Internal layer for a singularly perturbed equation with discontinuous right-hand side (Q2212353)

In this paper, the authors establish sufficient conditions for existence of the smooth solution with internal transition layer for the singularly perturbed Dirichlet boundary value problem \[ \mu y''=f(y,t,\mu),\ 0\leq t\leq 1, \] \[ y(0,\mu)=y^0, \ y(1,\mu)=y^1, \] where the right-hand side of the equation is discontinuous along some curve \(y=h(t).\) The authors construct a formal asymptotic approximation of the solution to the problem above that has an internal transition layer separately for \(t\in[0,t^*]\) and for \(t\in[t^* ,1],\) by continuously matching these expansions at the point \(t^*\) assuming that the solution \(y(t,\mu)\) takes the value \(h(t^*)\) at this point. The theory is illustrated by an example.