On a singular perturbation problem with two second-order turning points
The paper deals with the singularly perturbed equation \[ \varepsilon p(x)u_{xx} + 2(x-a)(b-x)u_x -2su = 0, \;x \in (0,1), \tag{1} \] where \(0 < a < b < 1,\) \(\varepsilon\) is small positive parameter and \(s = \text{const}.\) The functions \(p(x)\) and \(q(x)\) are strictly positive and sufficiently smooth on the interval \((0,1).\) Equation (1) has two second-order turning points \(x=a\) and \(x=b\) on \((0,1).\) The author gives uniform asymptotic solutions for (1) on the whole interval \((0,1).\) The solutions obtained are applied to the study of the boundary value problem \(u(0)=k_1, u(1)=k_2\) for equation (1).
scientific article; zbMATH DE number 5015182
scientific article