Appendix A
The functions in Eq. (13) are as follows:
$$\begin{aligned} R_{1} (\omega ,s)= & {} \frac{L_{3} (\omega ,s)V_{1} +L_{4} (\omega ,s)V_{2} +L_{5} (\omega ,s)V_{3} }{\kappa L_{1} (\omega ,s)V_{1} } \end{aligned}$$
(A1)
$$\begin{aligned} R_{2} (\omega ,s)= & {} \frac{L_{4} (\omega ,s)\mu _{110} -L_{5} (\omega ,s)d_{110} }{L_{2} (\omega ,s)V_{1} } \end{aligned}$$
(A2)
$$\begin{aligned} R_{3} (\omega ,s)= & {} \frac{L_{5} (\omega ,s)\varepsilon _{110} -L_{4} (\omega ,s)d_{110} }{L_{2} (\omega ,s)V_{1} } \end{aligned}$$
(A3)
where
$$\begin{aligned} V_{1}= & {} d_{110}^{2} -\mu _{110} \varepsilon _{110} \end{aligned}$$
(A4)
$$\begin{aligned} V_{2}= & {} d_{110} q_{150} -e_{150} \mu _{110} \end{aligned}$$
(A5)
$$\begin{aligned} V_{3}= & {} d_{110} e_{150} -q_{150} \varepsilon _{110} \end{aligned}$$
(A6)
$$\begin{aligned} L_{1} (\omega ,s)= & {} -\frac{(\gamma _{1} +\gamma )(\Delta _{1} -\gamma _{1} +\gamma )+(\gamma _{1} -\gamma )(\Delta _{1} +\gamma _{1} +\gamma )e^{-2\gamma _{1} h_{2} }}{\gamma _{1} +\gamma } \end{aligned}$$
(A7)
$$\begin{aligned} L_{2} (\omega ,s)= & {} -\frac{(\gamma _{2} +\gamma )(\Delta _{2} -\gamma _{2} +\gamma )+(\gamma _{2} -\gamma )(\Delta _{2} +\gamma _{2} +\gamma )e^{-2\gamma _{2} h_{2} }}{\gamma _{2} +\gamma } \end{aligned}$$
(A8)
$$\begin{aligned} L_{3} (\omega ,s)= & {} [{\Delta }'_{1} \kappa -{\Delta }'_{2} (e_{150} \alpha _{2} +q_{150} \alpha _{3} )]\,b_{z} +{\Delta }'_{2} (e_{150} b_{\phi } +q_{150} b_{\psi } ) \end{aligned}$$
(A9)
$$\begin{aligned} L_{4} (\omega ,s)= & {} {\Delta }'_{2} [(\varepsilon _{110} \alpha _{2} +d_{110} \alpha _{3} )b_{z} -\varepsilon _{110} b_{\phi } -d_{110} b_{\psi } ] \end{aligned}$$
(A10)
$$\begin{aligned} L_{5} (\omega ,s)= & {} {\Delta }'_{2} [(\alpha _{2} d_{110} +\alpha _{3} \mu _{110} )b_{z} -d_{110} b_{\phi } -\mu _{110} b_{\psi } ] \end{aligned}$$
(A11)
and
$$\begin{aligned} {\Delta }'_{1} (\omega ,s)= & {} \frac{(\gamma _{1}^{2} -\beta ^{2})(1-e^{-2\gamma _{1} h_{1} })}{\gamma _{1} (1+e^{-2\gamma _{1} h_{1} })\,-\beta (1-e^{-2\gamma _{1} h_{1} })} \end{aligned}$$
(A12)
$$\begin{aligned} {\Delta }'_{2} (\omega ,s)= & {} \frac{(\gamma _{2}^{2} -\beta ^{2})(1-e^{-2\gamma _{2} h_{1} })}{\gamma _{2} (1+e^{-2\gamma _{2} h_{1} })-\beta (1-e^{-2\gamma _{2} h_{1} })} \end{aligned}$$
(A13)
Appendix B
The kernels in Eq. (18) are:
