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

Authors: Gabil M. Amiraliyev; Aysenur Ucar
Journal: The Scientific World Journal
 (2013)

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## 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.


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## Body

## 1. Introduction and Preliminaries

In this paper we investigate the equation(1)Lu≡εu′′+a(t)u′+b(t)u=f(t),0<t<T,with the periodic conditions(2)u(0)=u(T),u′(0)=u′(T),whereε∈(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 parameterε 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 foru(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 paperC denotes the generic positive constants independent of ε. Such a subscripted constant is also independent of ε, but its value is fixed.Lemma 1.
Letδ(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
(3)δ′(t)+c0(t)δ(t)≤ρ(t),δ(0)≤μδ(T)+γ,
then(4)δ(t)≤(1-μe-∫0Tc0(s)ds)-1δ(t)≤×(γe-∫0Tc0(η)dη+∫0Tρ(s)e-∫sTc0(η)dηds)e-∫0Tc0(η)dηδ(t)≤+∫0tρ(s)e-∫sTc0(η)dηds
provided that(5)1-μe-∫0Tc0(s)ds>0.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 solutionu(t) of the problem (1)-(2) satisfies the bound
(6)ε|u′|2+|u|2≤C∫0t|f(s)|2ds,
provided that(7)γ=λ0min[0,T](2b(t)-a′(t))-b*->0,
where(8)b*-=max[0,T]b′(t),0<λ0<(α+α2+8β)4.Proof.
Consider the identity(9)Lu(u′+λu)=(u′+λu)f(t)
with parameterλ>0 which will be chosen later. By using the equalities
(10)εu′′u′=ε2[(u′)2]′,λu′′u=λ(u′u)′-λ(u′)2,λa(t)u′u=λ2a(t)(u2)′=λ2[a(t)u2]′-λ2a′(t)u2,b(t)u′u=12b(t)(u2)′=12[b(t)u2]′-12b′(t)u2,
and the inequalities(11)u′f(t)≤μ1(u′)2+14μ1f2(t),μ1>0,λuf(t)≤λμ2u2+λ4μ2f2(t),μ2>0,
in (9), we have
(12){εu′2+2ελu′u+λa(t)u2+b(t)u2}′≤-2{a(t)-ελ-μ1}u′2+{b′(t)+λa′(t)-2λb(t)+2λμ2}u2+{12μ1+λ2μ2}f2(t).
Denoting nowδ(t)=εu′2+2ελu′u+λa(t)u2+b(t)u2 and choosing μ=1/2, we arrive at
(13)δ(t)≥ε2u′2+{β+λ(α-2λε)}u2.
After takingλ=λ0<(α+α2+8β)/4, the last inequality reduces to
(14)δ(t)≥C0(εu′2+u2),
where(15)0<C0=min{12,β+λ0(α-2λ0ε)}.
On the other hand for the functionδ(t) holds the following inequality clearly:
(16)δ(t)≤ε(1+λ)u′2+(b*+ελ+λa*)u2δ(t)≤ε(1+λ0)u′2+(b*+λ0+λ0a*)u2.
For the right-hand side of inequality (12), we have
(17)2{a(t)-ελ-μ1}u′2+{-b′(t)-λa′(t)+2λb(t)-2λμ2}u2≥2ε{α-ελ0-μ1}u′2+{-b*--λ0a′(t)+2λ0b(t)-2λ0μ2}u2≥αεu′2+γ2u2.
Taking into accountε≤1 and γ>0, after choosing μ1=(α-2λ0)/2 and μ2=γ/4λ0, we have
(18)δ′(t)≤-C1δ(t)+ρ(t),δ(0)=δ(T),
where(19)0<C1=min{α1+λ0,γ2(b*+λ0+λ0a*)},ρ(t)={12μ1+λ2μ2}f2(t).
From (18) by using Lemma 1, we have
(20)δ(t)≤e-C1t1-e-C1T∫0Tρ(s)e-C1(t-s)ds+∫0tρ(s)e-C1(t-s)ds
which proves Theorem2.Note 2.
As it is seen from (6)
(21)|u(t)|≤C∥f∥2,
where(22)∥f∥2=∫0T|f2(s)|1/2ds.Theorem 3.
Under the assumptions of Theorem2, the following asymptotic estimates for the derivatives hold true:
(23)|u(k)(t)|≤C{1+ε1-ke-αt/ε},0≤t≤T,k=0,1,2.Proof.
The casek=0 directly follows from the identity (4).
Fork=1, the problem (1)-(2) can be rewritten as
(24)εu′′+a(t)u′=F(t),0<t<T,(25)|u(0)|≤C,(26)u′(0)=u′(T),
where(27)F(t)=f(t)-b(t)u
and by virtue of Theorem2(28)|F(t)|≤C.
The solution of (24)–(26) can be expressed as
(29)u′(t)=u′(0)e-(1/ε)∫0ta(s)ds+1ε∫0tF(s)e-(1/ε)∫sta(ξ)dξds,
and taking into account (26), we have
(30)u′(0)=(1-e-(1/ε)∫0Ta(s)ds)-11ε∫0TF(s)e-(1/ε)∫sTa(ξ)dξds.
Thus we get(31)|u′(0)|≤Cα-1(1-e-αT/ε)1-e-a*T/ε≤Cα-1.
The relation (29) along with (31) leads to (23) for k=1 immediately.
Next fork=2, from (1) we have
(32)|u′′(0)|=1ε|f(0)-b(0)u(0)-a(0)u′(0)|≤Cε.
Differentiating now (1), we obtain
(33)εu′′′+a(t)u′′=f′(t)-b′(t)u-b(t)u′-a′(t)u≡φ(t).
Under the smoothness conditions on data functions and boundness ofu(t) and u′(t), we deduce evidently
(34)|φ(t)|≤C.
The solution of (33) is
(35)u′′(t)=u′′(0)e-(1/ε)∫0ta(s)ds+1ε∫0tφ(s)e-(1/ε)∫sta(ξ)dξds.
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(36)a(t)=4,b(t)=3,f(t)=3t,T=1.The solution of this problem is given by(37)u(t)=A1e-((2-4-3ε)/ε)t+A2e-((2+4-3ε)/ε)t+t-43,where(38)A1=2+4-3ε24-3ε(1-e-(2-4-3ε)/ε),A2=4-3ε-224-3ε(1-e-(2+4-3ε)/ε).For the first derivative we have(39)u′(t)=-324-3ε(1-e-(2-4-3ε)/ε)e-((2-4-3ε)/ε)tu′(t)=-324-3ε(1-e-(2+4-3ε)/ε)e-((2+4-3ε)/ε)t+1from which it is clear that the first derivative ofu(t) is uniformly bounded but has a boundary layer near t=0 of thickness O(ε).The second derivative(40)u′′(t)=-3ε(2-4-3ε24-3ε(1-e-(2-4-3ε)/ε)e-((2-4-3ε)/ε)t+2+4-3ε24-3ε(1-e-(2+4-3ε)/ε)e-((2+4-3ε)/ε)t)is unbounded whileε values are tending to zero.Therefore we observe here the accordance in our theoretical results described above.



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*Source: 301609-2013-12-04.xml*