Asymptotic Estimates for Second-Order Parameterized Singularly Perturbed Problem ()
1. Introduction
In this paper, we are going to obtain the asymptotıc bounds for the following parameterized singularly perturbed boundary value problem (BVP):
(1)
(2)
where 
is a 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
. (3)
By a solution of (1), (2) we mean pair 
for which problem (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] , some approximating aspects of this kind of problems have also been considered. The qualitative analysis of singular perturbation situations have 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] [16] . In this note we establish the boundary layer behaviour for 
of the solution of (1)-(2) and its first and second derivatives. Example that agree with the analytical results is given.
Theorem 1.1. For 
the solution 
of the problem (1), (2) satisfies,
(4)
(5)
where
![]()
,
![]()
and
![]()
(6)
provided ![]()
and ![]()
for ![]()
and ![]()
![]()
.
Proof. We rewrite Equation (1) in the form
![]()
, (7)
where,
![]()
, ![]()
, ![]()
,![]()
![]()
―intermediate values.
From (7) for the first derivate, we have
![]()
(8)
Integrating this equality over ![]()
we get
![]()
(9)
from which by setting the boundary condition ![]()
we obtain,
![]()
(10)
Applying the mean value theorem for integrals, we deduce that,
![]()
(11)
and
![]()
(12)
Also, for first and second terms in right side of (10), for ![]()
values, we have
![]()
(13)
It then follows from (11)-(13)
![]()
(14)
Next from (9), we see that
![]()
Under the conditions ![]()
and ![]()
the operator ![]()
admits the following maximum principle: Suppose ![]()
be any function satisfiying ![]()
, ![]()
and ![]()
then ![]()
Using the maximum principle whith barrier functions ![]()
we have the inequality
![]()
(15)
The inequlities (14), (15) immediately leads to (4), (5). After taking into consideration the uniformly boundnees in ![]()
of ![]()
and ![]()
it then follows from (8) that,
![]()
,
which proves (6) for ![]()
To obtain (6) for![]()
, first from Equation (1) we have
![]()
,
from which after taking into consideration here ![]()
and (4)
![]()
(16)
Next, differentiation (1) gives
![]()
(17)
![]()
(18)
with
![]()
![]()
,
and due to our assumptions clearly,
![]()
Consequently, from (17), (18) we have
![]()
which proves (6) for![]()
.
Example. Consider the particular problem
![]()
![]()
where, ![]()
and ![]()
selected so that the solution is
![]()
with
![]()
First and second derivatives have the form
![]()
Therefore we observe here the accordance in our theoretical results described above.