A turning point asymptotic expansion for a rigid-dumbbell polymer fluid probability configurational equation for fast shear flows

Abstract

This paper establishes an asymptotic expansion for a second order differential equation with a small diffusion coefficient, which generalizes the configurational probability diffusion equation of the Rigid Dumbbell Model (RDM) of diluted polymer solutions theory for fast shear flows. This is a singular perturbation problem with turning point.

Keywords

Turning point asymptotic problem rigid dumbbell polymer chain model fast shear flows configurational probability diffusion equation

1. Introduction

The study of polymer fluid models – and of the resulting constitutive equations, be they viscoelastic or non-Newtonian in nature – is an effervescing activity due to tremendously important industrial applications. In particular, the models accounting for the dominant molecular interactions – although more complex compared to those originating from continuum physics – are increasingly used for predicting experimentally observed flow patterns. One such a model is called the “Rigid Dumbbell Model (RDM)”. It is the idealization of the real polymer chain in the form of an inflexible dumbbell with the chain mass concentrated at its extremities. It is remarkable that such a simple model answers many wants of polymer scientists despite of being a crude rendition of reality; its generalization consists of a concatenation of freely articulated rigid segments (basically a multibead-rod chain).

The RDM belongs to the realm of kinetical theories [3], which means it is made-up of a configurational probability diffusion equation and a corresponding constitutive equation/law needed to be inserted into the system of momentum balance equations. Given its performances in modeling the rheological behavior of dilute polymer solutions, it has been put to test in a variety of flow situations, including that of fast shear flows because the later are very common in processing technologies of complex liquids.

In this paper the object of our study is the configurational probability diffusion equation which in spherical coordinates $θ \in (0, π)$ , and with ϕ restricted to $ϕ \in (- π / 2, π / 2)$ by symmetry arguments, reads (see e.g. equations (14.4-1) and (14.6-13) in [3] and equation (1) in [13]; see also [4]) $\begin{array}{l} (1.1) & \frac{1}{\dot{γ}} \frac{\partial f}{\partial t} = - \frac{1}{2} \frac{sin 2 ϕ}{S} \frac{\partial}{\partial θ} (S^{2} C f) + \frac{\partial}{\partial ϕ} ({sin}^{2} ϕ f) + \frac{1}{6 λ \dot{γ}} [\frac{1}{S} \frac{\partial}{\partial θ} (S \frac{\partial f}{\partial θ}) + \frac{1}{S^{2}} \frac{\partial^{2} f}{\partial ϕ^{2}}] . \end{array}$

In the above $f = f (t, θ, ϕ)$ is the configurational probability function, $\dot{γ}$ the shear rate (see e.g. [2,5]), $λ > 0$ a model parameter (a relaxation time), and $S = sin θ$ and $C = cos θ$ are shorthand notations.

In [14] an intuitive attempt to solve (1.1) for arbitrary shear rates has been carried-out by Stewart and Sorensen, assuming the solution may be expressible with the help of an eigenfunction expansion of spherical harmonics. No formal existence/uniqueness proofs do appear in [14] and, based on a seemingly convergence of the corresponding numerical algorithm, they positively concluded on the convergence of the corresponding series. Later, using educated physical intuition and guesswork, Öttinger conjectured in [13] that, for high shear rates (i.e. for $λ \dot{γ}$ or, simply, for $\dot{γ} \to + \infty$ ) and for steady-state version (i.e. setting $\frac{\partial P}{\partial t} = 0$ ),

the position of the rod dumbbells is “concentrated” in a region for which $| ϕ | ≪ 1$ , $θ ≃ π / 2$ ;

a further simplification is obtained by setting $S = 1$ and integrating (1.1) w.r.t. θ and obtaining a simpler equation for a new unknown called “contracted distribution” function $P = P (ϕ)$ : $\begin{matrix} (1.2) & \frac{\partial}{\partial ϕ} ({sin}^{2} ϕ P) + \frac{1}{6 λ \dot{γ}} \frac{\partial^{2} P}{\partial ϕ^{2}} = 0 . \end{matrix}$

Assuming the boundary conditions $P (- \frac{π}{2}) = P (\frac{π}{2})$ , Öttinger obtained the following asymptotic behavior for $λ \dot{γ} \to + \infty$ : $\begin{array}{l} (1.3) & P (ϕ) = N (λ \dot{γ}) g ({[λ \dot{γ}]}^{1 / 3} ϕ) for ϕ = O ({[λ \dot{γ}]}^{1 / 3}), λ \dot{γ} \to + \infty, \\ (1.4) & g (u) = \int_{0}^{+ \infty} exp [- \frac{s^{3}}{2} - 6 s {(u - \frac{s}{2})}^{2}] d s, \\ (1.5) & N (λ \dot{γ}) = {(λ \dot{γ})}^{1 / 3} \frac{\sqrt{3}}{\sqrt{π} 2^{2 / 3} Γ (7 / 6)} . \end{array}$

No convergence results are given in [13] either, however the numerical results for the viscometric functions (steady state viscosity and first and second normal stress differences) are found to be in very good agreement with those of [14]. This compatibility indicates that the heuristic assumptions $(a_{1})$ , $(a_{2})$ of [13] are correct.

