The asymptotics of solutions of a singularly perturbed equation with a of fractional turning point (Q1709115)

The paper deals with constructing uniform asymptotic expansions of solutions for two singularly perturbed problems (\(0<\varepsilon\ll1\)): \[ \varepsilon y_{\varepsilon}'(x)+x^{\alpha}p(x)y_{\varepsilon}(x)=f(x),\;x\in(0,T], \] \[ y_{\varepsilon}(0)=y^0\in\mathbb{R} \] and \[ \varepsilon y_{\varepsilon}''(x)-x^{\alpha}p(x)y_{\varepsilon}(x)=f(x),\;x\in(0,1), \] \[ y_{\varepsilon}(0)=0,\;\;y_{\varepsilon}(1)=0 \] with \(p,f\in C^{\infty},\) \(p>0,\) \(0<\alpha\in\mathbb{Q},\) and whose solution for \(\varepsilon=0\) has a singularity at \(x=0.\) In the paper, the case \(p(x)\equiv1\) and \(\alpha=1/2\) is analyzed in detail. The main idea is based on the Vishik-Lyusternik-Vasil'eva-Imanaliev boundary function method. The appropriateness of the approach is confirmed by the remainder asymptotics analysis.