Geodesic
On singularly perturbed systems of~ODE with a~multiple root of~the degenerate equation
Izvestiya. Mathematics , Tome 84 (2020) no. 2, pp. 262-290.

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We consider a boundary-value problem for a system of two second-order ODE with distinct powers of a small parameter at the second derivative in the first and second equations. When one of the two equations of the degenerate system has a double root, the asymptotic behaviour of the boundary-layer solution of the boundary-value problem turns out to be qualitatively different from the known asymptotic behaviour in the case when those equations have simple roots. In particular, the scales of the boundary-layer variables and the very algorithm for constructing the boundary-layer series depend on the type of the boundary conditions for the unknown functions. We construct and justify asymptotic expansions of the boundary-layer solution for boundary conditions of a particular type. These expansions differ from those for other boundary conditions.
Keywords: singularly perturbed boundary-value problems, boundary layer, asymptotics in a small parameter, the case of a multiple root of the degenerate equation. 
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     title = {On singularly perturbed systems {of~ODE} with a~multiple root of~the degenerate equation},
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V. F. Butuzov. On singularly perturbed systems of~ODE with a~multiple root of~the degenerate equation. Izvestiya. Mathematics , Tome 84 (2020) no. 2, pp. 262-290. http://geodesic.mathdoc.fr/item/IM2_2020_84_2_a2/

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