## 3 Analytical Solutions
Furthermore, combining Equations (14) and (18) provides the exact expression of \(\mathbf{u}_{\text{d}}\):  
\[\mathbf{u}_{\text{d},zzt}=\rho\Big{(}\mathbf{u}_{\text{d},t}+\lambda\rho^{-1} \mathbf{u}_{\text{d},zz}-\lambda_{\text{q}}\mathbf{q}_{i}\Big{)} \tag{19}\]  
where, \(\lambda=\frac{2}{r_{e}^{2}}\mathbf{C}^{-1}\mathbf{C_{v_{r}}}\mathbf{F}^{-1}\), and \(\lambda_{\text{q}}=\mathbf{C}^{-1}\mathbf{C_{q}}\). It is noted that the only function \(\mathbf{u_{d}}\) to be solved appears in Equation (19), thus the solution of \(\mathbf{u_{d}}\) can be obtained explicitly hereinafter. Based on the Fourier series expansion theory and boundary conditions (i.e., Equations (9) and (10)), \(\mathbf{u_{d}}\) is expressed as:  
\[\mathbf{u_{d}}=\sum_{k=1}^{\infty}\mathbf{U_{d}}(\mathbf{t})\sin\biggl{(} \frac{K}{H}z\biggr{)} \tag{20}\]  
where, \(K=\frac{2k+1}{2}\pi,\ k=1,2,3\ldots\)  
By substituting Equations (19) into (20), one can obtain:  
\[\sum_{k=1}^{\infty}\mathbf{U_{d,t}}(\mathbf{t})\biggl{(}\frac{K}{H}\biggr{)}^ {2}\sin\biggl{(}\frac{K}{H}z\biggr{)}=-\rho\biggl{[}\sum_{k=1}^{\infty} \mathbf{U_{d,t}}(\mathbf{t})\sin\biggl{(}\frac{K}{H}z\biggr{)}-\lambda \mathbf{p}^{-1}\sum_{k=1}^{\infty}\mathbf{U_{d}}(\mathbf{t})\biggl{(}\frac{K} {H}\biggr{)}^{2}\sin\biggl{(}\frac{K}{H}z\biggr{)}-\lambda\mathbf{q}\mathbf{q }_{t}\biggr{]} \tag{21}\]  
By applying the orthogonality of trigonometric functions, Equation (21) can be converted into:  
\[\mathbf{U_{d,t}}(\mathbf{t})=\left[ \begin{array}{cc}A_{\mathrm{a}}&-A_{ \mathrm{w}}\\ -W_{\mathrm{a}}&W_{\mathrm{w}}\end{array} \right]\mathbf{U_{d}}(\mathbf{t})+ \left[ \begin{array}{cc}A_{\mathrm{q}}&0\\ 0&W_{\mathrm{q}}\end{array} \right]\mathbf{q}_{t} \tag{22}\]  
where,  
\[A_{\mathrm{a}}=\frac{\lambda_{\mathrm{a}}K^{2}}{K^{2}+\rho_{\mathrm{a}}H^{2}},\ A_{\mathrm{w}}=\frac{C_{\mathrm{a}}\lambda_{\mathrm{w}}\rho_{\mathrm{a}}K ^{2}}{\rho_{\mathrm{w}}(K^{2}+\rho_{\mathrm{a}}H^{2})},\ W_{\mathrm{a}}=\frac{ C_{\mathrm{w}}\lambda_{\mathrm{a}}\rho_{\mathrm{w}}K^{2}}{\rho_{\mathrm{a}}(K^{2}+ \rho_{\mathrm{w}}H^{2})};\]  
\[W_{\mathrm{w}}=\frac{\lambda_{\mathrm{w}}K^{2}}{K^{2}+\rho_{\mathrm{w}}H^{2}},\ A_{\mathrm{q}}=-\frac{2}{K}\frac{\rho_{\mathrm{a}}\lambda_{\mathrm{a}}^{ 4}H^{2}}{K^{2}+\rho_{\mathrm{a}}H^{2}},\ W_{\mathrm{q}}=-\frac{2}{K}\frac{ \rho_{\mathrm{w}}\lambda_{\mathrm{w}}^{4}H^{2}}{K^{2}+\rho_{\mathrm{w}}H^{2}}.\]  
It is appointed that the expression of \(\mathbf{U_{d}}(\mathbf{t})\), as shown is Equation (22), is arranged as the form in accordance with that of [15] due to the similarity of the partial differential equations, and the differences occur in the intermediate parameters (i.e., \(A_{\mathrm{a}}\), \(A_{\mathrm{w}}\), \(W_{\mathrm{a}}\), \(W_{\mathrm{w}}\), \(A_{\mathrm{q}}\) and \(W_{\mathrm{q}}\)). Furthermore, the solution for \(\mathbf{U_{d}}(\mathbf{t})\) according to the decoupling method proposed by [PERSON] et al. (2010), and the constant variation method by [PERSON] et al. (2018), is:  
\[\mathbf{U_{d}}(\mathbf{t})=\frac{1}{q_{12}q_{21}-1}\biggl{\{}\left[ \begin{array} []{cc}-C_{1}&q_{21}C_{2}\\ q_{12}C_{1}&-C_{2}\end{array} \right]\left[ \begin{array}{cc}e^{\mathcal{O}_{ 1}t}\\ e^{\mathcal{O}_{2}t}\end{array} \right]+\int_{0}^{t}\left[ \begin{array}{cc}- \beta_{1}&q_{21}\beta_{2}\\ q_{12}\beta_{1}&-\beta_{2}\end{array} \right]\left[ \begin{array}{cc}e^{ \mathcal{O}_{1}(t-\tau)}\\ e^{\mathcal{O}_{1}(t-\tau)}\end{array} \right]\frac{\partial q}{\partial\tau}d\tau\biggr{\}} \tag{23}\]  
where, \(q_{12}=\frac{W_{\mathrm{a}}}{A_{\mathrm{a}}-Q_{2}}=\frac{W_{\mathrm{w}}-Q_{2} }{A_{\mathrm{w}}},q_{21}=\frac{A_{\mathrm{a}}-Q_{1}}{W_{\mathrm{a}}}=\frac{A_{ \mathrm{w}}}{W_{\mathrm{w}}-Q_{1}},\ \beta_{1}=A_{\mathrm{q}}+q_{21}W_{\mathrm{q}},\ \beta_{2}=q_{12}A_{ \mathrm{q}}+W_{\mathrm{q}};\)  
\(Q_{1,2}=\frac{1}{2}\biggl{[}\left(A_{\mathrm{a}}+W_{\mathrm{w}}\right)\pm \sqrt{\left(A_{\mathrm{a}}-W_{\mathrm{w}}\right)^{2}+4W_{\mathrm{a}}A_{ \mathrm{w}}}\biggr{]}\). \(C_{1}\) and \(C_{2}\) are the unknown constants determined by initial conditions.