {\text{d}}S_{a} } , \, Li_{211} = \int_{{S_{a} }} {L_{211} \phi_{m} \psi_{n}^{\prime } {\text{d}}S_{a} } ,$$

(D.5)

$$Li_{212} = \int_{{S_{a} }} {L_{212} \phi_{m} \psi_{n}^{\prime } {\text{d}}S_{a} } , \, Li_{213} = \int_{{S_{a} }} {L_{213} \phi_{m} \psi_{n}^{\prime } {\text{d}}S_{a} } ,$$

(D.6)

$$Li_{31} = \int_{{S_{a} }} {L_{31} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{32} = \int_{{S_{a} }} {L_{32} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{33} = \int_{{S_{a} }} {L_{33} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{34} = \int_{{S_{a} }} {L_{34} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.7)

$$Li_{35} = \int_{{S_{a} }} {L_{35} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{36} = \int_{{S_{a} }} {L_{36} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{37} = \int_{{S_{a} }} {L_{37} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{38} = \int_{{S_{a} }} {L_{38} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.8)

$$Li_{39} = \int_{{S_{a} }} {L_{39} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{390} = \int_{{S_{a} }} {L_{390} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{391} = \int_{{S_{a} }} {L_{391} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{392} = \int_{{S_{a} }} {L_{392} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.9)

$$Li_{393} = \int_{{S_{a} }} {L_{393} \phi_{m} \psi_{n} d{\text{d}}_{a} } , \, Li_{394} = \int_{{S_{a} }} {L_{394} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,Li_{395} = \int_{{S_{a} }} {L_{395} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,Li_{311} = \int_{{S_{a} }} {L_{311} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.10)

$$Li_{312} = \int_{{S_{a} }} {L_{312} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{313} = \int_{{S_{a} }} {L_{313} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{314} = \int_{{S_{a} }} {L_{314} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,Li_{315} = \int_{{S_{a} }} {L_{315} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.11)

$$Li_{316} = \int_{{S_{a} }} {L_{316} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{q} = \int_{{S_{a} }} {\phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.12)

$$Li_{41} = \int_{{S_{a} }} {L_{41} \phi_{m} \psi_{n} d{\text{d}}_{a} } , \, Li_{42} = \int_{{S_{a} }} {L_{42} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{43} = \int_{{S_{a} }} {L_{43} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{44} = \int_{{S_{a} }} {L_{44} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.13)

$$Li_{45} = \int_{{S_{a} }} {L_{45} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{46} = \int_{{S_{a} }} {L_{46} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{47} = \int_{{S_{a} }} {L_{47} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{48} = \int_{{S_{a} }} {L_{48} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.14)

$$Li_{49} = \int_{{S_{a} }} {L_{49} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{490} = \int_{{S_{a} }} {L_{490} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{491} = \int_{{S_{a} }} {L_{491} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{492} = \int_{{S_{a} }} {L_{492} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.15)

$$Li_{493} = \int_{{S_{a} }} {L_{493} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{494} = \int_{{S_{a} }} {L_{494} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,Li_{495} = \int_{{S_{a} }} {L_{495} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,Li_{411} = \int_{{S_{a} }} {L_{411} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.16)

$$Li_{412} = \int_{{S_{a} }} {L_{412} \phi_{m} \psi_{n} {\text{d}}S_{a} } , \, Li_{413} = \int_{{S_{a} }} {L_{413} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,Li_{414} = \int_{{S_{a} }} {L_{414} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,Li_{415} = \int_{{S_{a} }} {L_{415} \phi_{m} \psi_{n} {\text{d}}S_{a} } ,$$

(D.17)

$$Li_{416} = \int_{{S_{a} }} {L_{416} \phi_{m} \psi_{n} {\text{d}}S_{a} }.$$

(D.18)