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

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 𝐿𝑢≡𝜀𝑢^′′+𝑎(𝑡)𝑢^′+𝑏(𝑡)𝑢=𝑓(𝑡),0<𝑡<𝑇, (1)

with the periodic conditions 𝑢(0)=𝑢(𝑇),𝑢^′(0)=𝑢^′(𝑇), (2)

where 𝜀∈(0,1] is the perturbation parameter, 0<𝛼≤𝑎(𝑡)≤𝑎^∗, 0<𝛽≤𝑏(𝑡)≤𝑏^∗, and 𝑓(𝑡) are the 𝑇-periodic functions satisfying 𝑎,𝑏,𝑓∈𝐶¹[0,𝑇].

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 𝜀 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 𝑢(𝑡) 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 𝐶 denotes the generic positive constants independent of 𝜀. Such a subscripted constant is also independent of 𝜀, but its value is fixed.

Lemma 1. Let 𝛿(𝑡)≥0 be the continuous function defined on [0,𝑇] and 𝑐₀(𝑡),𝜌(𝑡)∈𝐶[0,𝑇] and 𝛾, 𝜇 are given constants. If 𝛿^′(𝑡)+𝑐₀(𝑡)𝛿(𝑡)≤𝜌(𝑡),𝛿(0)≤𝜇𝛿(𝑇)+𝛾, (3)
then 𝛿(𝑡)≤1−𝜇𝑒^{−∫𝑇0𝑐0(𝑠)𝑑𝑠−1}×𝛾𝑒^{−∫𝑇0𝑐0(𝜂)𝑑𝜂}+^𝑇₀𝜌(𝑠)𝑒^{−∫𝑇𝑠𝑐0(𝜂)𝑑𝜂}𝑑𝑠𝑒^{−∫𝑇0𝑐0(𝜂)𝑑𝜂}+^𝑡₀𝜌(𝑠)𝑒^{−∫𝑇𝑠𝑐0(𝜂)𝑑𝜂}𝑑𝑠 (4)
provided that 1−𝜇𝑒^{−∫𝑇0𝑐0(𝑠)𝑑𝑠}>0. (5)

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 𝑢(𝑡) of the problem (1)-(2) satisfies the bound 𝜀𝑢^′2+|𝑢|²≤𝐶^𝑡₀𝑓(𝑠)²𝑑𝑠, (6)
provided that 𝛾=𝜆₀min_[]0,𝑇2𝑏(𝑡)−𝑎^′(𝑡)−𝑏^∗>0, (7)
where 𝑏^∗=max_[]0,𝑇𝑏^′(𝑡),0<𝜆₀<𝛼+𝛼²+8𝛽4. (8)

