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The Scientific World Journal
Volume 2013 (2013), Article ID 301609, 4 pages
http://dx.doi.org/10.1155/2013/301609
Research Article

Asymptotic Bounds for the Time-Periodic Solutions to the Singularly Perturbed Ordinary Differential Equations

Department of Mathematics, Sinop University, 57000 Sinop, Turkey

Received 3 October 2013; Accepted 24 October 2013

Academic Editors: F. Mukhamedov, G. Tsiatas, and H. Yang

Copyright © 2013 Gabil M. Amiraliyev and Aysenur Ucar. 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.

Abstract

The periodical in time problem for singularly perturbed second order linear ordinary differential equation is considered. The boundary layer behavior of the solution and its first and second derivatives have been established. An example supporting the theoretical analysis is presented.

1. Introduction and Preliminaries

In this paper we investigate the equation

with the periodic conditions

where is the perturbation parameter, , , and are the -periodic functions satisfying .

Periodical in time problems arise in many areas of mathematical physics and fluid mechanics [13]. Various properties of periodical in time problems in the absence of boundary layers have been investigated earlier by many authors (see, e.g., [4, 5] and references therein).

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; that is, there are thin transition layers where the solution varies rapidly while away from layers and it behaves regularly and varies slowly [68].

We note that periodical in space variable problems and also their approximate solutions were investigated by many authors (see, e.g., [913]).

In this note we establish the boundary layer behaviour for of the solution of (1)-(2) and its first and second derivatives. The maximum principle, which is usually used for periodical boundary value problems, is not applicable here; because of this we use another approach which is convenient for this type of problems. The approach used here is similar to those in [9, 14, 15].

Note 1. Throughout the paper denotes the generic positive constants independent of . Such a subscripted constant is also independent of , but its value is fixed.

Lemma 1. Let be the continuous function defined on and and , are given constants. If
then
provided that

Proof. Inequality (4) can be easily obtained by using first order differential inequality containing initial condition.

2. Asymptotic Estimate

We now give a priori bounds on the solution and its derivatives for problem (1)-(2).

Theorem 2. The solution of the problem (1)-(2) satisfies the bound
provided that
where

Proof. Consider the identity
with parameter which will be chosen later. By using the equalities
and the inequalities
in (9), we have
Denoting now and choosing , we arrive at
After taking , the last inequality reduces to
where
On the other hand for the function holds the following inequality clearly:
For the right-hand side of inequality (12), we have
Taking into account and , after choosing and , we have
where
From (18) by using Lemma 1, we have
which proves Theorem 2.

Note 2. As it is seen from (6)
where

Theorem 3. Under the assumptions of Theorem 2, the following asymptotic estimates for the derivatives hold true:

Proof. The case directly follows from the identity (4).
For , the problem (1)-(2) can be rewritten as
where
and by virtue of Theorem 2
The solution of (24)–(26) can be expressed as
and taking into account (26), we have
Thus we get
The relation (29) along with (31) leads to (23) for immediately.
Next for , from (1) we have
Differentiating now (1), we obtain
Under the smoothness conditions on data functions and boundness of and , we deduce evidently
The solution of (33) is
The validity of (23) for now easily can be seen by using (32)–(34) in (35).

3. Example

Consider the particular problem with

The solution of this problem is given by

where

For the first derivative we have

from which it is clear that the first derivative of is uniformly bounded but has a boundary layer near of thickness .

The second derivative

is unbounded while values are tending to zero.

Therefore we observe here the accordance in our theoretical results described above.

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