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Author: Butuzov, Valentin F. × Subject: asymptotic expansion ×

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    On a singularly perturbed initial value problem in case of a double root of the degenerate equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Butuzov, Valentin F.; Nefedov, Nikolai N.; Recke, Lutz; Schneider, Klaus R.
    We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptotic solution will be used to construct lower and upper solutions guaranteeing the existence of a unique solution and justifying its asymptotic expansion.
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    Asymptotics and stability of a periodic solution to a singularly perturbed parabolic problem in case of a double root of the degenerate equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Butuzov, Valentin F.; Nefedov, Nikolai N.; Recke, Lutz; Schneider, Klaus R.
    For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in the small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction.