## 3 Results
#### 3.1.2 The Transition Layer Functions
The systems of equations for the transition layer functions are obtained in the standard way [17] by equating the coefficients at the same powers of \(\varepsilon\) in the Taylor expansion of the equalities  
\[\left(\varepsilon^{2}\frac{\partial^{2}}{\partial\tau^{2}}- \varepsilon^{3}k(\mathbf{y})\frac{\partial}{\partial\tau}+\sum_{i\geq 0} \varepsilon^{i+4}L_{i}(\tau,\mathbf{y})\right)\left[\sum_{i=0}^{n}\varepsilon^{i} Q_{i}^{(\mp)}u(\tau,\mathbf{y})\ \right] =Q^{(\mp)}f(\tau,\mathbf{y},\varepsilon), \tag{20}\] \[\left(\frac{\partial^{2}}{\partial\tau^{2}}-\varepsilon k( \mathbf{y})\frac{\partial}{\partial\tau}+\sum_{i\geq 0}\varepsilon^{i+2}L_{i}(\tau, \mathbf{y})\right)\left[\sum_{i=0}^{n}\varepsilon^{i}Q_{i}^{(\mp)}v(\tau, \mathbf{y})\ \right] =Q^{(\mp)}g(\tau,\mathbf{y},\varepsilon),\] (21) \[\left(\frac{\partial^{2}}{\partial\sigma^{2}}-\varepsilon^{2}k( \mathbf{y})\frac{\partial}{\partial\sigma}+\sum_{i\geq 0}\varepsilon^{2i+4}L_{i}(\sigma, \mathbf{y})\right)\left[\sum_{i=0}^{n}\varepsilon^{i}M_{i}^{(\mp)}u(\sigma, \mathbf{y})\ \right] =M^{(\mp)}f(\sigma,\mathbf{y},\varepsilon),\] (22) \[\left(\frac{1}{\varepsilon^{2}}\frac{\partial^{2}}{\partial\sigma^ {2}}-k(\mathbf{y})\frac{\partial}{\partial\sigma}+\sum_{i\geq 0}\varepsilon^{2i+2}L_{i}(\sigma, \mathbf{y})\right)\left[\sum_{i=0}^{n+2}\varepsilon^{i}M_{i}^{(\mp)}v(\sigma, \mathbf{y})\right] =M^{(\mp)}g(\sigma,\mathbf{y},\varepsilon). \tag{23}\]Here, we denote  
\[Q^{(\mp)}f(\tau,\mathbf{y},\epsilon):=f^{(\mp)}\Big{(}\sum_{i=0}^{n }\epsilon^{i}\Big{(}\bar{u}_{i}^{(\mp)}(\mathbf{y}+\varepsilon\tau\mathbf{n}_{ \gamma}(\mathbf{y}))+Q_{i}^{(\mp)}u(\tau,\mathbf{y})\Big{)},\] \[\sum_{i=0}^{n}\epsilon^{i}\Big{(}\bar{u}_{i}^{(\mp)}(\mathbf{y}+ \varepsilon\tau\mathbf{n}_{\gamma}(\mathbf{y}))+Q_{i}^{(\mp)}v(\tau,\mathbf{y })\Big{)},\mathbf{y}+\varepsilon\tau\mathbf{n}_{\gamma}(\mathbf{y}),\epsilon \Big{)}-\] \[-f^{(\mp)}\Bigg{(}\sum_{i=0}^{n}\epsilon^{i}\bar{u}_{i}^{(\mp)}( \mathbf{y}+\varepsilon\tau\mathbf{n}_{\gamma}(\mathbf{y})),\sum_{i=0}^{n} \epsilon^{i}\bar{u}_{i}^{(\mp)}(\mathbf{y}+\varepsilon\tau\mathbf{n}_{\gamma} (\mathbf{y})),\mathbf{y}+\varepsilon\tau\mathbf{n}_{\gamma}(\mathbf{y}), \epsilon\Bigg{)},\]  
and we define the functions \(Q^{(\mp)}g(\tau,\mathbf{y},\epsilon)\) in the same way;  
\[M^{(\mp)}f(\sigma,\mathbf{y},\epsilon):=f^{(\mp)}\Big{(}\sum_{ i=0}^{n}\epsilon^{i}\Big{(}\bar{u}_{i}^{(\mp)}(\mathbf{y}+\epsilon^{2} \sigma\mathbf{n}_{\gamma}(\mathbf{y}))+Q_{i}^{(\mp)}u(\varepsilon\tau, \mathbf{y})+M_{i}^{(\mp)}u(\sigma,\mathbf{y})\Big{)},\] \[\Delta =\frac{1}{\varepsilon^{4}}\frac{\partial^{2}}{\partial\sigma^{2}}- \frac{1}{\varepsilon^{2}}k(\mathbf{y})\frac{\partial}{\partial\sigma}+\sum_{i \geq 0}\varepsilon^{2i}L_{i}(\sigma,\mathbf{y}),\]0 \[\Delta =\frac{1}{\varepsilon^{4}}\frac{\partial^{2}}{\partial\sigma^{2}}- \frac{1}{\varepsilon^{2}}k(\mathbf{y})\frac{\partial}{\partial\sigma}+\sum_{i \geq 0}\varepsilon^{2i}L_{i}(\sigma,\mathbf{y}),\]1 \[\Delta =\frac{1}{\varepsilon^{4}}\frac{\partial^{2}}{\partial\sigma^{2}}- \frac{1}{\varepsilon^{2}}k(\mathbf{y})\frac{\partial}{\partial\sigma}+\sum_{i \geq 0}\varepsilon^{2i}L_{i}(\sigma,\mathbf{y}),\]2  
The functions \(M^{(\mp)}g(\sigma,\mathbf{y},\epsilon)\) are defined by analogy with \(M^{(\mp)}f(\sigma,\mathbf{y},\epsilon)\). Proceeding in this way, we will obtain differential equations for the functions \(Q_{i}^{(\mp)}v\), \(M_{i}^{(\mp)}u\), \(M_{i}^{(\mp)}v\) and finite equations for the functions \(Q_{i}^{(\mp)}u\), \(i=0,1,\,\dots\). We set the supplementary condition of decreasing to zero at infinity for the differential equations and also the conditions at \(\tau=0\) and \(\sigma=0\) for the functions \(Q_{i}^{(\mp)}v\), \(M_{i}^{(\mp)}u\), in such a way that the equalities in Equation (15) are satisfied in the \(i\)th order. The functions \(M_{0}^{(\mp)}v(\sigma,\mathbf{y})\) and \(M_{1}^{(\mp)}v(\sigma,\mathbf{y})\) turn out to be trivial.