Appendix A
$$\begin{aligned} a^{11}_{mn{\bar{n}}}= & {} -\delta _{n{\bar{n}}}\times \frac{\left[ I_{m}'\left( \frac{n\pi }{\beta _{1}}\gamma \right) K_{m}'\left( \frac{n\pi }{\beta _{1}}\right) -I_{m}'\left( \frac{n\pi }{\beta _{1}}\right) K_{m}'\left( \frac{n\pi }{\beta _{1}}\gamma \right) \right] }{K_{m}'\left( \frac{n\pi }{\beta _{1}}\right) } \times \int ^{\beta _{1}}_{0}\cos ^{2}\left( \frac{n\pi }{\beta _{1}}\eta \right) d\eta \end{aligned}$$
(A.1)
$$\begin{aligned} a^{12}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\times \frac{\left[ I_{m}'\left( \frac{n\pi }{\beta _{1}}\gamma \right) K_{m}'\left( \frac{n\pi }{\beta _{1}}\alpha \right) -I_{m}'\left( \frac{n\pi }{\beta _{1}}\alpha \right) K_{m}'\left( \frac{n\pi }{\beta _{1}}\gamma \right) \right] }{K_{m}'\left( \frac{n\pi }{\beta _{1}}\alpha \right) } \times \int ^{\beta _{1}}_{0}\cos ^{2}\left( \frac{n\pi }{\beta _{1}}\eta \right) d\eta \end{aligned}$$
(A.2)
$$\begin{aligned} a^{21}_{mn{\bar{n}}}= & {} -\delta _{n{\bar{n}}}\times \frac{\left[ I_{m}\left( \frac{n\pi }{\beta _{1}}\gamma \right) K_{m}'\left( \frac{n\pi }{\beta _{1}}\right) -I_{m}'\left( \frac{n\pi }{\beta _{1}}\right) K_{m}\left( \frac{n\pi }{\beta _{1}}\gamma \right) \right] }{K_{m}'\left( \frac{n\pi }{\beta _{1}}\right) } \times \int ^{\beta _{1}}_{0}\cos ^{2}\left( \frac{n\pi }{\beta _{1}}\eta \right) d\eta \end{aligned}$$
(A.3)
$$\begin{aligned} a^{22}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\times \frac{\left[ I_{m}\left( \frac{n\pi }{\beta _{1}}\gamma \right) K_{m}'\left( \frac{n\pi }{\beta _{1}}\alpha \right) -I_{m}'\left( \frac{n\pi }{\beta _{1}}\alpha \right) K_{m}\left( \frac{n\pi }{\beta _{1}}\gamma \right) \right] }{K_{m}'\left( \frac{n\pi }{\beta _{1}}\alpha \right) } \times \int ^{\beta _{1}}_{0}\cos ^{2}\left( \frac{n\pi }{\beta _{1}}\eta \right) d\eta \end{aligned}$$
(A.4)
$$\begin{aligned} a^{23}_{mn{\bar{n}}}= & {} \left[ J_{m}\left( k_{m{\bar{n}}}\gamma \right) Y'_{m}\left( k_{m{\bar{n}}}\alpha \right) -J_{m}\left( k_{m{\bar{n}}}\alpha \right) Y'_{m}\left( k_{m{\bar{n}}}\gamma \right) \right] \nonumber \\&\times \int ^{\beta _{1}}_{0}\left( e^{k_{m{\bar{n}}}\eta }+e^{-k_{m{\bar{n}}}\eta }\right) \cos \left( \frac{n\pi }{\beta _{1}}\eta \right) d\eta \end{aligned}$$
(A.5)
$$\begin{aligned} a^{33}_{mn{\bar{n}}}= & {} -\delta _{n{\bar{n}}}\times \left( e^{k_{mn}\beta _{1}}-e^{-k_{mn}\beta _{1}}\right) \times \nonumber \\&\int ^{\gamma }_{\alpha } \xi {\left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] }^{2} d\xi \end{aligned}$$
(A.6)
$$\begin{aligned} a^{38}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\times \left( e^{k_{mn}\beta _{1}}+e^{k_{mn}(2\beta _{2}-\beta _{1})}\right) \times \nonumber \\&\int ^{\gamma }_{\alpha }\xi {\left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] }^{2} d\xi \end{aligned}$$
(A.7)
$$\begin{aligned} a^{42}_{mn{\bar{n}}}= & {} (-1)^{{\bar{n}}+1}\int ^{\gamma }_{\alpha }\xi \left[ \frac{\left[ I_{m}\left( \frac{{\bar{n}}\pi }{\beta _{1}}\xi \right) K_{m}'\left( \frac{{\bar{n}}\pi }{\beta _{1}}\alpha \right) -I_{m}'\left( \frac{{\bar{n}}\pi }{\beta _{1}}\alpha \right) K_{m}\left( \frac{{\bar{n}}\pi }{\beta _{1}}\xi \right) \right] }{K_{m}'\left( \frac{{\bar{n}}\pi }{\beta _{1}}\alpha \right) }\right] \nonumber \\&\times \left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] d\xi \end{aligned}$$
