An interface problem with singular perturbation on a subinterval
© Xie; licensee Springer 2014
Received: 3 July 2014
Accepted: 13 August 2014
Published: 25 September 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.
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 .
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
where is double-valued at .
where , and is a large enough number.
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).
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.
has a solution .
has a solution.
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′).
at for the left problem. We have the following proposition.
has a solution, where.
has a solution .
The following proposition concerns the asymptotic behavior of the boundary layer term, whose proof is substantially similar to that of Lemma 3.1 in .
whereandare defined in Proposition 3, andis a constant independent of ε.
where , and .
on condition that ε is sufficiently small. Thus is a lower solution of (2).
which agrees with the exact solution accurate to order ε.
The author read and approved the final manuscript.
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.
- 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.MathSciNetGoogle Scholar
- Aitbayev R: Existence and uniqueness for a two-point interface boundary value problems. Electron. J. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-242Google Scholar
- Chern I, Shu Y: A coupling interface method for elliptic interface problems. J. Comput. Phys. 2007, 225: 2138-2174. 10.1016/j.jcp.2007.03.012MathSciNetView ArticleGoogle Scholar
- 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.2007.12.003MathSciNetView ArticleGoogle Scholar
- Huang Z: Tailored finite point method for the interface problem. Netw. Heterog. Media 2009, 4: 91-106. 10.3934/nhm.2009.4.91MathSciNetView ArticleGoogle Scholar
- Aguilar G, Lisbona F: Singular perturbation on a subdomain. J. Math. Anal. Appl. 1997, 210: 292-307. 10.1006/jmaa.1997.5404MathSciNetView ArticleGoogle Scholar
- Aguilar G, Lisbona F: On the coupling of elliptic and hyperbolic nonlinear differential equations. Math. Model. Numer. Anal. 1994, 28(4):399-417.MathSciNetGoogle Scholar
- Coster CD, Habets P: Two-Point Boundary Value Problems: Lower and Upper Solutions. Elsevier, New York; 2006.Google Scholar
- de Jager EM, Jiang F: The Theory of Singular Perturbations. North-Holland, Amsterdam; 1996.Google Scholar
- O’Malley RE: Singular Perturbation Methods for Ordinary Differential Equations. Springer, New York; 1991.View ArticleGoogle Scholar
- Zeidler E: Applied Functional Analysis: Applications to Mathematical Physics. Springer, New York; 1995.Google Scholar
- Kelley WG, Peterson AC: The Theory of Differential Equations. Springer, New York; 2010.View ArticleGoogle Scholar
- 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)90084-2MathSciNetView ArticleGoogle Scholar
- Xie F: On a class of singular boundary value problems with singular perturbation. J. Differ. Equ. 2012, 252: 2370-2387. 10.1016/j.jde.2011.10.003View ArticleGoogle Scholar
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