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The Scientific World Journal
Volume 2013, Article ID 301609, 4 pages
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.


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
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

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
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
From (18) by using Lemma 1, we have
which proves Theorem 2.

Note 2. As it is seen from (6)

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
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


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.


  1. 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
  2. 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
  3. 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 · View at Google Scholar · View at Scopus
  4. 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 · View at Scopus
  5. 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 · View at Google Scholar · View at Scopus
  6. 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 · View at Google Scholar · View at Scopus
  7. 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 · View at Google Scholar · View at Scopus
  8. 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 · View at Google Scholar · View at Scopus
  9. 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.
  10. 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
  11. 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 · View at Google Scholar · View at Scopus
  12. 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 · View at Google Scholar · View at Scopus
  13. 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
  14. 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 · View at Scopus
  15. 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