Appendix
$$\begin{aligned} \rho _{11}= & {} \left( 1-\phi _{t}\right) \rho _{s}+\left( \tau _{t}-1\right) \phi _{t}\rho _{l},~\rho _{33}=\tau _{f}v_{f}\phi _{f}\rho _{l},~2\rho _{12}\\= & {} \left[ \left( \tau _{f}-1\right) v_{f}\phi _{f}-\left( \tau _{m}-1\right) v_{m}\phi _{m}-\left( \tau _{t}-1\right) \phi _{t}\right] \rho _{l},\\ 2\rho _{13}= & {} \left[ \left( \tau _{m}-1\right) v_{m}\phi _{m}-\left( \tau _{f}-1\right) v_{f}\phi _{f}-(\tau _{t}-1)\phi _{t}\right] \rho _{l},~2\rho _{23}\\= & {} \left[ \left( \tau _{t}-1\right) \phi _{t}-\left( \tau _{m}-1\right) v_{m}\phi _{m}-\left( \tau _{f}-1\right) v_{f}\phi _{f}\right] \rho _{l},\\ \rho _{22}= & {} \tau _{m}v_{m}\phi _{m}\rho _{l},~\kappa =\kappa ^{(11)}\kappa ^{(22)}-\kappa ^{(12)}\kappa ^{(21)},~\Upsilon _{12}\\= & {} \frac{\eta _{l}v_{m}\phi _{m}\left( v_{m}\phi _{m}\kappa ^{(22)}-v_{f}\phi _{f}\kappa ^{(21)}\right) }{\kappa },\\ \Upsilon _{23}= & {} \frac{\eta _{l}v_{f}v_{m}\phi _{m}\phi _{f}\kappa ^{(12)}}{\kappa },~\Upsilon _{13}\\= & {} \frac{\eta _{l}v_{f}\phi _{f}\left( v_{f}\phi _{f}\kappa ^{(11)}-v_{m}\phi _{m}\kappa ^{(12)}\right) }{\kappa },~\bar{\zeta _{0}}=k_{1}\left( \alpha _{0}-\delta \beta _{0}\right) ,\\ \zeta _{0}^{'}= & {} k_{1}(\delta \alpha _{0}+\beta _{0}),~\alpha _{0}=\sqrt{\frac{\sqrt{\alpha _{0}^{*^{2}}+\beta _{0}^{*^{2}}}+\alpha _{0}^{*}}{2}},~\beta _{0}\\= & {} \sqrt{\frac{\sqrt{\alpha _{0}^{*^{2}}+\beta _{0}^{*^{2}}}-\alpha _{0}^{*}}{2}},~\alpha _{0}^{*}=1-\frac{B^{*}c\delta }{c_{s2}k_{1}H_{1}\left( 1+\delta ^{2}\right) }\\ \beta _{0}^{*}= & {} -\frac{B^{*}c}{c_{s2}k_{1}H_{1}(1+\delta ^{2})},~B^{*}=\frac{c_{s2}\rho _{0}H_{1}}{\eta _{0}},~c_{s2}=\left[ \frac{\overline{M}\alpha }{\beta }\left( \overline{\rho _{11}}\right. \right. \\&\left. \left. +\frac{2\overline{\Upsilon _{12}}^{2}\overline{\rho _{12}}+\overline{\Upsilon _{12}}^{2}\overline{\rho _{22}} -\omega ^{2}\overline{\rho _{12}}^{2}\overline{\rho _{22}}}{s_{1}}\right. \right. \\&\left. \left. +\frac{2\overline{\Upsilon _{13}}^{2}\overline{\rho _{13}}+\overline{\Upsilon _{13}}^{2}\overline{\rho _{33}}-\omega ^{2}\overline{\rho _{13}}^{2}\overline{\rho _{33}}}{s_{2}}\right) \right] ^{\frac{1}{2}},\\ s_{1}= & {} \overline{\Upsilon _{12}}^{2}+\omega ^{2}\overline{\rho _{22}}^{2},~s_{2}=\overline{\Upsilon _{13}}^{2}+\omega ^{2}\overline{\rho _{33}}^{2},~\delta _{1}\\= & {} \frac{\omega c_{s2}^{2}\beta }{2\overline{M}\alpha }\left[ \frac{\overline{\Upsilon _{12}}\left( \overline{\rho _{12}}+\overline{\rho _{22}}\right) ^{2}}{s_{1}}\right. \\&\left. +\frac{\overline{\Upsilon _{13}}\left( \overline{\rho _{13}}+\overline{\rho _{33}}\right) ^{2}}{s_{2}}\right] ,~\bar{\zeta _{3}}=k_{1}\sqrt{1-\frac{c^{2}}{c_{3}^{2}}},~\zeta _{3}^{'}=\delta \bar{\zeta _{3}}, \overline{\rho _{d_{r}}}\\= & {} \overline{\rho _{11}}-\frac{\overline{\rho _{22}}\left( \overline{\rho _{12}}^{2}-\frac{\overline{\Upsilon _{12}}^{2}}{\omega ^{2}}\right) -\frac{2\overline{\rho _{12}}\overline{\Upsilon _{12}}^{2}}{\omega ^{2}}}{\overline{\rho _{22}}^{2}+\frac{\overline{\Upsilon _{12}}^{2}}{\omega ^{2}}}\\&-\frac{\overline{\rho _{33}}\left( \overline{\rho _{13}}^{2}-\frac{\overline{\Upsilon _{13}}^{2}}{\omega ^{2}}\right) -\frac{2\overline{\rho _{13}}\overline{\Upsilon _{13}}^{2}}{\omega ^{2}}}{\overline{\rho _{33}}^{2}+\frac{\overline{\Upsilon _{13}}^{2}}{\omega ^{2}}},~q_{1}\\= & {} \sqrt{\frac{\sqrt{\alpha _{2}^{2}+\beta _{2}^{2}}+\alpha _{2}}{2}}, \overline{\rho _{d_{i}}}\\= & {} \frac{\overline{\Upsilon _{12}}+\overline{\Upsilon _{13}}}{\omega }+\frac{\frac{\overline{\Upsilon _{12}}}{\omega }\left( \overline{\rho _{12}}^{2}-\frac{\overline{\Upsilon _{12}}^{2}}{\omega ^{2}}\right) +\frac{2\overline{\rho _{12}}\overline{\rho _{22}}\overline{\Upsilon _{12}}}{\omega }}{\overline{\rho _{22}}^{2} +\frac{\overline{\Upsilon _{12}}^{2}}{\omega ^{2}}}\\&+\frac{\frac{\overline{\Upsilon _{13}}}{\omega }\left( \overline{\rho _{13}}^{2}-\frac{\overline{\Upsilon _{13}}^{2}}{\omega ^{2}}\right) +\frac{2\overline{\rho _{13}}\overline{\rho _{33}}\overline{\Upsilon _{13}}}{\omega }}{\overline{\rho _{33}}^{2}+\frac{\overline{\Upsilon _{13}}^{2}}{\omega ^{2}}},~q_{2}=\sqrt{\frac{\sqrt{\alpha _{2}^{2}+\beta _{2}^{2}}-\alpha _{2}}{2}},\\ \alpha _{2}= & {} \frac{c^{2}}{c_{s2}^{2}}-(1-\delta ^{2}),~\beta _{2}=2\left[ \frac{\delta _{1}c^{2}}{c_{s2}^{2}}-\delta \right] ,~A=\frac{c^{2}(\alpha -\beta )}{2\alpha \beta c_{s2}^{2}\sqrt{\alpha _{2}^{2}+\beta _{2}^{2}}},~r_{1}\\= & {} \sqrt{\frac{\sqrt{\alpha _{3}^{2}+\beta _{3}^{2}}+\alpha _{3}}{2}},~r_{2}=\sqrt{\frac{\sqrt{\alpha _{3}^{2}+\beta _{3}^{2}}-\alpha _{3}}{2}},\\ \alpha _{3}= & {} (1-4\delta _{1}^{2})\alpha _{2}+4\delta _{1}\beta _{2},~\beta _{3}\\= & {} 4\delta _{1}\alpha _{2}-\left( 1-4\delta _{1}^{2}\right) \beta _{2},~D_{11}^{n}\\= & {} {\text {e}}^{k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\cos \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] ,\\ D_{12}^{n}= & {} -{\text {e}}^{k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\sin \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] ,~\bar{a}^{n}\\= & {} -2k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) ,~\bar{b}^{n}\\= & {} 2k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) ,~\bar{c}^{n}=-k_{1}Ar_{2},\\ \bar{d}^{n}= & {} k_{1}Ar_{1},~f^{n}=\bar{c}^{n}\log \sqrt{\bar{a}^{n^{2}}+\bar{b}^{n^{2}}}\\&-\bar{d}^{n}\tan ^{-1}\left( \frac{\bar{b}^{n}}{\bar{a}^{n}}\right) ,~g^{n}=\bar{d}^{n}\log \sqrt{\bar{a}^{n^{2}}+\bar{b}^{n^{2}}}+\bar{c}^{n}\tan ^{-1}\left( \frac{\bar{b}^{n}}{\bar{a}^{n}}\right) ,~D_{21}^{n}=e^{f^{n}}\cos g^{n},\\ D_{22}^{n}= & {} {\text {e}}^{f^{n}}\sin g^{n},~D_{31}^{n} =1\\&+\frac{2k_{1}Ar_{1}q_{1}(k_{1}Ar_{2}+0.5)-k_{1}^{2}A^{2}r_{1}^{2}q_{2} +q_{2}(k_{1}Ar_{2}+0.5)^{2}}{2k_{1}(q_{1}^{2}+q_{2}^{2})\left( \frac{1+\alpha H_{1}}{\alpha }\right) },\\ D_{32}^{n}= & {} \frac{-k_{1}^{2}A^{2}r_{1}^{2}q_{1}+\left( k_{1}Ar_{2}+0.5\right) ^{2}q_{1} -2k_{1}Ar_{1}q_{2}\left( k_{1}Ar_{2}+0.5\right) }{2k_{1} \left( q_{1}^{2}+q_{2}^{2}\right) \left( \frac{1+\alpha H_{1}}{\alpha }\right) },\\ B_{11}^{n}= & {} D_{31}^{n}(D_{11}^{n}D_{21}^{n}-D_{12}^{n}D_{22}^{n})-D_{32}^{n}(D_{11}^{n}D_{22}^{n}+D_{12}^{n}D_{21}^{n}),~B_{12}^{n}\\&=D_{32}^{n}\left( D_{11}^{n}D_{21}^{n}-D_{12}^{n}D_{22}^{n}\right) +D_{31}^{n}\left( D_{11}^{n}D_{22}^{n}+D_{12}^{n}D_{21}^{n}\right) ,\\ E_{11}^{n}= & {} {\text {e}}^{-k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\cos \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] ,~E_{12}^{n}\\= & {} {\text {e}}^{-k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\sin \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] ,\\ \bar{a_{1}}^{n}= & {} 2k_{1}q_{2}\left( 1+\alpha H_{1}\right) \alpha ^{-1},~\bar{b_{1}}^{n}\\= & {} -2k_{1}q_{1}\left( 1+\alpha H_{1}\right) \alpha ^{-1},~\bar{c_{1}}^{n}=k_{1}Ar_{2},~\bar{d_{1}}^{n}=-k_{1}Ar_{1}, \\ f_{1}^{n}= & {} \bar{c_{1}}^{n}\log \sqrt{\bar{a_{1}}^{n^{2}}+\bar{b_{1}}^{n^{2}}}-\bar{d_{1}}^{n}\tan ^{-1}\left( \frac{\bar{b_{1}}^{n}}{\bar{a_{1}}^{n}}\right) ,~g_{1}^{n}\\= & {} \bar{d_{1}}^{n}\log \sqrt{\bar{a_{1}}^{n^{2}}+\bar{b_{1}}^{n^{2}}}+\bar{c_{1}}^{n}\tan ^{-1}\left( \frac{\bar{b_{1}}^{n}}{\bar{a_{1}}^{n}}\right) ,~E_{21}^{n}=e^{f_{1}^{n}}\cos g_{1}^{n},\\ E_{22}^{n}= & {} {\text {e}}^{f_{1}^{n}}\sin g_{1}^{n},~E_{31}^{n}\\= & {} 1-\frac{2k_{1}Ar_{1}q_{1}(k_{1}Ar_{2}-0.5)-k_{1}^{2}A^{2}r_{1}^{2}q_{2}+q_{2}(k_{1}Ar_{2}-0.5)^{2}}{2k_{1}(q_{1}^{2}+q_{2}^{2})\left( \frac{1+\alpha H_{1}}{\alpha }\right) },\\ E_{32}^{n}= & {} \frac{k_{1}^{2}A^{2}r_{1}^{2}q_{1}-\left( k_{1}Ar_{2}-0.5\right) ^{2}q_{1} +2k_{1}Ar_{1}q_{2}\left( k_{1}Ar_{2}-0.5\right) }{2k_{1}\left( q_{1}^{2}+q_{2}^{2}\right) \left( \frac{1+\alpha H_{1}}{\alpha }\right) },\\ B_{21}^{n}= & {} E_{31}^{n}(E_{11}^{n}E_{21}^{n}-E_{12}^{n}E_{22}^{n})-E_{32}^{n}(E_{11}^{n}E_{22}^{n}+E_{12}^{n}E_{21}^{n}),~B_{22}^{n}\\= & {} E_{32}^{n}\left( E_{11}^{n}E_{21}^{n}-E_{12}^{n}E_{22}^{n}\right) +E_{31}^{n}\left( E_{11}^{n}E_{22}^{n}+E_{12}^{n}E_{21}^{n}\right) ,\\ F_{11}^{n}= & {} {\text {e}}^{\frac{k_{1}q_{2}}{\alpha }}\cos \left( \frac{k_{1}q_{1}}{\alpha }\right) ,~F_{12}^{n}\\= & {} -{\text {e}}^{\frac{k_{1}q_{2}}{\alpha }}\sin \left( \frac{k_{1}q_{1}}{\alpha }\right) ,~\bar{a_{2}}^{n}\\= & {} -2k_{1}q_{2}\alpha ^{-1},~\bar{b_{2}}^{n}=2k_{1}q_{1}\alpha ^{-1},~\bar{c_{2}}^{n}\\= & {} -k_{1}Ar_{2},~\bar{d_{2}}^{n}=k_{1}Ar_{1}, \end{aligned}$$
$$\begin{aligned} f_{2}^{n}= & {} \bar{c_{2}}^{n}\log \sqrt{\bar{a_{2}}^{n^{2}}+\bar{b_{2}}^{n^{2}}}\\&-\bar{d_{2}}^{n}\tan ^{-1}\left( \frac{\bar{b_{2}}^{n}}{\bar{a_{2}}^{n}}\right) ,~g_{2}^{n}\\= & {} \bar{d_{2}}^{n}\log \sqrt{\bar{a_{2}}^{n^{2}}+\bar{b_{2}}^{n^{2}}}+\bar{c_{2}}^{n}\tan ^{-1}\left( \frac{\bar{b_{2}}^{n}}{\bar{a_{2}}^{n}}\right) ,~F_{21}^{n}\\= & {} {\text {e}}^{f_{2}^{n}}\cos g_{2}^{n},\\ F_{22}^{n}= & {} {\text {e}}^{f_{2}^{n}}\sin g_{2}^{n},~F_{31}^{n}\\= & {} 1+\frac{2k_{1}Ar_{1}q_{1}\left( k_{1}Ar_{2}+0.5\right) -k_{1}^{2}A^{2}r_{1}^{2}q_{2}+(k_{1}Ar_{2}+0.5)^{2}q_{2}}{2k_{1}\alpha ^{-1}(q_{1}^{2}+q_{2}^{2})},\\ F_{32}^{n}= & {} \frac{-k_{1}^{2}A^{2}r_{1}^{2}q_{1} +\left( k_{1}Ar_{2}+0.5\right) ^{2}q_{1}-2k_{1}Ar_{1}q_{2}\left( k_{1}Ar_{2}+0.5\right) }{2k_{1}\alpha ^{-1}\left( q_{1}^{2}+q_{2}^{2}\right) },\\ B_{31}^{n}= & {} F_{31}^{n}\left( F_{11}^{n}F_{21}^{n}-F_{12}^{n}E_{22}^{n}\right) \\&-F_{32}^{n}\left( F_{11}^{n}F_{22}^{n}+F_{12}^{n}F_{21}^{n}\right) ,~B_{32}^{n}\\= & {} F_{32}^{n}\left( F_{11}^{n}F_{21}^{n}-F_{12}^{n}F_{22}^{n}\right) +F_{31}^{n}\left( F_{11}^{n}F_{22}^{n}+F_{12}^{n}F_{21}^{n}\right) ,\\ I_{11}^{n}= & {} {\text {e}}^{-\frac{k_{1}q_{2}}{\alpha }}\cos \left( \frac{k_{1}q_{1}}{\alpha }\right) ,~I_{12}^{n}={\text {e}}^{-\frac{k_{1}q_{2}}{\alpha }}\sin \left( \frac{k_{1}q_{1}}{\alpha }\right) ,\\ \bar{a_{3}}^{n}= & {} 2k_{1}q_{2}\alpha ^{-1},~\bar{b_{3}}^{n}\\= & {} -2k_{1}q_{1}\alpha ^{-1},~\bar{c_{3}}^{n}=k_{1}Ar_{2},~\bar{d_{3}}^{n}=-k_{1}Ar_{1},\\ f_{3}^{n}= & {} \bar{c_{3}}^{n}\log \sqrt{\bar{a_{3}}^{n^{2}}+\bar{b_{3}}^{n^{2}}}\\&-\bar{d_{3}}^{n}\tan ^{-1}\left(
\frac{\bar{b_{3}}^{n}}{\bar{a_{3}}^{n}}\right) ,~g_{3}^{n}\\= & {} \bar{d_{3}}^{n}\log \sqrt{\bar{a_{3}}^{n^{2}}+\bar{b_{3}}^{n^{2}}}+\bar{c_{3}^{n}}\tan ^{-1}\left( \frac{\bar{b_{3}}^{n}}{\bar{a_{3}}^{n}}\right) ,~I_{21}^{n}\\= & {} {\text {e}}^{f_{3}^{n}}\cos g_{3}^{n},\\ I_{22}^{n}= & {} {\text {e}}^{f_{3}^{n}}\sin g_{3}^{n},~I_{31}^{n}\\= & {} 1-\frac{2k_{1}Ar_{1}q_{1}\left( k_{1}Ar_{2}-0.5\right) -k_{1}^{2}A^{2}r_{1}^{2}q_{2} +(k_{1}Ar_{2}-0.5)^{2}q_{2}}{2k_{1}\alpha ^{-1}\left( q_{1}^{2} +q_{2}^{2}\right) },\\ I_{32}^{n}= & {} \frac{k_{1}^{2}A^{2}r_{1}^{2}q_{1}-\left( k_{1}Ar_{2}-0.5\right) ^{2}q_{1} +2k_{1}Ar_{1}q_{2}\left( k_{1}Ar_{2}-0.5\right) }{2k_{1}\alpha ^{-1}\left( q_{1}^{2}+q_{2}^{2}\right) },\\ B_{41}^{n}= & {} I_{31}^{n}(I_{11}^{n}I_{21}^{n}-I_{12}^{n}I_{22}^{n})-I_{32}^{n}(I_{11}^{n}I_{22}^{n}+I_{12}^{n}I_{21}^{n}),~B_{42}^{n}\\= & {} I_{32}^{n}\left( I_{11}^{n}I_{21}^{n}-I_{12}^{n}I_{22}^{n}\right) \\&+I_{31}^{n}\left( I_{11}^{n}I_{22}^{n}+I_{12}^{n}I_{21}^{n}\right) ,\\ \bar{a_{11}}^{n}= & {} D_{11}^{n},~a_{11}^{'^{n}}=D_{12}^{n},~\bar{a_{4}}^{n}\\= & {} \bar{a}^{n},~\bar{b_{4}}^{n}=\bar{b}^{n},~\bar{c_{4}}^{n}=\bar{c}^{n},~\bar{d_{4}}^{n}\\= & {} \bar{d}^{n},~f_{4}^{n}=f^{n},~g_{4}^{n}=g^{n},~\bar{a_{12}}^{n}=D_{21}^{n},~a_{12}^{'^{n}}=D_{22}^{n},\\ a_{12}^{'^{n}}= & {} D_{22}^{n},~\bar{a_{13}}^{n}\\= & {} \frac{-2k_{1}Ar_{1}q_{1}\left( k_{1}Ar_{2}+0.5\right) +k_{1}^{2}A^{2}r_{1}^{2}q_{2}-q_{2}(k_{1}Ar_{2}+0.5)^{2}}{2k_{1}(q_{1}^{2}+q_{2}^{2})\left( \frac{1+\alpha H_{1}}{\alpha }\right) ^{2}},\\ a_{13}^{'^{n}}= & {} \frac{k_{1}^{2}A^{2}r_{1}^{2}q_{1}-(k_{1}Ar_{2}+0.5)^{2}q_{1}+2k_{1}Ar_{1}q_{2}(k_{1}Ar_{2}+0.5)}{2k_{1}(q_{1}^{2}+q_{2}^{2})\left( \frac{1+\alpha H_{1}}{\alpha }\right) ^{2}}\\ J_{11}^{n}= & {} \bar{a_{13}}^{n}\left( \bar{a_{11}}^{n}\bar{a_{12}}^{n}-a_{11}^{'^{n}}a_{12}^{'^{n}}\right) \\&-a_{13}^{'^{n}}\left( \bar{a_{11}}^{n}a_{12}^{'^{n}}+a_{11}^{'^{n}}\bar{a_{12}}^{n}\right) ,~J_{12}^{n}\\= & {} a_{13}^{'^{n}}\left( \bar{a_{11}}^{n}\bar{a_{12}}^{n}-a_{11}^{'^{n}}a_{12}^{'^{n}}\right) \\&+\bar{a_{13}}^{n}\left( \bar{a_{11}}^{n}a_{12}^{'^{n}}+a_{11}^{'^{n}}\bar{a_{12}}^{n}\right) ,\\ \bar{b_{12}}^{n}= & {} D_{31}^{n},~b_{12}^{'^{n}}=D_{32}^{n},~f_{5}^{n}\\= & {} -(k_{1}Ar_{2}+1)log\left( 1+\alpha H_{1}\right) \\&-k_{1}Ar_{2}log\left( \sqrt{4k_{1}^{2}\alpha ^{-2}(q_{1}^{2}+q_{2}^{2})}\right) \\&-k_{1}Ar_{1}\tan ^{-1}\left( -\frac{q_{1}}{q_{2}}\right) ,\\ g_{5}^{n}= & {} k_{1}Ar_{1}log\left( 1+\alpha H_{1}\right) \\&+k_{1}Ar_{1}log\left( \sqrt{4k_{1}^{2}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) }\right) \\&-k_{1}Ar_{2}\tan ^{-1}\left( -\frac{q_{1}}{q_{2}}\right) ,~\bar{\gamma _{13}}^{n}\\= & {} {\text {e}}^{f_{5}^{n}}\cos g_{5}^{n},~\gamma _{13}^{'^{n}}={\text {e}}^{f_{5}^{n}}\sin g_{5}^{n},\\ \bar{b_{13}}^{n}= & {} -k_{1}A\alpha (r_{2}\bar{\gamma _{13}}^{n}+r_{1}\gamma _{13}^{'^{n}}),~b_{13}^{'^{n}}=k_{1}A\alpha (r_{1}\bar{\gamma _{13}}^{n}-r_{2}\gamma _{13}^{'^{n}}),~J_{21}^{n}\\= & {} \bar{b_{13}}^{n}\left( \bar{a_{11}}^{n}\bar{b_{12}}^{n}-a_{11}^{'^{n}}b_{12}^{'^{n}}\right) -b_{13}^{'^{n}}\left( \bar{a_{11}}^{n}b_{12}^{'^{n}}+a_{11}^{'^{n}}\bar{b_{12}}^{n}\right) ,\\ J_{22}^{n}= & {} b_{13}^{'^{n}}(\bar{a_{11}}^{n}\bar{b_{12}}^{n}-a_{11}^{'^{n}}b_{12}^{'^{n}})+\bar{b_{13}}^{n}(\bar{a_{11}}^{n}b_{12}^{'^{n}}+a_{11}^{'^{n}}\bar{b_{12}}^{n}),\\ \bar{c_{11}}^{n}= & {} k_{1}q_{1}{\text {e}}^{k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\sin \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] \\&-k_{1}q_{2}{\text {e}}^{k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\cos \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] ,\\ c_{11}^{'^{n}}= & {} k_{1}q_{1}{\text {e}}^{k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\cos \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] \\&+k_{1}q_{2}{\text {e}}^{k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\sin \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] ,\\ J_{31}^{n}= & {} \bar{b_{12}}^{n}(\bar{c_{11}}^{n}\bar{a_{12}}^{n}-c_{11}^{'^{n}}a_{12}^{'^{n}})-b_{12}^{'^{n}}(\bar{c_{11}}^{n}a_{12}^{'^{n}}+c_{11}^{'^{n}}\bar{a_{12}}^{n}),~J_{32}^{n}\\= & {} b_{12}^{'^{n}}\left( \bar{c_{11}}^{n}\bar{a_{12}}^{n}-c_{11}^{'^{n}}a_{12}^{'^{n}}\right) \\&+\bar{b_{12}}^{n}\left( \bar{c_{11}}^{n}a_{12}^{'^{n}}+c_{11}^{'^{n}}\bar{a_{12}}^{n}\right) ,\\ C_{11}^{n}= & {} J_{11}^{n}+J_{21}^{n}-J_{31}^{n},~C_{12}^{n}\\= & {} J_{12}^{n}+J_{22}^{n}-J_{32}^{n},~\bar{d_{11}}^{n}\\= & {} E_{11}^{n},~d_{11}^{'^{n}}=E_{12}^{n},~\bar{a_{5}}^{n}\\= & {} \bar{a_{1}}^{n},~\bar{b_{5}}^{n}=\bar{b_{1}}^{n},~\bar{c_{5}}^{n}=\bar{c_{1}}^{n},~\bar{d_{5}}^{n}=\bar{d_{1}}^{n},\\ f_{6}^{n}= & {} f_{1}^{n},~g_{6}^{n}=g_{1}^{n},~\bar{d_{12}}^{n}\\= & {} E_{21}^{n},~d_{12}^{'^{n}}=E_{22}^{n},~\bar{d_{13}}^{n}\\= & {} \frac{2k_{1}Ar_{1}q_{1}\left( k_{1}Ar_{2}-0.5\right) -k_{1}^{2}A^{2}r_{1}^{2}q_{2}+q_{2}\left( k_{1}Ar_{2}-0.5\right) ^{2}}{2k_{1}\left( q_{1}^{2}+q_{2}^{2}\right) \left( \frac{1+\alpha H_{1}}{\alpha }\right) ^{2}},\\ d_{13}^{'^{n}}= & {} \frac{-k_{1}^{2}A^{2}r_{1}^{2}q_{1}+\left( k_{1}Ar_{2}-0.5\right) ^{2}q_{1}-2k_{1}Ar_{1}q_{2}(k_{1}Ar_{2}-0.5)}{2k_{1}(q_{1}^{2}+q_{2}^{2})\left( \frac{1+\alpha H_{1}}{\alpha }\right) ^{2}},\\ L_{11}^{n}= & {} \bar{d_{13}}^{n}\left( \bar{d_{11}}^{n}\bar{d_{12}}^{n}-d_{11}^{'^{n}}d_{12}^{'^{n}}\right) \\&-d_{13}^{'^{n}}\left( \bar{d_{11}}^{n}d_{12}^{'^{n}}+d_{11}^{'^{n}}\bar{d_{12}}^{n}\right) ,~L_{12}^{n}\\= & {} d_{13}^{'^{n}}\left( \bar{d_{11}}^{n}\bar{d_{12}}^{n}-d_{11}^{'^{n}}d_{12}^{'^{n}}\right) +\bar{d_{13}}^{n}\left( \bar{d_{11}}^{n}d_{12}^{'^{n}}+d_{11}^{'^{n}}\bar{d_{12}}^{n}\right) ,\\ \bar{e_{12}}^{n}= & {} E_{31}^{n},~e_{12}^{'^{n}}=E_{32}^{n},~f_{7}^{n}\\= & {} \left( k_{1}Ar_{2}-1\right) log\left( 1+\alpha H_{1}\right) \\&+k_{1}Ar_{2}log\left( \sqrt{4k_{1}^{2}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) }\right) +k_{1}Ar_{1}\tan ^{-1}\left( -\frac{q_{1}}{q_{2}}\right) ,\\ g_{7}^{n}= & {} -k_{1}Ar_{1}log\left( 1+\alpha H_{1}\right) -k_{1}Ar_{1}log\left( \sqrt{4k_{1}^{2}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) }\right) \\&+k_{1}Ar_{2}\tan ^{-1}\left( -\frac{q_{1}}{q_{2}}\right) ,~\bar{\delta _{13}}^{n}\\= & {} {\text {e}}^{f_{7}^{n}}\cos g_{7}^{n},~\delta _{13}^{'^{n}}={\text {e}}^{f_{7}^{n}}\sin g_{7}^{n},\\ \bar{e_{13}}^{n}= & {} k_{1}A\alpha (r_{2}\bar{\delta _{13}}^{n}+r_{1}\delta _{13}^{'^{n}}),~e_{13}^{'^{n}}\\= & {} k_{1}A\alpha \left( -r_{1}\bar{\delta _{13}}^{n}+r_{2}\delta _{13}^{'^{n}}\right) ,~L_{21}^{n}\\= & {} \bar{e_{13}}^{n}\left( \bar{d_{11}}^{n}\bar{e_{12}}^{n}-d_{11}^{'^{n}}e_{12}^{'^{n}}\right) \\&-e_{13}^{'^{n}}\left( \bar{d_{11}}^{n}e_{12}^{'^{n}}+d_{11}^{'^{n}}\bar{e_{12}}^{n}\right) ,\\ L_{22}^{n}= & {} e_{13}^{'^{n}}\left( \bar{d_{11}}^{n}\bar{e_{12}}^{n}-d_{11}^{'^{n}}e_{12}^{'^{n}}\right) \\&+\bar{e_{13}}^{n}\left( \bar{d_{11}}^{n}e_{12}^{'^{n}}+d_{11}^{'^{n}}\bar{e_{12}}^{n}\right) ,\\ \bar{f_{11}}^{n}= & {} k_{1}q_{1}{\text {e}}^{-k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\sin \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] \\&+k_{1}q_{2}{\text {e}}^{-k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\cos \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] ,\\ f_{11}^{'^{n}}= & {} k_{1}q_{2}{\text {e}}^{-k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\sin \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right] \\&-k_{1}q_{1}{\text {e}}^{-k_{1}q_{2}\left( \frac{1+\alpha H_{1}}{\alpha }\right) }\cos \left[ k_{1}q_{1}\left( \frac{1+\alpha H_{1}}{\alpha }\right) \right],\\L_{31}^{n}= & {} \bar{e_{12}}^{n}\left( \bar{f_{11}}^{n}\bar{d_{12}}^{n}-f_{11}^{'^{n}}d_{12}^{'^{n}}\right) \\&-e_{12}^{'^{n}}\left( \bar{f_{11}}^{n}d_{12}^{'^{n}}+f_{11}^{'^{n}}\bar{d_{12}}^{n}\right) ,~L_{32}^{n}\\= & {} e_{12}^{'^{n}}\left( \bar{f_{11}}^{n}\bar{d_{12}}^{n}-f_{11}^{'^{n}}d_{12}^{'^{n}}\right) \\&+\bar{e_{12}}^{n}\left( \bar{f_{11}}^{n}d_{12}^{'^{n}}+f_{11}^{'^{n}}\bar{d_{12}}^{n}\right) , \end{aligned}$$