Before proceeding further let us observe that:

the coefficient of the first l.h.s. term in (1.2) vanishes for $ϕ = 0$ , therefore equation (1.2) is – in the parlance of differential equations – of a turning point type, and

although an explicit solution for the time independent version of (1.2) is readily obtainable, it is not convenient at all for establishing the asymptotic behavior for $λ \dot{γ} \to + \infty$ .

From here on, we shall denote $ϵ = 1 / (λ \dot{γ})$ and the corresponding asymptotic pattern is to be obtained for $ϵ \to 0$ .

However, because it is more appealing from a mathematical prospective and for sake of generality, our goal is to solve a more general problem which we introduce in (2.3), (2.4) and (2.5) below; then the solution for (1.2) and the result in [13] will emerge as a particular cases. We shall obtain a rigorous asymptotic expansion for $ϵ \to 0$ for the unique solution of (2.3)–(2.5). Since $a (0) = 0$ the problem under scrutiny enters the class of turning point equations (see e.g. [6–8,15] for general introductions to the matter). Moreover, turning point problems are sometimes encountered in the area of Newtonian fluid mechanics as is the case in [9] and in [10].

To the best of our knowledge, this paper is the first to solve a turning point problem having periodic (not Dirichlet) boundary conditions and with the requirement that the integral of its solution be a given number.

A survey of previously published results shows that most works on – say – equations of the type $\begin{matrix} ϵ u_{ϵ}^{″} (x) + a (x) u_{ϵ}^{'} + b (x) u_{ϵ} = f (x) \end{matrix}$ together with Dirichlet conditions $\begin{matrix} u_{ϵ} (- 1) = A, u_{ϵ} (- 1) = B \end{matrix}$ deal with smooth functions a such that $a (0) = 0$ , as one may reckon from [1,11,12,16] and [17], for instance. In all the aforementioned papers 0 is a simple root of $a (x) = 0$ and function b does not vanish at 0. Consequently, the solving methods used in them are inapplicable to our case which all the more deals not with Dirichlet conditions. A natural consequence of all these facts is that the approach we devised is different. Moreover, the main part of the solution asymptotic expansion is obtained in a easily exploitable – for further practical numerical/physical/engineering applications – form that reads $\begin{matrix} \frac{1}{ϵ^{1 / (2 n + 1)}} \sum_{j ⩾ 0} ϵ^{j} A_{j + 1} (\frac{x}{ϵ^{1 / (2 n + 1)}}) + ϵ^{\frac{2 n - 1}{2 n + 1}} \sum_{j ⩾ 0} ϵ^{j} q_{j + 1} (x) . \end{matrix}$

The paper is organized as follows: In Section 2 we state the problem studied in this paper, we rapidly prove by an elementary calculation the existence and uniqueness of a solution $u_{ϵ}$ and we define an operator $L_{ϵ}$ related to the problem. In Section 3 we give invertibility properties of $L_{ϵ}$ and a consequence which shows how construct an asymptotic expansion of $u_{ϵ}$ . Some definitions and results useful to built the asymptotic expansion are given in Section 4. The asymptotic expansion is given in Section 5 and, finally, in Section 6 we present an approximate solution for the particular case given in (1.2) (case that actually sparked the interest for this work).

2. The problem under scrutiny

To begin with, we introduce the mathematical problem.

Let $Ω = (- 1, 1)$ . Define $a : \overline{Ω} \to R$ , with $a \in C^{\infty} (\overline{Ω})$ , $a (x) > 0$ for any $x \in \overline{Ω} - {0}$ , such that $a (0) = a^{'} (0) = \dots = a^{(2 n - 1)} (0) = 0$ , and $a^{(2 n)} (0) > 0$ , where $n \in N^{*}$ is being given.

