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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