Singularly perturbed ordinary differential equation with turning point and interior layer (Q2062110)

The authors consider a linear two-point boundary value problem for singularly perturbed ordinary differential equation with a turning point at \(x=0\) of the form \[ \varepsilon y_{\varepsilon}''(x) + xy_{\varepsilon}'(x) - xy_{\varepsilon}(x) = f_{\varepsilon}(x), \quad -1\leq x\leq 1, \] \[ y_{\varepsilon}(-1) = a, \quad y_{\varepsilon}(1) = b, \] where \(0<\varepsilon\ll1\) and \(f_{\varepsilon}(x)=\sum\limits_{k=0}^{\infty}\varepsilon^k f_k(x),\) \(f_k\in C^{\infty}[-1,1]\). Using the boundary layer method of Goldenveizer-Vishik-Lyusternik-Vasilyeva-Imanalieva, the uniform asymptotic expansion in the parameter \(\varepsilon\) of the solutions with and inner layer to the Dirichlet problem formulated above was constructed.