Perturbation problems with quadratic dependence on the first derivative (Q698851)

This paper presents the singular perturbation problem \[ \varepsilon x''=a(t,x){x'}^2+b(t,x)x'+c(t,x), \quad x(0)=A, x(1)=B, \] where \(\varepsilon>0\) is a small parameter. Using the classical lower and upper solution method, the author proves the existence of the solution \[ x(t,\varepsilon)=w(t)+\theta(t,\varepsilon)+O(\varepsilon\ln \varepsilon), \] where \(w(t)\) is a reduced solution and \(\theta(t,\varepsilon)\) a boundary layer correction. Using an alternative boundary layer correction \(\psi(t,\varepsilon)\), which is more difficult to compute, he proves the estimate \[ x(t,\varepsilon)=w(t)+\psi(t,\varepsilon)+O(\varepsilon). \] The similar problems \[ \varepsilon (x''-a(t,x){x'}^2)=b(t,x)x'+c(t,x), \quad x(0)=A, x(1)=B, \] and \[ \varepsilon (x''-a(t,x){x'}^2-b(t,x)x')=c(t,x), \quad x(0)=A, x(1)=B, \] are investigated along the same lines. In each case, the results are illustrated with examples that have a physical background.