一类奇摄动时滞边值问题的激波解

一类奇摄动时滞边值问题的激波解
莫嘉琪
【摘要】本文研究了一类具有时滞参数的拟线性奇摄动问题的激波解.在适当的条件下,利用匹配法和微分不等式理论,构造和讨论了原边值问题激波解的存在性和渐近性态.%The shock solutions for the quasilinear singularly perturbed problems with time delay parameter are considered.Under suitable conditions,using the match methods and the theory of differential inequalities the existence and asymptotic behavior of the shock solution for the original boundary value problems are constructed and studied.【期刊名称】《安徽师范大学学报（自然科学版）》
【年(卷),期】2013(036)004
【总页数】5页(P314-318)
【关键词】激波解;边值问题;奇摄动
【作者】莫嘉琪
【作者单位】安徽师范大学数学计算机科学学院,安徽芜湖241003
【正文语种】中文
【中图分类】O175.14
Introduction
The nonlinear singularly perturbed problem is a very attractive object in
the international academic circles[1，2]. Recently, many scholars have done a great deal of work[3-9]. The author et al. considered also a class of singularly perturbed problems[10-19]. In this paper using a special and simple method, we construct the shock solution for a class of quasilinear time delay singularly perturbed boundary value problem and the existence and asymptotic behavior of the solution are study.
Consider the following singularly perturbed time delay boundary value problem:
(1)
u=α,-η≤x≤0,
(2)
u=β,x=1，
(3)
where ε is a positive small parameter, η>0 is a small time delay parameter, a nd α,β are constants, f is a sufficiently smooth function with respect to variables in corresponding ranges.
The reduced problem of problem(1)-(3) is
(4)
uo(1)=β.
(5)
We also assume that
[H1] There exists a positive constant δ, such as fu≥δ>0, and 0<uo(0)<α；
[H2] there exists a monotone solution uo(x) for the reduced problem (4),(5).
1 Outer solution
We first developing u(x-η) in η and we have
(6)
From Eq.(1) we have
(7)
Let the outer solution U(x,ε,η) of original time delay problem is
(8)
Substituting Eq.(8) into Eqs. (3) and (7), expanding the nonlinear terms in ε,η, and equating the coefficients of the same powers in ε,η. For the coefficient of ε0η0, we have
From Eqs.(4) and (5), we obtain
(9)
Substituting Eq.(8) into Eqs.(3) and (7), equating the coefficients of the same powers in ε,η. For the coefficients of ε1η0 and ε0η1, and considering the condition (3), we have
(10)
From Eqs.(10) and (11), it is easy to see that we can obtain solutions and Thus we have the outer solution Uo(x,ε,η) of o riginal singularly perturbed time delay problem (1)-(3) as follows
(12)
2 Shock solution
We set up the stretched variable[11]
(13)
and the interior solution for the original problem near x=0 is Ui. Substituting Eq.(13) into Eq.(1), we can obtain
(14)
Let
(15)
Substituting Eq.(15) into Eq.(14), expanding the nonlinear terms in ε,η, and equating the coefficients of the same powers in ε,η. For the coefficient of ε0η0, we have
(16)
From Eq.(16), we have the shock solution near x=0 that, as
(18)
where k>0 and d are constants, which will be determined below.
3 Interior solution
Substituting Eq.(15) into Eq.(14), equating the coefficients of the same powers in ε,η. For the coefficients of ε1η0 and ε0η1, we have
(19)
(20)
From Eqs. (19) and (20), it is easy to see that we can obtain solutions and where C10,C01 are constants. Thus we have the interior solution Ui(ξ,ε,η) of original singularly perturbed time delay problem (1)-(3) as follows
(21)
4 Match of outer and interior solutions
Since the outer solution Uo(x,ε,η) and interior layer solution Ui(ξ,ε,η) must be matched near x=0, we expand the outer solution Uo(x,ε,η) in ε and η as x substituting εξ. The zero order approximation is
(22)
And we expand the interior layer solution Ui(ξ,ε,η) in ε and ηas ξ
substituting x/ε. From Eqs.(17) and (18), the zero order approximation is
(23)
From matched principle, Eq.(22) is equal to Eq.(23), i.e.
Thus
(24)
And we have
or
(25)
From the behavior of hyperbolic function and assumption, we have
(26)
where d is decided by (25).
Using the same method, we match Eqs. and and and near x=0. And we can decide the constants C10 and C01.
Thus, from Eqs.(12) and (21), we obtain the compounding expansion of slution u(x,ε,η) as follows
u(x,ε,η)=
+O(max(ε,η)),0<ε,η≪1.
5 The main result
Now we prove Eq.(27) is a uniformly valid asymptotic expansion of solution u(x,ε,η) for the singularly perturbed time delay problem (1)-(3). Theorem. Under the assumptions [H1] and [H2], there exists a solution u of the singularly perturbed time delay boundary value problem (1)-(3) and which possesses uniformly valid asymptotic expansion (27) as 0<ε,η≪1. Proof. We first construct two auxiliary functions u and
u=Z(x,ε,η)-γμ，
(28)
(29)
where μ=max(ε,η) and γ is a positive constant large enough, which will be decide below, and
Z(x,ε,η)=
We now prove that u and are lower and upper solutions of the original singularly perturbed problem (1)-(3) respectivly.
Obviously, u and are twice continuous differentiable at 0≤x≤1, and from the assumptions we have
u
(30)
From the assumption and Eqs.(28) and (29), it is easy to see that, as γ larger enough，
(31)
u
(32)
Now we prove that
uu(x-η))≥0,0<x<1，
(33)
(34)
We prove the inequality Eq.(34) only. The proof of inequality Eq. (33) is the similar.
From the assumptions and the behavior of the hyperbolic contangent, it is no difficult to see that, there is a positive constant M, such that
+Mμ-fu(x,ζ)γμ≤(M-δγ)μ，
where ζ is a certain value. Thus as ε,η small enough and selecting the inequality (34) is true.
From (30)-(34), we have proved that u and are lower and upper solutions of the original problem (1)-(3). Thus from the comparison theorem, there exists a solution u(x,ε,η) of the original singularly perturbed time delay
problem (1)-(3), and holds
u
From Eqs.(28) and (29), we then obtain uniformly valid expansion (27) for ε and η in 0≤x≤1. The proof of theorem is completed.
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