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An interface problem with singular perturbation on a subinterval
Boundary Value Problems volume 2014, Article number: 201 (2014)
In this paper we investigate an interface problem with singular perturbation on a subinterval. We first establish a lemma of lower and upper solutions which is an extension of the classical theory of lower and upper solutions. Based on the basic lemma we obtain the existence of a solution to the proposed problem, and the asymptotic behavior of solution as the singular perturbation parameter as well.
MSC: 34E10, 34A36, 34B15.
Interface problems, like coupled elliptic-hyperbolic or parabolic-hyperbolic problems with discontinuous coefficients, arise in many fields, such as material sciences, fluid-solid interactions. If an interface problem is confined in a one dimensional domain, one gets a boundary value problem of ordinary differential equations with interface conditions. For example, in  de Falco and O’Riordan considered a one dimensional metal-oxide-semiconductor structure which is modeled by a two-point interface boundary value problem with singular perturbation. Recently, interface problems have attracted much attention as regards both theoretical and numerical aspects; see for instance – and references therein.
In  Aguilar and Lisbona investigated -smooth solution of the following interface boundary value problem with singular perturbation:
where is the piecewise constant function of the form
and the functions , , satisfy
This kind of problem arises from some simplified physical models such as the infiltration process in an inhomogeneous soil . In , by using inverse-monotone operator theorems, the authors proved that the unique -smooth solution converges almost everywhere to the solution of the corresponding reduced problem as .
In physical problems, several typical interface conditions, such as perfect contact, flux jump, and thermal resistance, are often encountered. Hence, it is interesting and of significance to study the problem (1) with general interface conditions. In the present paper, as a natural generalization, we consider the following boundary value problem with interface conditions:
which is a special form of (2) with .
In Section 2, we establish first a lemma of low and upper solutions for the problem (2), which is an extension of classical theory of lower and upper solutions. Lower and upper solutions theorems for -smooth solutions of two-point second-order boundary value problems with discontinuous coefficients have been established in  where -solutions (-smooth certainly) are considered. However, the theory of lower and upper solutions for boundary value problems with general interface conditions has not been formulated, to our knowledge.
In Section 3, based on the basic lemma established in Section 2 we analyze the asymptotic behavior of solution to the problem (2) in everywhere sense. The original problem can be viewed as the coupling of the left problem and the right singular perturbation problem satisfying the jump interface conditions. The solution of the right problem exhibits generally a boundary layer at either end, which depends on the sign of (see ,  for instance). Thereby two cases should be distinguished. We prove that under suitable conditions the problem (2) has a solution whose asymptotic behavior is fully described as on the whole interval . A simple linear example as an illustration is presented at the end.
Throughout this paper, we assume
(H1) The functions and are -smooth, and on .
2 Lower and upper solutions lemma
For , let be a vector space of functions defined on satisfying
where is double-valued at .
A function is called a lower solution of the problem (2) if
A function is called an upper solution of the problem (2) if
Let us define the following modifications of the functions in the right hand side of (2):
where , and is a large enough number.
Consider the modified problem
Using the method of variation of constants, we write the solution of (8) in the following form:
while the functions
solve the homogeneous equation . The function
is the unique solution of the problem
The integral equation (9) defines an operator T on , that is, a Banach space endowed with the norm . Since is uniformly bounded in , the set is a relatively compact subset of . Moreover, T is continuous. Hence, it follows from the Schauder fixed-point theorem (see, for instance, ) that the boundary value problem (8) has a solution. Note that any solution of (8) which lies between and and satisfies , , is a solution of (2).
Noting that satisfies a Nagumo condition with respect to and , it follows that , (see Theorems 7.33 and 7.34 in ). In what follows, we prove , . Suppose, on the contrary, that the function has a positive maximum at some . From (5) we see . If , then , , and . On the other hand,
3 Asymptotic estimates
In this section, we investigate asymptotic behavior of solutions of (2) by constructing suitable pairs of lower and upper solutions. As in , we distinguish two cases, and consider the asymptotic behavior under the assumptions (H2) and (H2′), respectively.
(H2): There exists a positive constant such that for .
(H2′): There exists a positive constant such that for .
Case (I). Assume (H2).
We also assume the following.
(H3): The reduced problem
has a solution .
Generally, the right problem
has a boundary layer at . However, taking the interface condition into consideration, the solution of (11) must have no boundary layer at . Thus we have
The left boundary value problem
has a solution.