$$\begin{aligned} K_{ij}^{11} (p,q,s)= & {} \frac{\kappa e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{(\gamma _{1} -\gamma )}{\omega }(1-e^{-2\gamma _{1} ((y_{i} -y_{j} )+h_{2} )})\Omega _{1} (\omega ,s)e^{\gamma _{1} (y_{i} -y_{j} )}\right. \nonumber \\&\left. +\frac{e^{\omega (y_{i} -y_{j} )}}{2}\right\} \sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega \nonumber \\&+\frac{e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{(\gamma _{2} -\gamma )}{\omega }(1-e^{-2\gamma _{2} ((y_{i} -y_{j} )+h_{2} )})(e_{150} \Omega _{2} (\omega ,s)\right. \nonumber \\&+\,q_{150} \Omega _{3} (\omega ,s))e^{\gamma _{2} (y_{i} -y_{j} )} \nonumber \\&\left. -\frac{e^{\omega (y_{i} -y_{j} )}}{2}(e_{150} \alpha _{2} +q_{150} \alpha _{3} )\right\} \sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega \nonumber \\&+\frac{e^{\gamma (y_{i} -y_{j} )}(-\kappa +\alpha _{2} e_{150} +\alpha _{3} q_{150} )}{2\pi }\frac{(x_{i} -x_{j} )}{(x_{i} -x_{j} )^{2}+(y_{i} -y_{j} )^{2}} \end{aligned}$$
(B1)
$$\begin{aligned} K_{ij}^{12} (p,q,s)= & {} \frac{\kappa e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{(\gamma _{1} -\gamma )}{\omega }(1-e^{-2\gamma _{1} ((y_{i} -y_{j} )+h_{2} )})\Delta _{1} (\omega ,s)e^{\gamma _{1} (y_{i} -y_{j} )}\right\} \nonumber \\&\sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega \nonumber \\&+\frac{e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{(\gamma _{2} -\gamma )}{\omega }(1-e^{-2\gamma _{2} ((x_{i} -x_{j} )+h_{2} )})(e_{150} \Delta _{2} (\omega ,s)\right. \nonumber \\&+\,q_{150} \Delta _{3} (\omega ,s))e^{\gamma _{2} (y_{i} -y_{j} )} \nonumber \\&\left. +\frac{e_{150} e^{\omega (y_{i} -y_{j} )}}{2}\right\} \sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega -\frac{e^{\gamma (y_{i} -y_{j} )}e_{150} }{2\pi }\frac{(x_{i} -x_{j} )}{(x_{i} -x_{j} )^{2}+(y_{i} -y_{j} )^{2}} \end{aligned}$$
(B2)
$$\begin{aligned} K_{ij}^{13}= & {} \frac{\kappa e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{(\gamma _{1} -\gamma )}{\omega }(1-e^{-2\gamma _{1} ((y_{i} -y_{j} )+h_{2} )})\Lambda _{1} (\omega ,s)e^{\gamma _{1} (y_{i} -y_{j} )}\right\} \nonumber \\&\sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega \nonumber \\&+\frac{e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{(\gamma _{2} -\gamma )}{\omega }(1-e^{-2\gamma _{2} ((y_{i} -y_{j} )+h_{2} )})(e_{150} \Delta _{2} (\omega ,s)\right. \nonumber \\&+\,q_{150} \Delta _{3} (\omega ,s))e^{\gamma _{2} (y_{i} -y_{j} )}\nonumber \\&\left. +\frac{q_{150} e^{\omega (y_{i} -y_{j} )}}{2}\right\} \sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega -\frac{q_{150} }{2\pi }\frac{(x_{i} -x_{j} )}{(x_{i} -x_{j} )^{2}+(y_{i} -y_{j} )^{2}} \end{aligned}$$
