Asymptotics of the Solution of a Singularly Perturbed System of Equations with a Multizone Internal Layer

We consider a boundary value problem for a singularly perturbed system of two second-order ordinary differential equations with distinct powers of the small parameter multiplying the second derivatives. The problem has the specific feature that one of the two equations in the degenerate system has three disjoint roots, one being double and the other two being simple. We prove that for sufficiently small parameter values the problem has a solution with a rapid transition from the double root of the degenerate equation to a simple root in a neighborhood of some interior point of the interval. We construct and justify the complete asymptotic expansion of this solution. This expansion differs qualitatively from the well-known one for the case in which all roots of the degenerate equation are simple. In particular, the expansion is in fractional rather than integer powers of the small parameter, the boundary layer variables have a different scale, and the transition layer consists of six zones.