We also assume the following expansion of the function a around $x = 0$ and $x = - 1$ :

for any $k \in N$ , there exist $α_{0} > 0$ , $α_{1}, \dots, α_{k} \in R$ and $α_{k + 1} \in C^{\infty} (\overline{Ω})$ such that $\begin{matrix} (2.1) & a (x) = α_{0} x^{2 n} + α_{1} x^{2 n + 1} + \dots + α_{k} x^{2 n + k} + α_{k + 1} (x) x^{2 n + k + 1}, \forall x \in \overline{Ω}, \end{matrix}$

for any $k \in N$ , there exist $γ_{0} > 0, γ_{1}, \dots, γ_{k} \in R$ and $γ_{k + 1} \in C^{\infty} ([0, 2])$ such that $\begin{matrix} (2.2) & a (- 1 + s) = γ_{0} + γ_{1} s + \dots + γ_{k} s^{k} + γ_{k + 1} (s) s^{k + 1}, \forall s \in [0, 2] . \end{matrix}$

Problem 2.1.
Find the asymptotic behavior for $ϵ \to 0$ of the periodic function $u_{ϵ} : \overline{Ω} \to R$ that is solution to the equation $\begin{array}{l} (2.3) & ϵ u_{ϵ}^{″} + {(a u_{ϵ})}^{'} = 0 on Ω, \\ (2.4) & u_{ϵ} (- 1) = u_{ϵ} (1), \\ (2.5) & \int_{Ω} u_{ϵ} (x) d x = 1, \end{array}$ with $ϵ > 0$ being a small, given parameter.

Now, straightforward step by step integration leads to an explicit solution $u_{ϵ}$ (thus granting its existence and uniqueness): $\begin{array}{l} (2.6) & u_{ϵ}^{'} + \frac{a}{ϵ} u_{ϵ} = η_{1}, η_{1} \in R, \\ (2.7) & u_{ϵ} (x) = exp (- \frac{A (x)}{ϵ}) [η_{1} \int_{- 1}^{x} exp (\frac{A (y)}{ϵ}) d y + η_{2}], η_{2} \in R, \end{array}$ where $A (x) = \int_{0}^{x} a (y) d y$ , $A (0) = 0$ , is a strictly increasing function (see Figure 1 below for a plot of $u_{ϵ} (x)$ ). Compliance with the periodicity condition leads to $η_{2} = η_{1} η_{3}$ where $\begin{matrix} (2.8) & η_{3} = \frac{exp (- \frac{A (1)}{ϵ}) \int_{- 1}^{1} exp (\frac{A (y)}{ϵ}) d y}{exp (- \frac{A (- 1)}{ϵ}) - exp (- \frac{A (1)}{ϵ})} > 0 . \end{matrix}$

Next one gets $u_{ϵ} (x) = η_{1} \hat{u_{ϵ}} (x)$ , where $\hat{u_{ϵ}} (x) = exp (- \frac{A (x)}{ϵ}) [\int_{- 1}^{x} exp (\frac{A (y)}{ϵ}) d y + η_{3}]$ . The function $\hat{u_{ϵ}} \in C^{\infty} (\overline{Ω})$ is a strictly positive function. The integration constant $η_{1}$ is calculated upon using the normalization condition $\int_{Ω} u_{ϵ} (x) d x = 1$ .

Fig. 1.
The plot shows the solution $u_{ϵ}$ vs. x (see equations (2.7)–(2.8)) for several values of ϵ: solid line $ϵ = 10^{- 1}$ , dashed line $ϵ = 10^{- 2}$ , dash-dotted line $ϵ = 10^{- 3}$ .
Remark 2.1.
Observe that the real number 0 is a rank one eigenvalue to the differential operator $u \mapsto ϵ u^{″} + {(a u)}^{'}$ with periodic conditions, to which corresponds the eigenvector $u_{ϵ}$ , with $\int_{- 1}^{1} u_{ϵ} (x) d x = 1$ .

Although an explicit solution is obtained in (2.7), it is not an easy undertaking to use this result to get the asymptotic behavior of $u_{ϵ}$ for $ϵ \to 0$ . The goal of this paper is to solve this difficulty.

Let $\begin{matrix} D (L_{ϵ}) : = {v \in H^{2} (Ω) : v (- 1) = v (1), \int_{- 1}^{1} v (x) d x = 0} \end{matrix}$ and the operator $L_{ϵ} : D (L_{ϵ}) \subset L^{2} (Ω) \mapsto L^{2} (Ω)$ be defined as $\begin{matrix} L_{ϵ} (v) : = L_{ϵ} (v), \forall v \in D (L_{ϵ}), \end{matrix}$ where $\begin{matrix} L_{ϵ} v : = ϵ v^{″} + {(a v)}^{'} . \end{matrix}$ Next, for any $v \in H^{2} (Ω)$ , we call residue of v a function f such that $L_{ϵ} v = f^{'}$ . We define the periodicity defect of v the difference $v (- 1) - v (1)$ .