(B3)
$$\begin{aligned} K_{ij}^{21}= & {} \frac{e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{e^{\omega (y_{i} -y_{j} )}}{2}(\varepsilon _{110} \alpha _{2} +d_{110} \alpha _{3} )\right. \nonumber \\&-\frac{(\gamma _{2} -\gamma )}{\omega }[1-e^{-2\gamma _{2} ((y_{i} -y_{j} )+h_{2} )}] \nonumber \\&\left. \times (\varepsilon _{110} \Omega _{2} (\omega ,s)+d_{110} \Omega _{3} (\omega ,s))e^{\gamma _{2} (y_{i} -y_{j} )}\right\} \sin (\omega (y_{i} -y_{j} ))\hbox {d}\omega \, \nonumber \\&-\frac{e^{\gamma (y_{i} -y_{j} )}(\varepsilon _{110} \alpha _{2} +d_{110} \alpha _{3} )}{2\pi }\frac{(x_{i} -x_{j} )}{(x_{i} -x_{j} )^{2}+(y_{i} -y_{j} )^{2}} \end{aligned}$$
(B4)
$$\begin{aligned} K_{ij}^{22} (p,q,s)= & {} \,-\frac{e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{(\gamma _{2} -\gamma )}{\omega }(1-e^{-2\gamma _{2} ((y_{i} -y_{j} )+h_{2} )})(\varepsilon _{110} \Delta _{2} (\omega ,s)\right. \nonumber \\&+\,d_{110} \Delta _{3} (\omega ,s))e^{\gamma _{2} (y_{i} -y_{j} )}\nonumber \\&\left. +\frac{e^{\omega (y_{i} -y_{j} )}\varepsilon _{110} }{2}\right\} \sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega +\frac{e^{\gamma (y_{i} -y_{j} )}\varepsilon _{110} }{2\pi }\frac{(x_{i} -x_{j} )}{(x_{i} -x_{j} )^{2}+(y_{i} -y_{j} )^{2}} \end{aligned}$$
(B5)
$$\begin{aligned} K_{ij}^{23} (p,q,s)= & {} -\frac{e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{(\gamma _{2} -\gamma )}{\omega }(1-e^{-2\gamma _{2} ((y_{i} -y_{j} )+h_{2} )})(\varepsilon _{110} \Lambda _{2} (\omega ,s)\right. \nonumber \\&+\,d_{110} \Lambda _{3} (\omega ,s))e^{\gamma _{2} (y_{i} -y_{j} )} \nonumber \\&\left. +\frac{e^{\omega (y_{i} -y_{j} )}d_{110} }{2}\right\} \sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega \,+\frac{e^{\gamma (y_{i} -y_{j} )}d_{110} }{2\pi }\frac{(x_{i} -x_{j} )}{(x_{i} -x_{j} )^{2}+(y_{i} -y_{j} )^{2}} \end{aligned}$$
(B6)
$$\begin{aligned} K_{ij}^{31} (p,q,s)= & {} \frac{e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{e^{\omega (y_{i} -y_{j} )}}{2}(d_{110} \alpha _{2} +\mu _{110} \alpha _{3} )\right. \nonumber \\&-\frac{(\gamma _{2} -\gamma )}{\omega }[1-e^{-2\gamma _{2} ((y_{i} -y_{j} )+h_{2} )}] \nonumber \\&\left. \times (d_{110} \Omega _{2} (\omega ,s)+\mu _{110} \Omega _{3} (\omega ,s))e^{\gamma _{2} (y_{i} -y_{j} )}\right. \}\sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega \nonumber \\&-\frac{e^{\gamma (y_{i} -y_{j} )}(d_{110} \alpha _{2} +\mu _{110} \alpha _{3} )}{2\pi }\frac{(x_{i} -x_{j} )}{(x_{i} -x_{j} )^{2}+(y_{i} -y_{j} )^{2}} \end{aligned}$$
(B7)
$$\begin{aligned} K_{ij}^{32}= & {} -\frac{e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{(\gamma _{2} -\gamma )}{\omega }(1-e^{-2\gamma _{2} ((y_{i} -y_{j} )+h_{2} )})(d_{110} \Delta _{2} (\omega ,s)\right. \nonumber \\&+\mu _{110} \Delta _{3} (\omega ,s))e^{\gamma _{2} (y_{i} -y_{j} )} \nonumber \\&\left. +\frac{e^{\omega (y_{i} -y_{j} )}d_{110} }{2}\right\} \sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega +\frac{e^{\gamma (y_{i} -y_{j} )}d_{110} }{2\pi }\frac{(x_{i} -x_{j} )}{(x_{i} -x_{j} )^{2}+(y_{i} -y_{j} )^{2}} \end{aligned}$$
(B8)
$$\begin{aligned} K_{ij}^{33} (p,q,s)= & {} -\frac{e^{\gamma (y_{i} -y_{j} )}}{\pi }\int _0^{+\infty } \left\{ \frac{(\gamma _{2} -\gamma )}{\omega }(1-e^{-2\gamma _{2} ((y_{i} -y_{j} )+h_{2} )})(d_{110} \Lambda _{2} (\omega ,s)\right. \nonumber \\&+\mu _{110} \Lambda _{3} (\omega ,s))e^{\gamma _{2} (y_{i} -y_{j} )} \nonumber \\&\left. +\frac{e^{\omega (y_{i} -y_{j} )}\mu _{110} }{2}\right\} \sin (\omega (x_{i} -x_{j} ))\hbox {d}\omega +\frac{e^{\gamma (y_{i} -y_{j} )}\mu _{110} }{2\pi }\frac{(x_{i} -x_{j} )}{(x_{i} -x_{j} )^{2}+(y_{i} -y_{j} )^{2}} \end{aligned}$$
(B9)
In which:
$$\begin{aligned} \Omega _{1} (\omega ,s)= & {} \frac{a_{1} (\omega ,s)V_{1} +a_{2} (\omega ,s)V_{2} +a_{3} (\omega ,s)V_{3} }{\kappa L_{1} (\omega ,s)V_{1} } \end{aligned}$$
(B10)
$$\begin{aligned} \Omega _{2} (\omega ,s)= & {} \frac{a_{2} (\omega ,s)\mu _{110} -a_{3} (\omega ,s)d_{110} }{L_{2} (\omega ,s)V_{1} } \end{aligned}$$
(B11)
$$\begin{aligned} \Omega _{3} (\omega ,s)= & {} \frac{a_{3} (\omega ,s)\varepsilon _{110} -a_{2} (\omega ,s)d_{110} }{L_{2} (\omega ,s)V_{1} } \end{aligned}$$
(B12)
$$\begin{aligned} \Delta _{1} (\omega ,s)= & {} \frac{b_{1} (\omega ,s)V_{1} +b_{2} (\omega ,s)V_{2} +b_{3} (\omega ,s)V_{3} }{\kappa L_{1} (\omega ,s)V_{1} } \end{aligned}$$
(B13)
$$\begin{aligned} \Delta _{2} (\omega ,s)= & {} \frac{b_{2} (\omega ,s)\mu _{110} -b_{3} (\omega ,s)d_{110} }{L_{2} (\omega ,s)V_{1} } \end{aligned}$$
(B14)
$$\begin{aligned} \Delta _{3} (\omega ,s)= & {} \frac{b_{3} (\omega ,s)\varepsilon _{110} -b_{2} (\omega ,s)d_{110} }{L_{2} (\omega ,s)V_{1} } \end{aligned}$$
(B15)
$$\begin{aligned} \Lambda _{1} (\omega ,s)= & {} \frac{c_{1} (\omega ,s)V_{1} +c_{2} (\omega ,s)V_{2} +c_{3} (\omega ,s)V_{3} }{\kappa L_{1} (\omega ,s)V_{1} }\, \end{aligned}$$
(B16)
