Research Article | Open Access

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

**Academic Editor:**G. Tsiatas

#### 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 [1–3]. 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 [6–8].

We note that periodical in space variable problems and also their approximate solutions were investigated by many authors (see, e.g., [9–13]).

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.

#### References

- E. Chadwick and R. El-Mazuzi, “A coupled far-field formulation for time-periodic numerical problems in fluid dynamics,”
*Proceedings of the Indian Academy of Sciences*, vol. 122, no. 4, pp. 661–672, 2012. View at: Google Scholar - T. Iwabuchi and R. Takada, “Time periodic solutions to the Navier-Stokes equations in the rotational framework,”
*Journal of Evolution Equations*, vol. 12, no. 4, pp. 985–1000, 2012. View at: Google Scholar - S. Ji and Y. Li, “Time periodic solutions to the one-dimensional nonlinear wave equation,”
*Archive for Rational Mechanics and Analysis*, vol. 199, no. 2, pp. 435–451, 2011. View at: Publisher Site | Google Scholar - E. A. Butcher, H. Ma, E. Bueler, V. Averina, and Z. Szabo, “Stability of linear time-periodic delay-differential equations via Chebyshev polynomials,”
*International Journal for Numerical Methods in Engineering*, vol. 59, no. 7, pp. 895–922, 2004. View at: Google Scholar - H. Hartono and A. H. P. Van Der Burgh, “A linear differential equation with a time-periodic damping coefficient: stability diagram and an application,”
*Journal of Engineering Mathematics*, vol. 49, no. 2, pp. 99–112, 2004. View at: Publisher Site | Google Scholar - G. M. Amiraliyev and H. Duru, “A uniformly convergent difference method for the periodical boundary value problem,”
*Computers and Mathematics with Applications*, vol. 46, no. 5-6, pp. 695–703, 2003. View at: Publisher Site | Google Scholar - Z. Cen, “Uniformly convergent second-order difference scheme for a singularly perturbed periodical boundary value problem,”
*International Journal of Computer Mathematics*, vol. 88, no. 1, pp. 196–206, 2011. View at: Publisher Site | Google Scholar - L. Peng-cheng and J. Ben-xian, “A singular perturbation problem for periodic boundary differential equation,”
*Applied Mathematics and Mechanics*, vol. 8, no. 10, pp. 929–937, 1987. View at: Publisher Site | Google Scholar - G. M. Amiraliyev,
*Towards the Numerical Solution of the Periodical on Time Problem for Pseudo-Parabolic Equation*, Numerical Methods of Analysis Baku State University, 1988. - P. Giesl and H. Wendland, “Approximating the basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem,”
*Discrete and Continuous Dynamical Systems Supplements*, pp. 259–268, 2009, 7th AIMS Conference. View at: Google Scholar - J. J. Hench and A. J. Laub, “Numerical solution of the discrete-time periodic Riccati equation,”
*IEEE Transactions on Automatic Control*, vol. 39, no. 6, pp. 1197–1210, 1994. View at: Publisher Site | Google Scholar - C. V. Pao, “Numerical methods for time-periodic solutions of nonlinear parabolic boundary value problems,”
*SIAM Journal on Numerical Analysis*, vol. 39, no. 2, pp. 647–667, 2002. View at: Publisher Site | Google Scholar - E. V. Zemlyanaya and N. V. Alexeeva, “Numerical study of time-periodic solitons in the damped-driven NLS,”
*International Journal of Numerical Analysis & Modeling, Series B*, vol. 2, no. 2-3, pp. 248–261, 2011. View at: Google Scholar - G. M. Amiraliyev, “Difference method for a singularly perturbed initial value problem,”
*Turkish Journal of Mathematics*, vol. 22, no. 3, pp. 283–294, 1998. View at: Google Scholar - L. Herrmann, “Periodic solutions to a one-dimentional strongly nonlinear wave equation with strong dissipation,”
*Czechoslovak Mathematical Journal*, vol. 35, no. 2, pp. 278–293, 1985. View at: Google Scholar

#### Copyright

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.