On a double boundary layer in a nonlinear boundary value problem (Q1675325)

In this paper, the authors analyze the Dirichlet problem for the singularly perturbed second-order ordinary differential equation \[ \varepsilon^2 u_{xx}=u^2+\mu\left[ K(x,\mu)u+F(u,\varepsilon u_x,x,\mu)\right],\;0< x < L,\;0<\varepsilon\ll 1, \] \[ u(0)=\alpha(\mu),\;u(L)=\beta(\mu), \] where \(\mu=\varepsilon^{1/2}\) or \(\mu=\varepsilon,\) and \(K, F,\) \(\alpha\) and \(\beta\) are smooth functions in all variables. Using the asymptotic expansion techniques, the authors construct in the form of boundary layer expansion the formal (asymptotic) solution for the problem under consideration which provides its approximation for \(\varepsilon\rightarrow 0^+\) on the interval \([0,L]\).