$$\begin{aligned} \Lambda _{2} (\omega ,s)= & {} \frac{c_{2} (\omega ,s)\mu _{110} -c_{3} (\omega ,s)d_{110} }{L_{2} (\omega ,s)V_{1} } \end{aligned}$$
(B17)
$$\begin{aligned} \Lambda _{3} (\omega ,s)= & {} \frac{c_{3} (\omega ,s)\varepsilon _{110} -c_{2} (\omega ,s)d_{110} }{L_{2} (\omega ,s)V_{1} } \end{aligned}$$
(B18)
where
$$\begin{aligned} a_{1} (\omega ,s)= & {} \frac{(\gamma _{2}^{2} -\beta ^{2})(e^{-2\gamma _{2} h_{1} }-1)(e_{150} \alpha _{2} +q_{150} \alpha _{3} b)}{[(\gamma _{2} -\beta )+(\gamma _{2} +\beta )e^{-2\gamma _{2} h_{1} }]}-\frac{\kappa (\gamma _{1}^{2} -\beta ^{2})(e^{-2\gamma _{1} h_{1} }-1)}{[(\gamma _{1} -\beta )\,+(\gamma _{1} +\beta )e^{-2\gamma _{1} h_{1} }]} \end{aligned}$$
(B19)
$$\begin{aligned} a_{2} (\omega ,s)= & {} \frac{(\gamma _{2}^{2} -\beta ^{2})(1-e^{-2\gamma _{2} h_{1} })(\varepsilon _{110} \alpha _{2} +d_{110} \alpha _{3} )}{[(\gamma _{2} -\beta )+(\gamma _{2} +\beta )e^{-2\gamma _{2} h_{1} }]} \end{aligned}$$
(B20)
$$\begin{aligned} a_{3} (\omega ,s)= & {} \frac{(\gamma _{2}^{2} -\beta ^{2})(1-e^{-2\gamma _{2} h_{1} })(d_{110} \alpha _{2} +\mu _{110} \alpha _{3} )}{[(\gamma _{2} -\beta )+(\gamma _{2} +\beta )e^{-2\gamma _{2} h_{1} }]} \end{aligned}$$
(B21)
$$\begin{aligned} b_{1} (\omega ,s)= & {} -\frac{e_{150} (\gamma _{2}^{2} -\beta ^{2})(e^{-2\gamma _{2} h_{1} }-1)}{[(\gamma _{2} -\beta )+(\gamma _{2} +\beta )e^{-2\gamma _{2} h_{1} }]} \end{aligned}$$
(B22)
$$\begin{aligned} b_{2} (\omega ,s)= & {} -\frac{\varepsilon _{110} (\gamma _{2}^{2} -\beta ^{2})(1-e^{-2\gamma _{2} h_{1} })}{[(\gamma _{2} -\beta )+(\gamma _{2} +\beta )e^{-2\gamma _{2} h_{1} }]} \end{aligned}$$
(B23)
$$\begin{aligned} b_{3} (\omega ,s)= & {} -\frac{d_{110} (\gamma _{2}^{2} -\beta ^{2})(1-e^{-2\gamma _{2} h_{1} })}{[(\gamma _{2} -\beta )+(\gamma _{2} +\beta )e^{-2\gamma _{2} h_{1} }]} \end{aligned}$$
(B24)
$$\begin{aligned} c_{1} (\omega ,s)= & {} -\frac{q_{150} (\gamma _{2}^{2} -\beta ^{2})(e^{-2\gamma _{2} h_{1} }-1)}{[(\gamma _{2} -\beta )+(\gamma _{2} +\beta )e^{-2\gamma _{2} h_{1} }]} \end{aligned}$$
(B25)
$$\begin{aligned} c_{2} (\omega ,s)= & {} -\frac{d_{110} (\gamma _{2}^{2} -\beta ^{2})(1-e^{-2\gamma _{2} h_{1} })}{[(\gamma _{2} -\beta )+(\gamma _{2} +\beta )e^{-2\gamma _{2} h_{1} }]} \end{aligned}$$
(B26)
$$\begin{aligned} c_{3} (\omega ,s)= & {} -\frac{\mu _{110} (\gamma _{2}^{2} -\beta ^{2})(1-e^{-2\gamma _{2} h_{1} })}{[(\gamma _{2} -\beta )+(\gamma _{2} +\beta )e^{-2\gamma _{2} h_{1} }]} \end{aligned}$$
(B27)