## 4 Asymptotic Solution to the Boundary-Value Problem (17)
### First-Order Formal Asymptotic Solution to Problems (26) and (27)  
Based on the Boundary Functions Method (see, e.g., [66; 67]), we look for the first-order asymptotic solution  
\[\mathrm{col}\big{(}z_{1}^{1}(t,\varepsilon),z_{2}^{1}(t,\varepsilon),\lambda_{ \mathrm{u1}}^{1}(t,\varepsilon),\lambda_{\mathrm{u2}}^{1}(t,\varepsilon), \lambda_{\mathrm{v}}^{1}(t,\varepsilon),\lambda_{\mathrm{u,v}}^{1}(t, \varepsilon)\big{)}\]  
of Problems (26) and (27) in the form  
\[ \begin{split} z_{1}^{1}(t,\varepsilon)&=\sum_{k=0 }^{1}\varepsilon^{k}\big{[}z_{1,k}(t)+z_{1,k}^{0}(\xi)+z_{1,k}^{f}(\eta)\big{]},\\ z_{2}^{1}(t,\varepsilon)&=\sum_{k=0}^{1}\varepsilon ^{k}\big{[}z_{2,k}(t)+z_{2,k}^{0}(\xi)+z_{2,k}^{f}(\eta)\big{]},\\ \lambda_{\mathrm{u1}}^{1}(t,\varepsilon)&=\sum_{k=0 }^{1}\varepsilon^{k}\big{[}\bar{\lambda}_{\mathrm{u1},k}(t)+\lambda_{\mathrm{ u1},k}^{0}(\xi)+\lambda_{\mathrm{u1},k}^{f}(\eta)\big{]},\\ \lambda_{\mathrm{u2}}^{1}(t,\varepsilon)&=\sum_{k=0 }^{1}\varepsilon^{k}\big{[}\bar{\lambda}_{\mathrm{u2},k}(t)+\lambda_{\mathrm{ u2},k}^{0}(\xi)+\lambda_{\mathrm{u2},k}^{f}(\eta)\big{]},\\ \lambda_{\mathrm{v}}^{1}(t,\varepsilon)&=\sum_{k=0 }^{1}\varepsilon^{k}\big{[}\bar{\lambda}_{\mathrm{v},k}(t)+\lambda_{\mathrm{ v},k}^{0}(\xi)+\lambda_{\mathrm{v},k}^{f}(\eta)\big{]},\\ \lambda_{\mathrm{u,v}}^{1}(t,\varepsilon)&=\sum_{k=0 }^{1}\varepsilon^{k}\big{[}\bar{\lambda}_{\mathrm{u,v},k}(t)+\lambda_{\mathrm{ u,v},k}^{0}(\xi)+\lambda_{\mathrm{u,v},k}^{f}(\eta)\big{]},\end{split}  \tag{28}\]  
where  
\[\xi=\frac{t}{\varepsilon},\quad\eta=\frac{t-t_{f}}{\varepsilon}. \tag{29}\]  
**Remark 5**. : _In Equation (28), the terms with the overbar constitute the so-call outer solution, the terms with the superscript \(0\) are the boundary corrections in the right-hand neighborhood of \(t=0\), the terms with the superscript \(f\) are the boundary corrections in the left-hand neighborhood of \(t=t_{f}\). Equations and boundary conditions for the asymptotic solution terms are obtained substituting \(z_{1}^{1}(t,\varepsilon)\), \(z_{2}^{1}(t,\varepsilon)\), \(\lambda_{\mathrm{u1}}^{1}(t,\varepsilon)\), \(\lambda_{\mathrm{u2}}^{1}(t,\varepsilon)\), \(\lambda_{\mathrm{v}}^{1}(t,\varepsilon)\), and \(\lambda_{\mathrm{u,v}}^{1}(t,\varepsilon)\) from (28) into Problems (26) and (27), instead of \(z_{1}(t)\), \(z_{2}(t)\), \(\lambda_{\mathrm{u1}}(t)\), \(\lambda_{\mathrm{u2}}(t)\), \(\Lambda_{\mathrm{v}}(t)\), and \(\lambda_{\mathrm{u,v}}(t)\), respectively, and equating the coefficients for the same power of \(e\) on both sides of the resulting equations, separately depending on \(t\), \(\bar{\xi}\), and \(\eta\). Additionally, the following should be noted: if \(\varepsilon\to+0\), then, for any \(t\in(0,t_{f}]\), \(\bar{\xi}\to+\infty\), while, for any \(t\in[0,t_{f})\), \(\eta\to-\infty\)._  
2.1 Obtaining the Boundary Corrections \(z_{1,0}^{0}(\xi)\), \(\lambda_{\mathrm{u1},0}^{0}(\xi)\), \(\lambda_{\mathrm{v},0}^{0}(\xi)\) and \(\lambda_{\mathrm{u,v}0}^{0}(\xi)\)  
The equations for obtaining these boundary corrections are  
\[ \begin{split}\frac{dz_{1,0}^{0}(\xi)}{d\xi}=0,\quad\xi\geq 0, \\ \frac{d\lambda_{\mathrm{u1},0}^{0}(\xi)}{d\xi}=0,\quad\xi\geq 0, \\ \frac{d\lambda_{\mathrm{v},0}^{0}(\xi)}{d\xi}=0,\quad\xi\geq 0, \\ \frac{d\lambda_{\mathrm{u,v}0}^{0}(\xi)}{d\xi}=0,\quad\xi\geq 0. \end{split}  \tag{30}\]  
To obtain the unique solutions of these equations, one should give some additional conditions for the unknown functions.