$$\begin{aligned} C_{21}^{n}= & {} L_{11}^{n}+L_{21}^{n}-L_{31}^{n},~C_{22}^{n}\\= & {} L_{12}^{n}+L_{22}^{n}-L_{32}^{n},~\bar{g_{11}}^{n}=F_{11}^{n},~g_{11}^{'^{n}}\\= & {} F_{12}^{n},~\bar{a_{6}}^{n}=\bar{a_{2}}^{n},~\bar{b_{6}}^{n}\\= & {} \bar{b_{2}}^{n},~\bar{c_{6}}^{n}=\bar{c_{2}}^{n},~\bar{d_{6}}^{n}=\bar{d_{2}}^{n},\\ f_{8}^{n}= & {} f_{2}^{n},~g_{8}^{n}=g_{2}^{n},~\bar{g_{12}}^{n}\\= & {} F_{21}^{n},~g_{12}^{'^{n}}=F_{22}^{n},~\bar{g_{13}}^{n}\\= & {} \frac{-2k_{1}Ar_{1}q_{1}\left( k_{1}Ar_{2}+0.5\right) +k_{1}^{2}A^{2}r_{1}^{2}q_{2} -\left( k_{1}Ar_{2}+0.5\right) ^{2}q_{2}}{2k_{1}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) },\\ g_{13}^{'^{n}}= & {} \frac{k_{1}^{2}A^{2}r_{1}^{2}q_{1} -\left( k_{1}Ar_{2}+0.5\right) ^{2}q_{1}+2k_{1}Ar_{1}q_{2}\left( k_{1}Ar_{2}+0.5\right) }{2k_{1}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) },\\ M_{11}^{n}= & {} \bar{g_{13}}^{n}\left( \bar{g_{11}}^{n}\bar{g_{12}}^{n}-g_{11}^{'^{n}}g_{12}^{'^{n}}\right) \\&-g_{13}^{'^{n}}\left( \bar{g_{11}}^{n}g_{12}^{'^{n}}+g_{11}^{'^{n}}\bar{g_{12}}^{n}\right) ,~M_{12}^{n}\\= & {} g_{13}^{'^{n}}\left( \bar{g_{11}}^{n}\bar{g_{12}}^{n}-g_{11}^{'^{n}}g_{12}^{'^{n}}\right) \\&+\bar{g_{13}}^{n}\left( \bar{g_{11}}^{n}g_{12}^{'^{n}}+g_{11}^{'^{n}}\bar{g_{12}}^{n}\right) ,\\ \bar{h_{12}}^{n}= & {} 1-\frac{2k_{1}Ar_{1}q_{1}\left( k_{1}Ar_{2}+0.5\right) -k_{1}^{2}A^{2}r_{1}^{2}q_{2}+\left( k_{1}Ar_{2}+0.5\right) ^{2}q_{2}}{2k_{1}\alpha ^{-1}\left( q_{1}^{2}+q_{2}^{2}\right) },\\ h_{12}^{'^{n}}= & {} \frac{-k_{1}^{2}A^{2}r_{1}^{2}q_{1} +\left( k_{1}Ar_{2}+0.5\right) ^{2}q_{1}-2k_{1}Ar_{1}q_{2}\left( k_{1}Ar_{2}+0.5\right) }{2k_{1}\alpha ^{-1}\left( q_{1}^{2}+q_{2}^{2}\right) },\\ f_{9}^{n}= & {} -k_{1}Ar_{2}log\left( \sqrt{4k_{1}^{2}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) }\right) \\&-k_{1}Ar_{1}\tan ^{-1}\left( -\frac{q_{1}}{q_{2}}\right) ,~g_{9}^{n}\\= & {} k_{1}Ar_{1}log\left( \sqrt{4k_{1}^{2}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) }\right) \\&-k_{1}Ar_{2}\tan ^{-1}\left( -\frac{q_{1}}{q_{2}}\right) ,\\ \bar{\eta _{13}}^{n}= & {} {\text {e}}^{f_{9}^{n}}\cos g_{9}^{n},~\eta _{13}^{'^{n}}\\= & {} {\text {e}}^{f_{9}^{n}}\sin g_{9}^{n},~\bar{h_{13}}^{n}\\= & {} -k_{1}A\alpha \left( r_{2}\bar{\eta _{13}}^{n}+r_{1}\eta _{13}^{'^{n}}\right) ,~h_{13}^{'^{n}}\\= & {} k_{1}A\alpha \left( r_{1}\bar{\eta _{13}}^{n}-r_{2}\eta _{13}^{'^{n}}\right) ,\\ M_{21}^{n}= & {} \bar{h_{13}}^{n}\left( \bar{g_{11}}^{n}\bar{h_{12}}^{n}-g_{11}^{'^{n}}h_{12}^{'^{n}}\right) \\&-h_{13}^{'^{n}}\left( \bar{g_{11}}^{n}h_{12}^{'^{n}}+g_{11}^{'^{n}}\bar{h_{12}}^{n}\right) ,~M_{22}^{n}\\= & {} h_{13}^{'^{n}}\left( \bar{g_{11}}^{n}\bar{h_{12}}^{n}-g_{11}^{'^{n}}h_{12}^{'^{n}}\right) \\&+\bar{h_{13}}^{n}\left( \bar{g_{11}}^{n}h_{12}^{'^{n}}+g_{11}^{'^{n}}\bar{h_{12}}^{n}\right) ,\\ \bar{j_{11}}^{n}= & {} -k_{1}q_{2}{\text {e}}^{k_{1}q_{2}\alpha ^{-1}}\cos \left( \frac{k_{1}q_{1}}{\alpha }\right) \\&+k_{1}q_{1}{\text {e}}^{k_{1}q_{2}\alpha ^{-1}}\sin \left( \frac{k_{1}q_{1}}{\alpha }\right) ,~j_{11}^{'^{n}}\\= & {} k_{1}q_{1}{\text {e}}^{k_{1}q_{2}\alpha ^{-1}}\cos \left( \frac{k_{1}q_{1}}{\alpha }\right) \\&+k_{1}q_{2}{\text {e}}^{k_{1}q_{2}\alpha ^{-1}}\sin \left( \frac{k_{1}q_{1}}{\alpha }\right) ,\\ M_{31}^{n}= & {} \bar{h_{12}}^{n}\left( \bar{j_{11}}^{n}\bar{g_{12}}^{n}-j_{11}^{'^{n}}g_{12}^{'^{n}}\right) \\&-h_{12}^{'^{n}}\left( \bar{j_{11}}^{n}g_{12}^{'^{n}}+j_{11}^{'^{n}}\bar{g_{12}}^{n}\right) ,~M_{32}^{n}\\= & {} h_{13}^{'^{n}}\left( \bar{j_{11}}^{n}\bar{g_{12}}^{n}-j_{11}^{'^{n}}g_{12}^{'^{n}}\right) \\&+\bar{h_{12}}^{n}\left( \bar{j_{11}}^{n}g_{12}^{'^{n}}+j_{11}^{'^{n}}\bar{g_{12}}^{n}\right) ,~~~~\\ C_{31}^{n}= & {} M_{11}^{n}+M_{21}^{n}-M_{31}^{n},~C_{32}^{n}\\= & {} M_{12}^{n}+M_{22}^{n}-M_{32}^{n},~\bar{l_{11}}^{n}\\= & {} I_{11}^{n},~l_{11}^{'^{n}}=I_{12}^{n},~\bar{a_{7}}^{n}=\bar{a_{3}}^{n},~\bar{b_{7}}^{n}\\= & {} \bar{b_{3}}^{n},~\bar{c_{7}}^{n}=\bar{c_{3}}^{n},~~~~~~~~~~~\\ \bar{d_{7}}^{n}= & {} \bar{d_{3}}^{n},~f_{10}^{n}=f_{3}^{n},~g_{10}^{n}\\= & {} g_{3}^{n},~\bar{l_{12}}^{n}=I_{21}^{n},~l_{12}^{'^{n}}=I_{22}^{n},~\bar{l_{13}}^{n}\\= & {} \frac{2k_{1}Ar_{1}q_{1}\left( k_{1}Ar_{2}-0.5\right) -k_{1}^{2}A^{2}r_{1}^{2}q_{2} +\left( k_{1}Ar_{2}-0.5\right) ^{2}q_{2}}{2k_{1}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) },\\ l_{13}^{'^{n}}= & {} \frac{-k_{1}^{2}A^{2}r_{1}^{2}q_{1} +\left( k_{1}Ar_{2}-0.5\right) ^{2}q_{1}-2k_{1}Ar_{1}q_{2}\left( k_{1}Ar_{2}-0.5\right) }{2k_{1}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) },~~~~~~~~~~~~~~~~~~\\ N_{11}^{n}= & {} \bar{l_{13}}^{n}\left( \bar{l_{11}}^{n}\bar{l_{12}}^{n}-l_{11}^{'^{n}}l_{12}^{'^{n}}\right) -l_{13}^{'^{n}}(\bar{l_{11}}^{n}l_{12}^{'^{n}}+l_{11}^{'^{n}}\bar{l_{12}}^{n}),~N_{12}^{n}\\= & {} l_{13}^{'^{n}}\left( \bar{l_{11}}^{n}\bar{l_{12}}^{n}-l_{11}^{'^{n}}l_{12}^{'^{n}}\right) \\&+\bar{l_{13}}^{n}\left( \bar{l_{11}}^{n}l_{12}^{'^{n}}+l_{11}^{'^{n}}\bar{l_{12}}^{n}\right) ,~~~~~~~~~~~~~~~~\\ \bar{m_{12}}^{n}= & {} I_{31}^{n},~m_{12}^{'^{n}}=I_{32}^{n},~f_{0}^{n}\\= & {} k_{1}Ar_{2}log\left( \sqrt{4k_{1}^{2}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) }\right) \\&+k_{1}Ar_{1}\tan ^{-1}\left( -\frac{q_{1}}{q_{2}}\right) ,\\ g_{0}^{n}= & {} -k_{1}Ar_{1}log\left( \sqrt{4k_{1}^{2}\alpha ^{-2}\left( q_{1}^{2}+q_{2}^{2}\right) }\right) \\&+k_{1}Ar_{2}\tan ^{-1}\left( -\frac{q_{1}}{q_{2}}\right) ,\\ \bar{\xi _{13}}^{n}= & {} {\text {e}}^{f_{0}^{n}}\cos g_{0}^{n},~\xi _{13}^{'^{n}}\\= & {} {\text {e}}^{f_{0}^{n}}\sin g_{0}^{n},~\bar{m_{13}}^{n}=k_{1}A\alpha (r_{2}\bar{\xi _{13}}^{n}+r_{1}\xi _{13}^{'^{n}}),~m_{13}^{'^{n}}\\= & {} k_{1}A\alpha \left( -r_{1}\bar{\xi _{13}}^{n}+r_{2}\xi _{13}^{'^{n}}\right) ,\\ N_{21}^{n}= & {} \bar{m_{13}}^{n}\left( \bar{l_{11}}^{n}\bar{m_{12}}^{n}-l_{11}^{'^{n}}m_{12}^{'^{n}}\right) \\&-m_{13}^{'^{n}}(\bar{l_{11}}^{n}m_{12}^{'^{n}}+l_{11}^{'^{n}}\bar{m_{12}}^{n}),~N_{22}^{n}\\= & {} m_{13}^{'^{n}}\left( \bar{l_{11}}^{n}\bar{m_{12}}^{n}-l_{11}^{'^{n}}m_{12}^{'^{n}}\right) \\&+\bar{m_{13}}^{n}\left( \bar{l_{11}}^{n}m_{12}^{'^{n}}+l_{11}^{'^{n}}\bar{m_{12}}^{n}\right) ,\\ \bar{n_{11}}^{n}= & {} k_{1}q_{2}{\text {e}}^{-k_{1}q_{2}\alpha ^{-1}}\cos \left( \frac{k_{1}q_{1}}{\alpha }\right) \\&+k_{1}q_{1}{\text {e}}^{-k_{1}q_{2}\alpha ^{-1}}\sin \left( \frac{k_{1}q_{1}}{\alpha }\right) ,~~~~~~~~~~~~~~~~~\\ n_{11}^{'^{n}}= & {} -k_{1}q_{1}{\text {e}}^{-k_{1}q_{2}\alpha ^{-1}}\cos \left( \frac{k_{1}q_{1}}{\alpha }\right) \\&+k_{1}q_{2}{\text {e}}^{-k_{1}q_{2}\alpha ^{-1}}\sin \left( \frac{k_{1}q_{1}}{\alpha }\right) ,~~~~~~~~~~~~~~~~\\ N_{31}^{n}= & {} \bar{m_{12}}^{n}(\bar{n_{11}}^{n}\bar{l_{12}}^{n}-n_{11}^{'^{n}}l_{12}^{'^{n}})\\&-m_{12}^{'^{n}}(\bar{n_{11}}^{n}l_{12}^{'^{n}}+n_{11}^{'^{n}}\bar{l_{12}}^{n}),~N_{32}^{n}\\= & {} m_{13}^{'^{n}}(\bar{n_{11}}^{n}\bar{l_{12}}^{n}-n_{11}^{'^{n}}l_{12}^{'^{n}})\\&+\bar{m_{12}}^{n}(\bar{n_{11}}^{n}l_{12}^{'^{n}}+n_{11}^{'^{n}}\bar{l_{12}}^{n}),~~\\ C_{41}^{n}= & {} N_{11}^{n}+N_{21}^{n}-N_{31}^{n},~C_{42}^{n}=N_{12}^{n}+N_{22}^{n}-N_{32}^{n},\\ T_{11}^{n}= & {} \sqrt{\overline{M}}\left[ \alpha \left( B_{31}^{n}B_{41}^{n}-B_{32}^{n}B_{42}^{n}\right) \right. \\&\left. -\left( C_{31}^{n}B_{41}^{n}-C_{32}^{n}B_{42}^{n}\right) -\left( C_{41}^{n}B_{31}^{n}-C_{42}^{n}B_{32}^{n}\right) \right] ,\\ T_{12}^{n}= & {} \sqrt{\overline{M}}\left[ \alpha \left( B_{32}^{n}B_{41}^{n}+B_{31}^{n}B_{42}^{n}\right) \right. \\&\left. -\left( C_{32}^{n}B_{41}^{n}+C_{31}^{n}B_{42}^{n}\right) -\left( C_{42}^{n}B_{31}^{n}+C_{41}^{n}B_{32}^{n}\right) \right] ,~\\ T_{13}^{n}= & {} \left( -\zeta _{0}^{'}-\delta \bar{\zeta _{0}}\right) \left( B_{31}^{n}B_{41}^{n}-B_{32}^{n}B_{42}^{n}\right) \\&-\left( \bar{\zeta _{0}}-\delta \zeta _{0}^{'}\right) \left( B_{32}^{n}B_{41}^{n}+B_{31}^{n}B_{42}^{n}\right) ,\\ T_{14}^{n}= & {} \left( \bar{\zeta _{0}}-\delta \zeta _{0}^{'}\right) \left( B_{31}^{n}B_{41}^{n}-B_{32}^{n}B_{42}^{n}\right) \\&+\left( -\zeta _{0}^{'}-\delta \bar{\zeta _{0}}\right) \left( B_{32}^{n}B_{41}^{n}+B_{31}^{n}B_{42}^{n}\right) ,\\ T_{15}^{n}= & {} \frac{\tanh \left( \bar{\zeta _{0}}H_{2}\right) +\tan ^{2}\left( \zeta _{0}^{'}H_{2}\right) \tanh \left( \bar{\zeta _{0}}H_{2}\right) }{1+\tanh ^{2}\left( \bar{\zeta _{0}}H_{2}\right) \tan ^{2}\left( \zeta _{0}^{'}H_{2}\right) },~T_{16}^{n}\\= & {} \frac{\tan (\zeta _{0}^{'}H_{2})-\tan ^{2}\left( \zeta _{0}^{'}H_{2}\right) \tanh ^{2}\left( \bar{\zeta _{0}}H_{2}\right) }{1+\tanh ^{2}\left( \bar{\zeta _{0}}H_{2}\right) \tan ^{2}\left( \zeta _{0}^{'}H_{2}\right) },\\ T_{17}^{n}= & {} \frac{2\eta _{0}\omega (1+\gamma H_{2})}{\sqrt{\overline{M}}}(T_{13}^{n}T_{15}^{n}-T_{14}^{n}T_{16}^{n}),~T_{18}^{n}\\= & {} \frac{2\eta _{0}\omega (1+\gamma H_{2})}{\sqrt{\overline{M}}}(T_{14}^{n}T_{15}^{n}\\&+T_{13}^{n}T_{16}^{n}),~T_{19}^{n}=T_{11}^{n}-T_{17}^{n},~T_{20}^{n} =T_{12}^{n}-T_{18}^{n},~\\ T_{21}^{n}= & {} \sqrt{\overline{M}}\left( C_{41}^{n}B_{31}^{n}-C_{42}^{n}B_{32}^{n}\right. \\&\left. -C_{31}^{n}B_{41}^{n}+C_{32}^{n}B_{42}^{n}\right) ,~T_{22}^{n}\\= & {} \sqrt{\overline{M}}\left( C_{42}^{n}B_{31}^{n}+C_{41}^{n}B_{32}^{n}\right. \\&\left. -C_{32}^{n}B_{41}^{n}-C_{31}^{n}B_{42}^{n}\right) ,\\ T_{23}^{n}= & {} \overline{M}\left( 1+\alpha H_{1}\right) \left( B_{41}^{n}C_{11}^{n}-B_{42}^{n}C_{12}^{n}+B_{31}^{n}C_{21}^{n}-B_{32}^{n}C_{22}^{n}\right) \\&+\left( \mu _{3}\bar{\zeta _{3}}-0.5\alpha \overline{M}\right) \left( B_{11}^{n}B_{41}^{n}-B_{12}^{n}B_{42}^{n}\right. \\&\left. +B_{21}^{n}B_{31}^{n}-B_{22}^{n}B_{32}^{n}\right) ~\\&-\mu _{3}\zeta _{3}^{'}\left( B_{11}^{n}B_{42}^{n}+B_{12}^{n}B_{41}^{n}+B_{21}^{n}B_{32}^{n}+B_{22}^{n}B_{31}^{n}\right) ,\\ T_{24}^{n}= & {} \overline{M}(1+\alpha H_{1})\left( B_{42}^{n}C_{11}^{n}+B_{41}^{n}C_{12}^{n}\right. \\&\left. +B_{31}^{n}C_{22}^{n}+B_{32}^{n}C_{21}^{n}\right) \\&+\left( \mu _{3}\bar{\zeta _{3}}-0.5\alpha \overline{M}\right) \left( B_{11}^{n}B_{42}^{n}\right. \\&\left. +B_{12}^{n}B_{41}^{n}+B_{21}^{n}B_{32}^{n}+B_{22}^{n}B_{31}^{n}\right) ~\\&+\mu _{3}\zeta _{3}^{'}(B_{11}^{n}B_{41}^{n}-B_{12}^{n}B_{42}^{n}+B_{21}^{n}B_{31}^{n}-B_{22}^{n}B_{32}^{n}),\\ T_{25}^{n}= & {} \overline{M}(1+\alpha H_{1})\left( B_{41}^{n}C_{11}^{n}-B_{42}^{n}C_{12}^{n}-B_{31}^{n}C_{21}^{n}+B_{32}^{n}C_{22}^{n}\right) \\&+\left( \mu _{3}\bar{\zeta _{3}}-0.5\alpha
\overline{M}\right) \left( B_{11}^{n}B_{41}^{n}-B_{12}^{n}B_{42}^{n}-B_{21}^{n}B_{31}^{n}+B_{22}^{n}B_{32}^{n}\right) ~\\&-\mu _{3}\zeta _{3}^{'}(B_{11}^{n}B_{42}^{n}+B_{12}^{n}B_{41}^{n}\\&-B_{21}^{n}B_{32}^{n}-B_{22}^{n}B_{31}^{n}),~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ T_{26}^{n}= & {} \overline{M}(1+\alpha H_{1})(B_{42}^{n}C_{11}^{n}+B_{41}^{n}C_{12}^{n}\\&-B_{31}^{n}C_{22}^{n}-B_{32}^{n}C_{21}^{n})+(\mu _{3}\bar{\zeta _{3}}\\&-0.5\alpha \overline{M})(B_{11}^{n}B_{42}^{n}+B_{12}^{n}B_{41}^{n}-B_{21}^{n}B_{32}^{n}-B_{22}^{n}B_{31}^{n})~\\&+\mu _{3}\zeta _{3}^{'}\left( B_{11}^{n}B_{41}^{n}-B_{12}^{n}B_{42}^{n}-B_{21}^{n}B_{31}^{n}+B_{22}^{n}B_{32}^{n}\right) ,\\ n= & {} 0,1,2 \end{aligned}$$