Proof. Consider the identity 𝐿𝑢𝑢^′+𝜆𝑢=𝑢^′+𝜆𝑢𝑓(𝑡) (9)
with parameter 𝜆>0 which will be chosen later. By using the equalities 𝜀𝑢^′′𝑢^′=𝜀2𝑢^′2′,𝜆𝑢^′′𝑢=𝜆𝑢^′𝑢^′−𝜆𝑢^′2,𝜆𝑎(𝑡)𝑢^′𝜆𝑢=2𝑎(𝑡)𝑢^2′=𝜆2𝑎(𝑡)𝑢^2′−𝜆2𝑎^′(𝑡)𝑢²,𝑏(𝑡)𝑢^′1𝑢=2𝑏(𝑡)𝑢^2′=12𝑏(𝑡)𝑢^2′−12𝑏^′(𝑡)𝑢², (10)
and the inequalities 𝑢^′𝑓(𝑡)≤𝜇₁𝑢^′2+14𝜇₁𝑓²(𝑡),𝜇₁(𝑡)>0,𝜆𝑢𝑓≤𝜆𝜇₂𝑢²+𝜆4𝜇₂𝑓²(𝑡),𝜇₂>0, (11)
in (9), we have 𝜀𝑢^′2+2𝜀𝜆𝑢^′𝑢+𝜆𝑎(𝑡)𝑢²+𝑏(𝑡)𝑢^2′≤−2𝑎(𝑡)−𝜀𝜆−𝜇₁𝑢^′2+𝑏^′(𝑡)+𝜆𝑎^′(𝑡)−2𝜆𝑏(𝑡)+2𝜆𝜇₂𝑢²+12𝜇₁+𝜆2𝜇₂𝑓²(𝑡). (12)
Denoting now 𝛿(𝑡)=𝜀𝑢^′2+2𝜀𝜆𝑢^′𝑢+𝜆𝑎(𝑡)𝑢²+𝑏(𝑡)𝑢² and choosing 𝜇=1/2, we arrive at 𝛿(𝑡)≥𝜀2𝑢^′2+𝛽+𝜆(𝛼−2𝜆𝜀)𝑢². (13)
After taking 𝜆=𝜆₀<(𝛼+𝛼²+8𝛽)/4, the last inequality reduces to 𝛿(𝑡)≥𝐶₀𝜀𝑢^′2+𝑢², (14)
where 0<𝐶₀1=min2,𝛽+𝜆₀𝛼−2𝜆₀𝜀. (15)
On the other hand for the function 𝛿(𝑡) holds the following inequality clearly: 𝛿(𝑡)≤𝜀(1+𝜆)𝑢^′2+𝑏^∗+𝜀𝜆+𝜆𝑎^∗𝑢²≤𝜀1+𝜆₀𝑢^′2+𝑏^∗+𝜆₀+𝜆₀𝑎^∗𝑢². (16)
For the right-hand side of inequality (12), we have 2𝑎(𝑡)−𝜀𝜆−𝜇₁𝑢^′2+−𝑏^′(𝑡)−𝜆𝑎^′(𝑡)+2𝜆𝑏(𝑡)−2𝜆𝜇₂𝑢²≥2𝜀𝛼−𝜀𝜆₀−𝜇₁𝑢^′2+−𝑏^∗−𝜆₀𝑎^′(𝑡)+2𝜆₀𝑏(𝑡)−2𝜆₀𝜇₂𝑢²≥𝛼𝜀𝑢^′2+𝛾2𝑢². (17)
Taking into account 𝜀≤1 and 𝛾>0, after choosing 𝜇₁=(𝛼−2𝜆₀)/2 and 𝜇₂=𝛾/4𝜆₀, we have 𝛿^′(𝑡)≤−𝐶₁𝛿(𝑡)+𝜌(𝑡),𝛿(0)=𝛿(𝑇), (18)
where 0<𝐶₁𝛼=min1+𝜆₀,𝛾2𝑏^∗+𝜆₀+𝜆₀𝑎^∗,𝜌(𝑡)=12𝜇₁+𝜆2𝜇₂𝑓²(𝑡). (19)
From (18) by using Lemma 1, we have 𝛿(𝑡)≤𝑒^−𝐶1𝑡1−𝑒^{−𝐶1𝑇𝑇}₀𝜌(𝑠)𝑒^{−𝐶1(𝑡−𝑠)}𝑑𝑠+^𝑡₀𝜌(𝑠)𝑒^{−𝐶1(𝑡−𝑠)}𝑑𝑠 (20)
which proves Theorem 2.

Note 2. As it is seen from (6) |𝑢(𝑡)|≤𝐶𝑓₂, (21)
where 𝑓₂=^𝑇₀𝑓²(𝑠)^1/2𝑑𝑠. (22)

Theorem 3. Under the assumptions of Theorem 2, the following asymptotic estimates for the derivatives hold true: 𝑢^(𝑘)(𝑡)≤𝐶1+𝜀^1−𝑘𝑒^{−𝛼𝑡/𝜀},0≤𝑡≤𝑇,𝑘=0,1,2. (23)

