Asymptotic solution of the Robin problem with a regularly singular point
In this paper, the authors investigate the Robin boundary-value problem for a singularly perturbed linear inhomogeneous problem \[\begin{aligned} \varepsilon y_{\varepsilon}''(x)+xp(x)y_{\varepsilon}'(x)-q(x)y_{\varepsilon}(x)&=f(x), \quad &&0\leq x\leq 1, \ 0<\varepsilon\ll 1,\\ y_{\varepsilon}(0)-h_1y_{\varepsilon}'(0)&=a, &&0\leq h_1,\\ y_{\varepsilon}(1)+h_2y_{\varepsilon}'(1)&=b, &&0\leq h_2, \end{aligned} \] where \(p(x),q(x)>0\) on \([0,1],\) \(p,q,f\) are of the class \(C^{\infty},\) \(p(0)=1,\) \(q(0)=2,\) and \(a,b,h_1, h_2\) are known constants, \(p(1)+h_2q(1)\neq 0.\) The authors construct a uniform asymptotic expansion of the solution \(y_{\varepsilon}(x)\) of the Robin problem with an arbitrary degree of accuracy on the interval \([0,1]\), when \(\varepsilon\to 0^+\).
scientific article; zbMATH DE number 7350139
scientific article