It is easy to verify that and are a pair of lower and upper solutions of (12), where
Let the conditions (H1), (H2), and (H3) hold. Moreover, we assume that
Then for sufficiently smallthe boundary value problem (2) has a solutionsatisfying
From the assumptions (H1) and (H3) it follows that there is a positive constant M such that for sufficiently small
We construct the barrier functions as follows:
where , are positive constants such that
which is a solution of the differential equation
It follows from the construction of Φ and Ψ that
and for ,
provided that ε is small enough, where , and .
For sufficiently small the inequality (6) can be verified in a similar way. Thus we have proved that and are lower and upper solutions of (2), respectively. The conclusion immediately follows from Lemma 1. □
Case (II). Assume (H2′).
In this case, the solution to the right problem (11) exhibits a boundary layer at . Hence, we need first to establish a solution of the left problem. Considering the interface conditions we impose the following nonlinear boundary condition:
at for the left problem. We have the following proposition.
Then the left boundary value problem
has a solution, where.
(H3′): Assume that the right reduced problem
has a solution .
In general, , and thereby we need to construct a corrected boundary layer term. To this end, substituting
into the right boundary value problem
and letting , we obtain
Considering the continuity of we introduce
The following proposition concerns the asymptotic behavior of the boundary layer term, whose proof is substantially similar to that of Lemma 3.1 in .
The boundary value problem (17) has a solutionwith the exponential estimates
Let the conditions (H1), (H2′), (H3′), and (14) hold. Moreover, we assume that
Then for sufficiently smallthe boundary value problem (2) has a solutionsuch that for
whereandare defined in Proposition 3, andis a constant independent of ε.
It follows from the assumptions (H1) and (H3′) that there exists a positive constant such that for sufficiently small
where , and .
Select the bounding functions as follows:
is a solution of the equation
and the function
Here we check the inequality (3) only for , since the equality on can be verified by following similar lines as in the proof of Theorem 1. From the definition of we have for
on condition that ε is sufficiently small. Thus is a lower solution of (2).
Finally, as an illustration, let us consider a linear interface boundary value problem,
From Theorem 2 it follows that (21) has a solution with the following asymptotic estimate:
which agrees with the exact solution accurate to order ε.
The author read and approved the final manuscript.
de Falco C, O’Riordan E: Interior layers in a reaction-diffusion equation with a discontinuous diffusion coefficient. Int. J. Numer. Anal. Model. 2010, 7: 444-461.
Aitbayev R: Existence and uniqueness for a two-point interface boundary value problems. Electron. J. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-242
Chern I, Shu Y: A coupling interface method for elliptic interface problems. J. Comput. Phys. 2007, 225: 2138-2174. 10.1016/j.jcp.2007.03.012
Loubenets A, Ali T, Hanke M: Highly accurate finite element method for one-dimensional elliptic interface problems. Appl. Numer. Math. 2009, 59: 119-134. 10.1016/j.apnum.2007.12.003
Huang Z: Tailored finite point method for the interface problem. Netw. Heterog. Media 2009, 4: 91-106. 10.3934/nhm.2009.4.91
Aguilar G, Lisbona F: Singular perturbation on a subdomain. J. Math. Anal. Appl. 1997, 210: 292-307. 10.1006/jmaa.1997.5404
Aguilar G, Lisbona F: On the coupling of elliptic and hyperbolic nonlinear differential equations. Math. Model. Numer. Anal. 1994, 28(4):399-417.
Coster CD, Habets P: Two-Point Boundary Value Problems: Lower and Upper Solutions. Elsevier, New York; 2006.
de Jager EM, Jiang F: The Theory of Singular Perturbations. North-Holland, Amsterdam; 1996.
O’Malley RE: Singular Perturbation Methods for Ordinary Differential Equations. Springer, New York; 1991.
Zeidler E: Applied Functional Analysis: Applications to Mathematical Physics. Springer, New York; 1995.
Kelley WG, Peterson AC: The Theory of Differential Equations. Springer, New York; 2010.
Fabry C, Habets P: Upper and lower solutions for second-order boundary value problems with nonlinear boundary conditions. Nonlinear Anal. TMA 1986, 10: 985-1007. 10.1016/0362-546X(86)90084-2
Xie F: On a class of singular boundary value problems with singular perturbation. J. Differ. Equ. 2012, 252: 2370-2387. 10.1016/j.jde.2011.10.003
The author wants to thank the referees for valuable comments and suggestions. The author was supported by the National Natural Science Foundation of China (No. 11371087), in part by the Natural Science Foundation of Shanghai (No. 12ZR1400100), and by the Fundamental Research Funds for the Central Universities.
The author declares that he has no competing interests.
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Cite this article
Xie, F. An interface problem with singular perturbation on a subinterval. Bound Value Probl 2014, 201 (2014). https://doi.org/10.1186/s13661-014-0201-8
- lower and upper solutions
- singular perturbation
- interface conditions