The stages we go through are listed below:

We prove that $L_{ϵ}$ is invertible and obtain suitable estimates for $L_{ϵ}^{- 1}$ in specific normed spaces to be introduced in the next section.

Next, we prove that if we have a function $v_{ϵ}$ with small residue and periodicity defect then we can built a function depending on $v_{ϵ}$ which furnishes a good approximation of $u_{ϵ}$ . As a matter of fact most of the efforts recorded in this paper are devoted to obtaining a such function $v_{ϵ}$ which will be later called $v_{app}^{ϵ}$ .
3. The invertibility of the operator $L_{ϵ}$ and consequences of it

Let the function $g : Ω \to R$ be defined a.e. on Ω, be measurable, and $g > 0$ a.e. on Ω. Then, for any such function g, we introduce the functional space: $\begin{matrix} (3.1) & L^{2} (g d x) : = {v : Ω \to R, v is measurable, \int_{Ω} v^{2} g d x < \infty} . \end{matrix}$

The space $L^{2} (g d x)$ is a Banach space endowed with the norm $\begin{matrix} (3.2) & ‖ v ‖_{L^{2} (g d x)} = {(\int_{Ω} v^{2} g d x)}^{1 / 2} = ‖ \sqrt{g} v ‖_{L^{2} (Ω)} . \end{matrix}$

Now we prove the following statement:

Lemma 3.1.

The operator $L_{ϵ}$ is invertible.

Let $f \in H^{1} (Ω) \cap L^{2} (a^{- 1} d x)$ , and $v_{ϵ} \in D (L_{ϵ})$ such that $L_{ϵ} (v_{ϵ}) = f^{'}$ . Then, one has: $\begin{array}{l} (3.3) & ‖ v_{ϵ} ‖_{L^{2} (a d x)} ⩽ ‖ f ‖_{L^{2} (a^{- 1} d x)}, \\ (3.4) & ‖ v_{ϵ} ‖_{H^{1} (Ω)} ⩽ \frac{c}{ϵ} ‖ f ‖_{L^{2} (a^{- 1} d x)}, \end{array}$ where c is a constant independent of ϵ and f.

Proof.
A calculation along the lines of that leading to equation (2.7) proves the invertibility of $L_{ϵ}$ . The following is devoted to proving (ii). One has: $\begin{array}{l} (3.5) & ϵ v_{ϵ}^{″} + {(a v_{ϵ})}^{'} = f^{'}, \\ (3.6) & v_{ϵ} (- 1) = v_{ϵ} (1), \\ (3.7) & \int_{Ω} v_{ϵ} d x = 0 . \end{array}$

From (3.5) we infer there exists $d_{ϵ} \in R$ such that $\begin{matrix} (3.8) & ϵ v_{ϵ}^{'} + a v_{ϵ} = f + d_{ϵ} \end{matrix}$ from which, multiplying by $v_{ϵ}$ and integrating on Ω one gets with the help of (3.6)–(3.7) $\begin{matrix} (3.9) & \int_{Ω} a v_{ϵ}^{2} d x = \int_{Ω} f v_{ϵ} d x . \end{matrix}$

From the estimates $\begin{matrix} (3.10) & \int_{Ω} f v_{ϵ} d x ⩽ ‖ f ‖_{L^{2} (a^{- 1} d x)} ‖ v_{ϵ} ‖_{L^{2} (a d x)} \end{matrix}$ one gets (3.3). Next, $\begin{matrix} (3.11) & ‖ a v_{ϵ} ‖_{L^{2} (Ω)} ⩽ ‖ \sqrt{a} ‖_{L^{\infty} (Ω)} ‖ v_{ϵ} ‖_{L^{2} (a d x)} ⩽ ‖ \sqrt{a} ‖_{L^{\infty} (Ω)} ‖ f ‖_{L^{2} (a^{- 1} d x)} . \end{matrix}$

Next, integrate (3.8) on Ω, then using the periodicity property (3.6) gives $\begin{matrix} (3.12) & | d_{ϵ} | ⩽ \frac{1}{2} [‖ a v_{ϵ} ‖_{L^{1} (Ω)} + ‖ f ‖_{L^{1} (Ω)}] . \end{matrix}$

Now, with the help of the below estimate: $\begin{matrix} (3.13) & ‖ f ‖_{L^{1} (Ω)} ⩽ ‖ \sqrt{a} ‖_{L^{2} (Ω)} ‖ f ‖_{L^{2} (a^{- 1} d x)} \end{matrix}$ and of inequalities (3.11) and (3.12) we obtain that $\begin{matrix} (3.14) & | d_{ϵ} | ⩽ c ‖ f ‖_{L^{2} (a^{- 1} d x)} . \end{matrix}$

Equations (3.8), (3.11), (3.14) imply $\begin{matrix} (3.15) & ϵ {‖ v_{ϵ}^{'} ‖}_{L^{2} (Ω)} ⩽ c ‖ f ‖_{L^{2} (a^{- 1} d x)} . \end{matrix}$

Using now (3.7) and the Poincaré inequality leads to (3.4). □
Remark 3.1.

