System of singularly perturbed equations with differential turning point of the first kind (Q2356357)

A uniform asymptotic expansions of the solutions to a system of differential equations with a small parameter at the highest derivative and turning point is constructed. The system of singularly perturbed differential equations (SSPDE) under consideration is: \[ \varepsilon Y'(x,\varepsilon)-A(x,\varepsilon)Y(x,\varepsilon)=H(x), \] where \(\varepsilon>0\) is a small parameter, \(x\in[0,l]\), \(Y(x,\varepsilon)=(y_1(x,\varepsilon), y_2(x,\varepsilon), y_3(x, \varepsilon))^T\) (the superscript \(T\) stands for the transpose of a vector or a matrix)\break \(H(x)=(0, 0, h(x))^T\), \(A(x,\varepsilon)=A_0(x,\varepsilon)+\varepsilon A_1(x,\varepsilon)\) with \[ A_0(x,\varepsilon)=\left(\begin{matrix} 0 &0&0\\ 0&0&1\\ -b(x)&-a(x)&0\end{matrix}\right), A_1(x, \varepsilon)=\left(\begin{matrix} 0&1&0\\ 0&0&0\\ 0&0&0\end{matrix}\right) \] \(a(x), b(x)\) being smooth functions with specific structure. First, a formal solution of the homogeneous equation is constructed and further a formal particular solution of the inhomogeneous extended equation is derived. Finally, an estimation of the remainders of the asymptotic expansions of the solution is provided. The specific feature of the SSPDE investigated in this paper, is that the spectrum of the boundary operator contains multiple elements and elements vanishing identically.