Proof. The case 𝑘=0 directly follows from the identity (4).
For 𝑘=1, the problem (1)-(2) can be rewritten as 𝜀𝑢^′′+𝑎(𝑡)𝑢^′(𝑡)|𝑢(0𝑢=𝐹,0<𝑡<𝑇,)|≤𝐶,_′(0)=𝑢^′(𝑇), (24)(25)(26)
where 𝐹(𝑡)=𝑓(𝑡)−𝑏(𝑡)𝑢 (27)
and by virtue of Theorem 2|𝐹(𝑡)|≤𝐶. (28)
The solution of (24)–(26) can be expressed as 𝑢^′(𝑡)=𝑢^′(0)𝑒^{−(1/𝜀)∫𝑡0𝑎(𝑠)𝑑𝑠}+1𝜀^𝑡₀𝐹(𝑠)𝑒^{−(1/𝜀)∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑑𝑠, (29)
and taking into account (26), we have 𝑢^′(0)=1−𝑒^{−(1/𝜀)∫𝑇0𝑎(𝑠)𝑑𝑠−1}1𝜀^𝑇₀𝐹(𝑠)𝑒^{−(1/𝜀)∫𝑇𝑠𝑎(𝜉)𝑑𝜉}𝑑𝑠. (30)
Thus we get 𝑢^′(0)≤𝐶𝛼⁻¹1−𝑒^{−𝛼𝑇/𝜀}1−𝑒^{−𝑎∗𝑇/𝜀}≤𝐶𝛼⁻¹. (31)
The relation (29) along with (31) leads to (23) for 𝑘=1 immediately.
Next for 𝑘=2, from (1) we have 𝑢^′′(0=1)𝜀𝑓(0)−𝑏(0)𝑢(0)−𝑎(0)𝑢^′(0≤𝐶)𝜀. (32)
Differentiating now (1), we obtain 𝜀𝑢^′′′+𝑎(𝑡)𝑢^′′=𝑓^′(𝑡)−𝑏^′(𝑡)𝑢−𝑏(𝑡)𝑢^′−𝑎^′(𝑡)𝑢≡𝜑(𝑡). (33)
Under the smoothness conditions on data functions and boundness of 𝑢(𝑡) and 𝑢^′(𝑡), we deduce evidently 𝜑(𝑡)≤𝐶. (34)
The solution of (33) is 𝑢^′′(𝑡)=𝑢^′′(0)𝑒^{−(1/𝜀)∫𝑡0𝑎(𝑠)𝑑𝑠}+1𝜀^𝑡₀𝜑(𝑠)𝑒^{−(1/𝜀)∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑑𝑠. (35)
The validity of (23) for 𝑘=2 now easily can be seen by using (32)–(34) in (35).

3. Example

Consider the particular problem with 𝑎(𝑡)=4,𝑏(𝑡)=3,𝑓(𝑡)=3𝑡,𝑇=1. (36)

The solution of this problem is given by 𝑢(𝑡)=𝐴₁𝑒^{√−((2−4−3𝜀)/𝜀)𝑡}+𝐴₂𝑒^{√−((2+4−3𝜀)/𝜀)𝑡}4+𝑡−3, (37)

where 𝐴₁=2+√4−3𝜀2√4−3𝜀1−𝑒^{√−(2−4−3𝜀)/𝜀},𝐴₂=√4−3𝜀−22√4−3𝜀1−𝑒^{√−(2+4−3𝜀)/𝜀}. (38)

For the first derivative we have 𝑢^′(𝑡)3=−2√4−3𝜀1−𝑒^{√−(2−4−3𝜀)/𝜀}𝑒^{√−((2−4−3𝜀)/𝜀)𝑡}−32√4−3𝜀1−𝑒^{√−(2+4−3𝜀)/𝜀}𝑒^{√−((2+4−3𝜀)/𝜀)𝑡}+1 (39)

from which it is clear that the first derivative of 𝑢(𝑡) is uniformly bounded but has a boundary layer near 𝑡=0 of thickness 𝑂(𝜀).

The second derivative 𝑢^′′(𝑡)3=−𝜀2−√4−3𝜀2√4−3𝜀1−𝑒^{√−(2−4−3𝜀)/𝜀}𝑒^{√−((2−4−3𝜀)/𝜀)𝑡}+2+√4−3𝜀2√4−3𝜀1−𝑒^{√−(2+4−3𝜀)/𝜀}𝑒^{√−((2+4−3𝜀)/𝜀)𝑡} (40)

is unbounded while 𝜀 values are tending to zero.

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