Equation (3.4) and the fact that $H^{1} (Ω) \subset L^{\infty} (Ω)$ trigger the estimate $\begin{matrix} (3.16) & ‖ v_{ϵ} ‖_{L^{\infty} (Ω)} ⩽ \frac{c}{ϵ} ‖ f ‖_{L^{2} (a^{- 1} d x)}, \end{matrix}$ where both $v_{ϵ}$ and f comply with the assumptions of Lemma 3.1.

We will require $f (x) \sim_{x \to 0} x^{n}$ so that $\begin{matrix} (3.17) & \int_{Ω} \frac{1}{a (x)} f^{2} (x) d x < \infty \end{matrix}$ holds true.

Remark 3.2.
If we consider a function $v_{ϵ}$ solving the system (3.5)–(3.6) and being such that $\int_{Ω} v_{ϵ} d x = 1$ (in place of (3.7)), and if f is small enough in the norm $L^{2} (a^{- 1} d x)$ , then $v_{ϵ}$ furnishes an approximation to the solution $u_{ϵ}$ (it suffices to apply Lemma 3.1 to $u_{ϵ} - v_{ϵ}$ ).

In practice it is difficult to construct a function $v_{ϵ}$ satisfying both the periodicity condition (3.6) and $\int_{Ω} v_{ϵ} d x = 1$ . Because of that we give in the following another result with relaxed periodicity and integral conditions.

We first take-on to built a function $u_{bd, q}^{ϵ}$ so that $L_{ϵ} u_{bd, q}^{ϵ}$ is close to 0 and $u_{bd, q}^{ϵ} (- 1) - u_{bd, q}^{ϵ} (1) = 1$ .

To do that we first consider for any $q \in N^{}$ a function $φ_{q, ϵ} : \overline{Ω} \to R$ in the form $\begin{matrix} (3.18) & φ_{q, ϵ} (x) = φ_{0} (\frac{x + 1}{ϵ}) + ϵ φ_{1} (\frac{x + 1}{ϵ}) + \dots + ϵ^{q} φ_{q} (\frac{x + 1}{ϵ}), \forall x \in \overline{Ω} \end{matrix}$ with $φ_{j} : R^{+} \to R, j \in N$ . Posing $z = \frac{x + 1}{ϵ}$ , one sees that $\begin{array}{rcl} (L_{ϵ} φ_{q, ϵ}) (z) & = & ϵ \frac{1}{ϵ^{2}} \frac{d}{d z} [φ_{0}^{'} (z) + ϵ φ_{1}^{'} (z) + \dots + ϵ^{q} φ_{q}^{'} (z)] \\ + \frac{1}{ϵ} \frac{d}{d z} {[γ_{0} + γ_{1} ϵ z + \dots + γ_{q} ϵ^{q} z^{q} + γ_{q + 1} (ϵ z) ϵ^{q + 1} z^{q + 1}] \\ \times [φ_{0} (z) + ϵ φ_{1} (z) + \dots + ϵ^{q} φ_{q} (z)]} . \end{array}$

Equating same order in ϵ terms leads to $\begin{array}{l} O (1 / ϵ) : φ_{0}^{'} (z) + γ_{0} φ_{0} (z) = 0, \\ O (1) : φ_{1}^{'} (z) + [γ_{0} φ_{1} (z) + γ_{1} z φ_{0} (z)] = 0, \\ ⋮ \\ O (ϵ^{j - 1}) : φ_{j}^{'} (z) + [γ_{0} φ_{j} (z) + γ_{1} z φ_{j - 1} (z) + \dots + γ_{j} z^{j} φ_{0} (z)] = 0, j \in N^{} . \end{array}$

The boundary conditions in $z = 0$ read: $φ_{0} (0) = 1$ , $φ_{j} (0) = 0$ for $j \in N^{}$ . It results: $\begin{matrix} (3.19) & φ_{0} (z) = e^{- γ_{0} z}, \forall z ⩾ 0 . \end{matrix}$

The next rank equation $\begin{matrix} φ_{1}^{'} (z) + γ_{0} φ_{1} (z) = - γ_{1} z e^{- γ_{0} z}, φ_{1} (0) = 0 \end{matrix}$ has the solution $\begin{matrix} (3.20) & φ_{1} (z) = - \frac{γ_{1} z^{2}}{2} e^{- γ_{0} z} = - \frac{γ_{1} z^{2}}{2} φ_{0} (z) . \end{matrix}$

