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, HumboldtUniversität zu Berlin (ISSN 08630976),
Oleh Omel'chenko
,
Lutz Recke
Preprint series:
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976),
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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