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

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.

Periodical in time problems arise in many areas of mathematical physics and fluid mechanics [1][2][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][7][8].
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. then The Scientific World Journal provided that Proof. Inequality (4) can be easily obtained by using first order differential inequality containing initial condition.

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

Note 2.
As it is seen from (6)
Next for = 2, 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 = 2 now easily can be seen by using (32)-(34) in (35).
The solution of this problem is given by For the first derivative we have is unbounded while values are tending to zero. Therefore we observe here the accordance in our theoretical results described above.