singular perturbation asymptotic expansion boundary layer implicit function theorem Boundary Layer Solutions to Singularly Perturbed Problems via the Implicit Function Theorem Oleh Omel'chenko Omel'chenko Oleh Lutz Recke Recke Lutz Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

Boundary Layer Solutions to Singularly Perturbed Problems via the Implicit Function Theorem

Oleh Omel'chenko , Lutz Recke

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

34E15 Singular perturbations, general theory
34B15 Nonlinear boundary value problems
47J07 Abstract inverse mapping and implicit function theorems
58C15 Implicit function theorems; global Newton methods

Abstract
We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed problems of the type $\varepsilon^2 u''=f(x,u,\varepsilon u',\varepsilon), 0< x <1$, with Dirichlet and Neumann boundary conditions. For that we assume that there is given a family of approximate solutions which satisfy the differential equation and the boundary conditions with certain low accuracy. Moreover, we show that, if this accuracy is high, then the closeness of the approximate solution to the exact solution is correspondingly high. The main tool of the proofs is a modification of an Implicit Function Theorem of R.Magnus. Finally we show how to construct approximate solutions under certain natural conditions.


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