(A.8)
$$\begin{aligned} a^{43}_{mn{\bar{n}}}= & {} -\delta _{n{\bar{n}}}\times \left( e^{k_{mn}\beta _{1}}+e^{-k_{mn}\beta _{1}}\right) \times \nonumber \\&\int ^{\gamma }_{\alpha }\xi {\left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] }^{2} d\xi \nonumber \\&-\delta ^{2}_{m}\frac{\gamma ^{m}-\gamma ^{-m}}{2\beta _{1}(1-\alpha ^{2m})} \left[ J_{m}\left( k_{m{\bar{n}}}\gamma \right) Y'_{m}\left( k_{m{\bar{n}}}\alpha \right) -J_{m}\left( k_{m{\bar{n}}}\alpha \right) Y'_{m}\left( k_{m{\bar{n}}}\gamma \right) \right] \times \nonumber \\&\int ^{\beta _{1}}_{0}\left( e^{k_{m{\bar{n}}}\eta }+e^{-k_{m{\bar{n}}}\eta }\right) d\eta \times \int ^{\gamma }_{\alpha }(\xi ^{m+1}+\alpha ^{2m}\xi ^{-m+1})\times \nonumber \\&\left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] d\xi \end{aligned}$$
(A.9)
$$\begin{aligned} a^{46}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\times 2e^{k_{mn}\beta _{1}} \int ^{\gamma }_{\alpha } \xi {\left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] }^{2} d\xi \end{aligned}$$
(A.10)
$$\begin{aligned} a^{47}_{mn{\bar{n}}}= & {} \int ^{\gamma }_{\alpha } \xi \left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] \times \nonumber \\&\frac{\left[ I_{m}\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\xi \right) K_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) -I_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) K_{m}\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\xi \right) \right] }{K_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) } d\xi \end{aligned}$$
(A.11)
$$\begin{aligned} a^{48}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\times \left( e^{k_{mn}\beta _{1}}-e^{-k_{mn}(2\beta _{2}-\beta _{1})}\right) \times \nonumber \\&\int ^{\gamma }_{\alpha } \xi {\left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] }^{2} d\xi \end{aligned}$$
(A.12)
$$\begin{aligned} a^{55}_{mn{\bar{n}}}= & {} -\delta _{n{\bar{n}}}\times \frac{\left[ I'_{m}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\gamma \right) K_{m}'\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\right) -I_{m}'\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\right) K'_{m}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\gamma \right) \right] }{K_{m}' \left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})} \right) } \nonumber \\&\times \int ^{\beta _{2}}_{\beta _{1}}\cos ^{2}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})\pi }(\eta -\beta _{1})\right) d\eta \end{aligned}$$
(A.13)
$$\begin{aligned} a^{57}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}} \frac{\left[ I'_{m}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\gamma \right) K_{m}'\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) -I_{m}'\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) K'_{m}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\gamma \right) \right] }{K_{m}' \left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) } \nonumber \\&\times \int ^{\beta _{2}}_{\beta _{1}}\cos ^{2}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}(\eta -\beta _{1})\right) d\eta \end{aligned}$$
(A.14)
$$\begin{aligned} a^{64}_{mn{\bar{n}}}= & {} -\left[ J_{m}\left( \lambda _{m{\bar{n}}}\gamma \right) Y'_{m}\left( \lambda _{m{\bar{n}}}\right) -J_{m}\left( \lambda _{m{\bar{n}}}\right) Y'_{m}\left( \lambda _{m{\bar{n}}}\gamma \right) \right] \nonumber \\&\times \int ^{\beta _{2}}_{\beta _{1}}\left( e^{\lambda _{m{\bar{n}}}\eta }+e^{\lambda _{m{\bar{n}}}(2\beta _{1}-\eta )}\right) \cos \left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}(\eta -\beta _{1})\right) d\eta \end{aligned}$$
(A.15)
$$\begin{aligned} a^{65}_{mn{\bar{n}}}= & {} -\delta _{n{\bar{n}}}\times \int ^{\beta _{2}}_{\beta _{1}}\cos ^{2}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}(\eta -\beta _{1})\right) d\eta \nonumber \\&\frac{\left[ I_{m}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\gamma \right) K_{m}'\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\right) -I_{m}'\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\right) K_{m}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\gamma \right) \right] }{K_{m}'\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\right) } \nonumber \\&+\delta ^{1}_{m}\frac{(2{\bar{n}}-1)(-1)^{{\bar{n}}+1}\pi }{\omega ^{*2}(\beta _{2}-\beta _{1})(1-\gamma ^{2})}\times \int ^{\beta _{2}}_{\beta _{1}}\cos \left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}(\eta -\beta _{1})\right) d\eta \times \nonumber \\&\int ^{1}_{\gamma } \xi \frac{\left[ I_{m}\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\xi \right) K_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\right) -I_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\right) K_{m}\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\xi \right) \right] }{K_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\right) }d\xi \end{aligned}$$
(A.16)
$$\begin{aligned} a^{66}_{mn{\bar{n}}}= & {} \left[ J_{m}\left( k_{m{\bar{n}}}\gamma \right) Y'_{m}\left( k_{m{\bar{n}}}\alpha \right) -J_{m}\left( k_{m{\bar{n}}}\alpha \right) Y'_{m}\left( k_{m{\bar{n}}}\gamma \right) \right] \nonumber \\&\times \int ^{\beta _{2}}_{\beta _{1}}\left( e^{k_{m{\bar{n}}}\eta }+e^{k_{m{\bar{n}}}(2\beta _{1}-\eta )}\right) \cos \left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}(\eta -\beta _{1})\right) d\eta \end{aligned}$$
(A.17)
$$\begin{aligned} a^{67}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}} \times \int ^{\beta _{2}}_{\beta _{1}}\cos ^{2}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}(\eta -\beta _{1})\right) d\eta \times \nonumber \\&\frac{\left[ I_{m}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\gamma \right) K_{m}'\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) -I_{m}'\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) K_{m}\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\gamma \right) \right] }{K_{m}'\left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) }\nonumber \\&+\delta ^{1}_{m}\frac{(2{\bar{n}}-1)(-1)^{{\bar{n}}+1}\pi }{\omega ^{*2}(\beta _{2}-\beta _{1})(\gamma ^{2}-\alpha ^{2})}\times \int ^{\beta _{2}}_{\beta _{1}}\cos \left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}(\eta -\beta _{1})\right) d\eta \times \nonumber \\&\int ^{\gamma }_{\alpha } \xi \frac{\left[ I_{m}\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\xi \right) K_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) -I_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) K_{m}\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\xi \right) \right] }{K_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) }d\xi \end{aligned}$$
(A.18)
