Boundary Layer Analysis for Nonlinear Singularly Perturbed Differential Equations

This paper focuses on the boundary layer phenomenon arising in the study of singularly perturbed differential equations. Our tools include the method of lower and upper solutions combined with analysis of the integral equation associated with the class of nonlinear equations under consideration. Key words and phrases: singularly perturbed systems, three–point boundary value problem, method of lower and upper solutions.


Introduction
This paper is devoted to study the second-order semilinear singularly perturbed differential equation subject to the three-point boundary value conditions where ǫ is a small perturbation parameter (0 < ǫ << 1).
In the past few years, much attention has been paid to the study of nonlocal boundary value problems, whose study for ordinary differential equations has been initiated by the work of Il'in and Moiseev [11,12].
In particular, existence of solutions for differential equation y ′′ + g(t)f (y(t)) = 0, 0 < t < 1 under one of the m-point boundary conditions as an important subclass of nonlocal boundary conditions has been thoroughly studied by Gupta et al., see, for example, [5,6,7,8,9].Eloe and Gao [3] discussed the quasilinearization method for a threepoint semilinear boundary value problem which provides an iterative scheme for approximating the solutions.The subject of multi-point nonlocal boundary value problems for singularly perturbed differential equations has been also addressed by many authors, see e.g.[1,2], and the references therein.For example, Du et al. [1] have studied a third-order multi-point singularly perturbed boundary value problem , applying differential inequalities technique (method of lower and upper solutions) and Leray-Schauder degree theory.This paper contains a large amount of material and can serve as an introduction to some of principles and methods of singular perturbation theory, not only for third-order nonlinear differential equations.
Singular perturbation problems can also arise in heat transfer problem with large Peclet numbers [14], Navier-Stokes flows with large Reynolds numbers, chemical reactor theory, aerodynamics, reactiondiffusion processes, quantum mechanics, optimal control [15], for example.
As far as we know, there is no paper related to the boundary layer analysis for nonlinear multi-point nonlocal singularly perturbed boundary value problems.
In the present paper, we focus our attention on the detailed analysis of the behavior of the solutions y ǫ for (1), (2) in the point t = b when a small parameter ǫ tends to zero.We show that the solutions y ǫ of (1), ( 2) remain close to u on K with an arising fast transient of ) and ǫ → 0 + ), which is the so-called boundary layer phenomenon ( [4,13]).Boundary layers are formed due to the nonuniform convergence of the exact solution y ǫ to the solution u of reduced problem in the neighborhood of the right end b.
We will assume that the following conditions are satisfied throughout this paper: It is instructive for the future to keep in mind that this assumption implies that u(c) = u(b) and f (c, y ǫ (c)) = f (b, y ǫ (b)) for every sufficiently small ǫ, say 0 < ǫ < ǫ 0 .

Boundary layer phenomenon at t = b
For an illustrative example we consider (1), (2) with f (t, y) = t 2 , a = 0, b = 2, c = 1 and its solution Hence we have We precede the main result of this article with the following important lemmas.
The Lemma 2.1 is proven.

Lemma 2.2 Let the assumptions (H1) and (H3) hold. Then the set
Proof.By Lagrange's Theorem and from Diff.Eq. ( 1) we obtain

Main result
Our main result is the following.
Theorem 3.1 Under the assumptions (H1)-(H3) the problem ( 1), ( 2) has for every ǫ, ǫ ∈ (0, ǫ 0 ] the unique solution y ǫ in D(u) which converges uniformly to the solution u of reduced problem for ǫ → 0 + on an arbitrary compact subset K of [a, b) and the set EJQTDE, 2011 No. 32, p. 6 Proof.The existence, uniqueness in D(u) and asymptotic behavior of the solutions for (1), (2) on the compact subset K ⊂ [a, b) has been proven in [16].It remains to prove (7), a boundary layer phenomenon at t = b.
Assume to the contrary that the set on [a, b], Cδ > 0 is constant.The problem (1), ( 2) is equivalent to the nonlinear integral equation where Differentiating the integral equation ( 9) with respect to the variable t we obtain Hence EJQTDE, 2011 No. 32, p. 7 Integrating all integrals in (10) by parts and after little algebraic arrangement we obtain Taking into consideration (8), the integrals Thus we have From (11) we can write where EJQTDE, 2011 No. 32, p. 8 Thus from (12) we have Using ( 13), we have Remark 3.2 As we can see from (14) the assumption (H2) is essential for an appearance the boundary layer phenomenon for singularly perturbed system (1), (2) at the point t = b.EJQTDE, 2011 No. 32, p. 9