DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

ASYMPTOTIC ANALYSIS OF TWO COUPLED
REACTION-DIFFUSION PROBLEMS
DIALLA KONATE
The current paper is a contribution to the attempt to have a systematic and mathematically
rigorous method to solve singularly perturbed equations using asymptotic expansions.
We are dealing with a system of coupled and singularly perturbed second-order
evolution equations. This problem has not collected much efforts in the literature yet.
Starting from the classical outer expansion (the current problem exhibits two disjoint
boundary layers), using the Hilbert spaces approach, we develop a systematic and rigorous
method to construct a higher-order corrector which results in a uniform approximation
solution to any prescribed order which is valid throughout the geometric domain of
interest, say Ω. To solve the given problem, the literature provides a dedicated numerical
method of Shiskin-type (cf. [9]). The system of equations under consideration is related
to actual problems arising in physics, chemistry, and engineering where the use of new
scientific instruments requires highly accurate approximation solutions which are out of
reach of known numerical methods. This makes it necessary and useful to develop analytical
solutions which are easy to computerize. The author has developed and successfully
applied the strategy used in the current paper to various classes of problems (cf. [5–7]).
This strategy works very well on linear problems of all types thanks to the availability of
a priori estimates. In this regard the current paper is self-contained. For more on a priori
estimates related to some other families of singularly perturbed problems, the reader
may refer to Gartland [2, 3], Geel [4]. The reader who is not very familiar with singular
perturbation may get some basics from Eckhaus [1], Lions [8], and O’Malley [10].
Copyright © 2006 Dialla Konate. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Consider the problem
⎛
ε U(x) = ⎜
⎝
−ε d2
⎞
dx 2 0
⎟
0 −ε d2 ⎠U(x)+A(x)U(x) = F,
dx 2
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 575–592