As a consequence we can prove the following statement:
Proposition 3.1.
For any* $j \in N^{}$ the solution* $φ_{j}$ to the equation $\begin{matrix} (3.21) & φ_{j}^{'} (z) + [γ_{0} φ_{j} (z) + γ_{1} z φ_{j - 1} (z) + \dots + γ_{j} z^{j} φ_{0} (z)] = 0, φ_{j} (0) = 0 \end{matrix}$ reads $φ_{j} (z) = e^{- γ_{0} z} Q_{j} (z)$ , with $Q_{j} (z)$ being a $2 j$ rank polynomial.
Proof.
The proof is recursive: for $j = 1$ the statement is true, see (3.20). Assume the statement holds true for $j - 1$ . From (3.21) $\begin{matrix} φ_{j}^{'} (z) + γ_{0} φ_{j} (z) = - [γ_{1} z φ_{j - 1} (z) + γ_{2} z^{2} φ_{j - 2} (z) + \dots + γ_{j} z^{j} φ_{0} (z)] . \end{matrix}$

Upon multiplying both sides by $e^{γ_{0} z}$ we get $\begin{matrix} \frac{d}{d z} (φ_{j} (z) e^{γ_{0} z}) = - [γ_{1} z Q_{j - 1} (z) + γ_{2} z^{2} Q_{j - 2} (z) + \dots + γ_{j} z^{j} Q_{0} (z)] \end{matrix}$ and clearly the above r.h.s. is a $2 j - 1$ rank polynomial. Integrating leads to $φ_{j} (z) e^{γ_{0} z} = Q_{j} (z)$ , with $Q_{j} (z)$ a $2 j$ rank polynomial. □