$$\begin{aligned} a^{68}_{mn{\bar{n}}}= & {} \left[ J_{m}\left( k_{m{\bar{n}}}\gamma \right) Y'_{m}\left( k_{m{\bar{n}}}\alpha \right) -J_{m}\left( k_{m{\bar{n}}}\alpha \right) Y'_{m}\left( k_{m{\bar{n}}}\gamma \right) \right] \nonumber \\&\times \int ^{\beta _{2}}_{\beta _{1}}\left( e^{k_{m{\bar{n}}}\eta }-e^{k_{m{\bar{n}}}(2\beta _{2}-\eta )}\right) \cos \left( \frac{(2n-1)\pi }{2(\beta _{2}-\beta _{1})}(\eta -\beta _{1})\right) d\eta \end{aligned}$$
(A.19)
$$\begin{aligned} a^{76}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}k_{mn}\left( e^{k_{mn}\beta _{2}}-e^{k_{mn}(2\beta _{1}-\beta _{2})}\right) \times \nonumber \\&\int ^{\gamma }_{\alpha }\xi \left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] ^{2} d\xi \nonumber \\&-\omega ^{*2}\delta _{n{\bar{n}}}\left( e^{k_{mn}\beta _{2}}+e^{k_{mn}(2\beta _{1}-\beta _{2})}\right) \times \nonumber \\&\int ^{\gamma }_{\alpha }\xi \left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] ^{2} d\xi \end{aligned}$$
(A.20)
$$\begin{aligned} a^{77}_{mn{\bar{n}}}= & {} \frac{(2{\bar{n}}-1)(-1)^{{\bar{n}}}\pi }{2(\beta _{2}-\beta _{1})}\int ^{\gamma }_{\alpha } \xi \times \left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] \nonumber \\&\frac{\left[ I_{m}\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\xi \right) K_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) -I_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) K_{m}\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\xi \right) \right] }{K_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\alpha \right) }d\xi \end{aligned}$$
(A.21)
$$\begin{aligned} a^{78}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}2k_{mn}e^{k_{mn}\beta _{2}}\int ^{\gamma }_{\alpha }\xi \left[ J_{m}\left( k_{mn}\xi \right) Y'_{m}\left( k_{mn}\alpha \right) -J_{m}\left( k_{mn}\alpha \right) Y'_{m}\left( k_{mn}\xi \right) \right] ^{2} d\xi \end{aligned}$$
(A.22)
$$\begin{aligned} a^{84}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\lambda _{mn}\left( e^{\lambda _{mn}\beta _{2}}-e^{\lambda _{mn}(2\beta _{1}-\beta _{2})}\right) \times \nonumber \\&\int ^{1}_{\gamma }\xi \left[ J_{m}\left( \lambda _{mn}\xi \right) Y'_{m}\left( \lambda _{mn}\right) -J_{m}\left( \lambda _{mn}\right) Y'_{m}\left( \lambda _{mn}\xi \right) \right] ^{2} d\xi \nonumber \\&-\omega ^{*2}\delta _{n{\bar{n}}}\left( e^{\lambda _{mn}\beta _{2}}+e^{\lambda _{mn}(2\beta _{1}-\beta _{2})}\right) \times \nonumber \\&\int ^{1}_{\gamma }\xi \left[ J_{m}\left( \lambda _{mn}\xi \right) Y'_{m}\left( \lambda _{mn}\right) -J_{m}\left( \lambda _{mn}\right) Y'_{m}\left( \lambda _{mn}\xi \right) \right] ^{2} d\xi \end{aligned}$$
(A.23)
$$\begin{aligned} a^{85}_{mn{\bar{n}}}= & {} \frac{(2{\bar{n}}-1)(-1)^{{\bar{n}}}\pi }{2(\beta _{2}-\beta _{1})}\int ^{1}_{\gamma } \xi \times \left[ J_{m}\left( \lambda _{mn}\xi \right) Y'_{m}\left( \lambda _{mn}\right) \nonumber \right. \\&\left. -J_{m}\left( \lambda _{mn}\right) Y'_{m}\left( \lambda _{mn}\xi \right) \right] \nonumber \\&\frac{\left[ I_{m}\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\xi \right) K_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\right) -I_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\right) K_{m}\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\xi \right) \right] }{K_{m}'\left( \frac{(2{\bar{n}}-1)\pi }{2(\beta _{2}-\beta _{1})}\right) } d\xi \end{aligned}$$
(A.24)
