A Priori Estimates of Solution of Parametrized Singularly Perturbed Problem ()
Received 10 September 2015; accepted 17 January 2016; published 20 January 2016
1. Introduction
In this paper, we are going to obtain the asymptotic bounds for the following parameterized singularly perturbed boundary value problem (BVP):
(1.1)
(1.2)
where 
is the perturbation parameter, 
are given constants and 
is a sufficiently smooth function in
. Further, the function 
is assumed to be sufficiently continuously differentiable for our purpose function in 
and
(1.3)
By a solution of (1.1), (1.2), we mean pair 
for which problem (1.1), (1.2) is satisfied.
An overview of some existence and uniqueness results and applications of parameterized equations may be obtained, for example, in [1] - [10] . In [11] - [14] have also been considered some approxi-mating aspects of this kind of problems. The qualitative analysis of singular perturbation situations has always been far from trivial because of the boundary layer behavior of the solution. In singular perturbation cases, problems depend on a small parameter 
in such a way that the solution exhibits a multiscale character, i.e., there are thin transition layers where the solution varies rapidly while away from layers it behaves regularly and varies slowly [15] -[18] . In this note, we establish the boundary layer behaviour for 
of the solution of (1.1)-(1.2) and its first and second derivatives. Example that agrees with the analytical results is given.
2. The Continuous Problem
Lemma 2.1. Let 
and 
be the continuous functions on
. Then, the solution of the boundary-value problem
![]()
(2.1)
![]()
(2.2)
satisfies the inequality
![]()
(2.3)
where
![]()
Proof. Under the above conditions, the operatör ![]()
admits the folloving maximum principle:
Suppose ![]()
be any function satisfiying ![]()
![]()
, ![]()
and![]()
. Then, ![]()
for all![]()
.
Now, for the barrier fonction
![]()
taking also into consideration that, ![]()
is a solution of the problem
![]()
it follows that,
![]()
![]()
![]()
therefore![]()
, which immediayely leads to (2.3).
Remark 1. The inequality (2.3) yields.
![]()
(2.4)
Theorem 2.1. For ![]()
and under conditions (1.3), the solution ![]()
of the problem (1.1), (1.2), satisfies,
![]()
(2.5)
![]()
(2.6)
where
![]()
![]()
and
![]()
(2.7)
provided ![]()
and ![]()
for ![]()
and![]()
.
Proof. We rewrite Equation (1.1) in form
![]()
(2.8)
where, ![]()
intermediate values.
From (2.8) for the first derivate, we have
![]()
(2.9)
from which, after using the initial condition![]()
, it follows that,
![]()
(2.10)
Applying the mean value theorem for integrals, we deduce that,
![]()
(2.11)
and
![]()
(2.12)
Also, for first and second terms in right side of (2.10) for ![]()
values, we have
![]()
(2.13)
It then follows from (2.11)-(2.13),
![]()
(2.14)
Further from (2.4) by taking ![]()
we get
![]()
(2.15)
The inequlities (2.14), (2.15) immediately leads to (2.5), (2.6). After taking into consideration the uniformly boundnees in ![]()
of ![]()
and![]()
, it then follows from (2.9) that,
![]()
which proves (2.7) for![]()
. To obtain (2.7) for![]()
, first from (1.1) we have
![]()
from which after taking into consideration here ![]()
and (2.5) we obtain
![]()
(2.16)
Next, differentiation (1.1) gives
![]()
(2.17)
![]()
(2.18)
with
![]()
![]()
and due to our assumptions clearly,
![]()
Consequently, from (2.17), (2.18) we have
![]()
which proves (2.7) for![]()
. □
Example. Consider the following parameterized singular perturbation problem:
![]()
![]()
with
![]()
and ![]()
selected so that the solution is
![]()
where,
![]()
First and second derivatives have the form
![]()
Therefore, we observe here the accordance in our theoretical results described above.