Consider now a cut-off function $ζ : R \to R$ with $ζ \in C (R)$ , such that $\begin{array}{l} ζ (z) = 1 for z \in [- 1, - 1 / 2], and \\ ζ (z) = 0 for z \in [- 1 / 4, 1] . \end{array}$ We now define, for any $q \in N^{}$ , the following “boundary function” $u_{bd, q}^{ϵ} : \overline{Ω} \to R$ as $\begin{matrix} (3.22) & u_{bd, q}^{ϵ} = ζ φ_{q, ϵ} \end{matrix}$ with $φ_{q, ϵ}$ defined in (3.18). We easily see that the periodicity defect of $u_{bd, q}^{ϵ}$ is equal to 1 and we shall see in the proof of the next Lemma that it has a small residue. The following result allows to give an approximation of $u_{ϵ}$ : Lemma 3.2.
Suppose that* $v_{ϵ} \in H^{2} (Ω)$ satisfies $\begin{matrix} L_{ϵ} v_{ϵ} = f_{ϵ}^{'} \end{matrix}$ with $f_{e} \in L^{2} (a^{- 1} d x)$ and denote $\begin{matrix} s_{ϵ} = v_{ϵ} (- 1) - v_{ϵ} (1) \end{matrix}$ and $\begin{matrix} λ_{ϵ} = \int_{Ω} (v_{ϵ} - s_{ϵ} u_{bd, q}^{ϵ}) d x \end{matrix}$ with $q \in N^{}$ . Suppose also that* $\begin{matrix} λ_{ϵ} \neq 0 for ϵ small enough . \end{matrix}$ Then we have for any $ϵ > 0$ small enough: $\begin{matrix} {‖ u_{ϵ} - \frac{1}{λ_{ϵ}} (v_{ϵ} - s_{ϵ} u_{bd, q}^{ϵ}) ‖}_{H^{1} (Ω)} ⩽ \frac{c}{| λ_{ϵ} | ϵ} [‖ f_{ϵ} ‖_{L^{2} (a^{- 1} d x)} + | s_{ϵ} | ϵ^{q + 1}], \end{matrix}$ where $c ⩾ 0$ is a constant independent of ϵ and $v_{ϵ}$ .
Proof.
We clearly have that $\begin{matrix} (3.23) & L_{ϵ} u_{bd, q}^{ϵ} = \frac{d}{d x} R_{bd, q, ϵ}, \end{matrix}$ where $\begin{matrix} (3.24) & R_{bd, q, ϵ} = ϵ ζ^{'} φ_{q, ϵ} + ζ (ϵ \frac{d}{d x} φ_{q, ϵ} + a φ_{q, ϵ}) . \end{matrix}$ Passing to z variable and capitalizing on the chosen form of $φ_{q, ϵ}$ leads to $\begin{matrix} (3.25) & ϵ \frac{d}{d x} φ_{q, ϵ} + a φ_{q, ϵ} = Θ_{q + 1, ϵ} (\frac{x + 1}{ϵ}), \end{matrix}$ where $\begin{matrix} (3.26) & Θ_{q + 1, ϵ} (z) = ϵ^{q + 1} Θ_{q + 1} (z) + ϵ^{q + 2} Θ_{q + 2} (z) + \dots + ϵ^{2 q} Θ_{2 q} (z) + ϵ^{2 q + 1} Θ_{2 q + 1} (z) \end{matrix}$ and: $\begin{array}{l} Θ_{q + 1} (z) = γ_{1} z φ_{q} (z) + γ_{2} z^{2} φ_{q - 1} (z) + \dots + γ_{q} z^{q} φ_{1} (z) + γ_{q + 1} (ϵ z) z^{q + 1} φ_{0} (z), \\ Θ_{q + 2} (z) = γ_{2} z^{2} φ_{q} (z) + γ_{3} z^{3} φ_{q - 1} (z) + \dots + γ_{q} z^{q} φ_{2} (z) + γ_{q + 1} (ϵ z) z^{q + 1} φ_{1} (z), \\ ⋮ \\ Θ_{2 q} (z) = γ_{q} z^{q} φ_{q} (z) + γ_{q + 1} (ϵ z) z^{q + 1} φ_{q - 1} (z), \\ Θ_{2 q + 1} (z) = γ_{q + 1} (ϵ z) z^{q + 1} φ_{q} (z) . \end{array}$ Now due to Proposition 3.1 we have, for any $j \in N$ , $\begin{matrix} | φ_{j} (z) | ⩽ c_{j} e^{- (γ_{0} / 2) z}, \forall z ⩾ 0 \end{matrix}$ and $\begin{matrix} | Θ_{j} (z) | ⩽ c_{j} e^{- (γ_{0} / 2) z}, \forall z ⩾ 0 \end{matrix}$ which implies $\begin{matrix} (3.27) & | Θ_{q + 1, ϵ} (z) | ⩽ c ϵ^{q + 1}, \forall z ⩾ 0 \end{matrix}$ and $\begin{array}{l} | φ_{q, ϵ} (x) | ⩽ c exp (- \frac{γ_{0} (x + 1)}{2 ϵ}), \forall x \in \overline{Ω} . \end{array}$ Given that $ζ^{'} (x) = 0$ for $x \notin (- 1 / 2, - 1 / 4)$ , we deduce $\begin{matrix} (3.28) & | ζ^{'} (x) φ_{q, ϵ} (x) | ⩽ c exp (- \frac{γ_{0}}{4 ϵ}), \forall x \in \overline{Ω} . \end{matrix}$ We then obtain from (3.24), (3.25), (3.27) and (3.28) $\begin{matrix} (3.29) & | R_{bd, q, ϵ} (x) | ⩽ c ϵ^{q + 1} | x |^{n}, \forall x \in \bar{Ω} \end{matrix}$ (since $R_{bd, q, ϵ} (x) = 0$ for $x > - 1 / 4$ ) where $c ⩾ 0$ is a constant independent of ϵ.

On the other hand, denoting $\begin{matrix} r_{ϵ} = u_{ϵ} - \frac{1}{λ_{ϵ}} (v_{ϵ} - s_{ϵ} u_{bd, q, ϵ}) \end{matrix}$ we observe that $r_{ϵ}$ satisfies $\begin{array}{l} ϵ r_{ϵ}^{″} + {(a r_{ϵ})}^{'} = - \frac{1}{λ_{ϵ}} (f_{ϵ}^{'} - s_{ϵ} R_{bd, ϵ}^{'}) on Ω, \\ r_{ϵ} (- 1) = r_{ϵ} (1), \\ \int_{Ω} r_{ϵ} d x = 0 . \end{array}$ Now applying Lemma 3.1 and using (3.29) we have the result (observe that the function $x \in Ω \to | x |^{n} \in R$ belongs to $L^{2} (a^{- 1} d x)$ ). □
Remark 3.3.
Lemma 3.2 will be applied with $λ_{ϵ} \to λ_{0} \neq 0$ .

4. Preliminary definitions and lemmas

Define $θ (y) = \frac{α_{0}}{2 n + 1} y^{2 n + 1}$ as a primitive of $α_{0} y^{2 n}$ .

