On an internal boundary layer problem
The authors consider the singular perturbation problem \[ \varepsilon y''+a(x)y'+b(x)y=0, \quad y(x_-)=A,\;y(x_+)=B, \tag{1} \] where \(x_-\leq 0\leq x_+\) and \(\varepsilon>0\) is a small parameter. They investigate a turning point problem such that \(a(x)=0\) if and only if \(x=0\), and near \(x=0\), \(a(x)\sim \alpha x\), \(b(x)\sim \beta\), with \(\alpha >0\) and \(\beta/\alpha\neq 1,2,\ldots\). The existence of a unique solution for small values of \(\varepsilon>0\) together with its asymptotic behavior is established. The idea of the approach is to use a change of variables so as to consider equation (1) as a perturbation of \[ \varepsilon u''+\alpha tu'+\beta u=0, \] the general solution to which can be written using parabolic cylinder functions. Problem (1) is then written as an integral equation and, for small values of \(\varepsilon\), its solution can be obtained using successive approximations.
scientific article; zbMATH DE number 1787665