TURK 2025

Consider the Cauchy problem for a system of ordinary differential equations with a small parameter:
\begin{equation}
L_\varepsilon u(x,\varepsilon) \equiv \varepsilon u'(x) + A(x)u(x) = f(x), \quad x \in (0,1], \quad u(0,\varepsilon) = u^0 \tag{1}
\end{equation}
under the following assumptions: the matrix $A(x) \in C^\infty\left([0,1], \mathbb{C}^{n \times n}\right)$ is positive definite, and $f(x) \in C^\infty\left([0,1], \mathbb{C}^n\right)$.

Let us perform an extension of problem (1) by introducing the regularizing variable $\xi = \frac{x}{\varepsilon}$ and then we obtain the extended problem:
\begin{equation}
\widetilde{L}\varepsilon \widetilde{u}(x,\xi,\varepsilon) \equiv \varepsilon \partial_x \widetilde{u} + \partial\xi \widetilde{u} + A(x)\widetilde{u} = f(x), \quad (x,\xi) \in \Omega, \quad \widetilde{u}(0,0,\varepsilon) = u^0. \tag{2}
\end{equation}