Next we list several useful notations:

For $\forall m \in Z$ , $\forall l, q \in N^{*}$ , $\forall b \in R^{q}$ with $b = (b_{1}, b_{2}, \dots, b_{q})$ , we denote $P_{m, l, b}$ the polynomial defined on $R^{*}$ such that: $\begin{matrix} (4.1) & P_{m, l, b} (y) = b_{1} y^{m} + b_{2} y^{m - l} + \dots + b_{q} y^{m - (q - 1) l} . \end{matrix}$

For any $m \in N$ we define the function $B_{m} : R \to R$ as: $\begin{matrix} (4.2) & B_{m} (y) = e^{- θ (y)} \int_{- \infty}^{y} s^{m} e^{θ (s)} d s, \forall y \in R \end{matrix}$ and for $\forall m \in Z - N$ , we introduce $B_{m} : R^{*} \to R$ as $\begin{array}{l} B_{m} (y) = e^{- θ (y)} \int_{- \infty}^{y} s^{m} e^{θ (s)} d s, for y < 0, \\ (4.3) & B_{m} (y) = e^{- θ (y)} \int_{1}^{y} s^{m} e^{θ (s)} d s, for y > 0 \end{array}$ (this function cannot be defined at $y = 0$ for $m \in Z - N$ ).

We now study the behavior of $B_{m} (y)$ for $| y | \to + \infty$ ; the result is embedded in the following statement:

Lemma 4.1.
For any $m \in Z$ there exist $c_{m}, d_{m} > 0$ such that $\begin{matrix} | B_{m} (y) | ⩽ c_{m} | y |^{m - 2 n}, \forall y \in R with | y | ⩾ d_{m} . \end{matrix}$
Proof.
Step 1:

Remark that for $y > 0$ we have $\begin{matrix} (4.4) & B_{m} (y) = e^{- θ (y)} [\int_{1}^{y} s^{m} e^{θ (s)} d s + r_{m}], \end{matrix}$ where $r_{m} = 0$ for $m \in Z - N$ , and $r_{m} = \int_{- \infty}^{1} s^{m} e^{θ (s)} d s$ for $m \in N$ .

We first prove that there exists $d_{m}^{1} > 0$ such that $\begin{matrix} (4.5) & | B_{m} (y) | ⩽ | y |^{m + 1}, \forall y \in R with | y | ⩾ d_{m}^{1} . \end{matrix}$ Indeed, suppose first that $y < 0$ . Then: $\begin{matrix} | B_{m} (y) | ⩽ e^{- θ (y)} \int_{- \infty}^{y} \frac{1}{s^{2}} | s |^{m + 2} e^{θ (s)} d s . \end{matrix}$

Observe there exists $d_{m}^{1} > 0$ such that the mapping $s \mapsto | s |^{m + 2} e^{θ (s)}$ is an increasing function on $(- \infty, - d_{m}^{1}]$ . Consequently, for $| y |$ large enough, $\begin{matrix} | B_{m} (y) | ⩽ e^{- θ (y)} (\int_{- \infty}^{y} \frac{1}{s^{2}} d s) | y |^{m + 2} e^{θ (y)} = | y |^{m + 1} \end{matrix}$ which gives (4.5).

Let now $y > 0$ . Observe that for y large enough, the mapping $s \in [1, y] \mapsto s^{m} e^{θ (s)}$ reaches its maximum at $s = y$ . Therefore, for y large enough, $\begin{matrix} | B_{m} (y) | ⩽ e^{- θ (y)} [y^{m} e^{θ (y)} (y - 1) + r_{m}] ⩽ y^{m + 1} \end{matrix}$ which ends the proof of (4.5).

Step 2: Let first $y < 0$ . Then: $\begin{matrix} B_{m} (y) = e^{- θ (y)} \int_{- \infty}^{y} s^{m} e^{θ (s)} d s = e^{- θ (y)} \int_{- \infty}^{y} s^{m - 2 n} \frac{1}{α_{0}} {(e^{θ (s)})}^{'} d s \end{matrix}$ which gives $\begin{matrix} (4.6) & B_{m} (y) = \frac{1}{α_{0}} y^{m - 2 n} - \frac{m - 2 n}{α_{0}} B_{m - 2 n - 1} (y) \end{matrix}$ and upon using (4.5) it leads to the result.

Let now $y > 0$ large enough. Using (4.4) we obtain, in a similar way as in the precedent case where $y < 0$ , that: $\begin{matrix} (4.7) & B_{m} (y) = \frac{1}{α_{0}} y^{m - 2 n} - \frac{1}{α_{0}} e^{θ (1) - θ (y)} - \frac{m - 2 n}{α_{0}} [B_{m - 2 n - 1} (y) - r_{m - 2 n - 1} e^{- θ (y)}] + r_{m} e^{- θ (y)} \end{matrix}$ from which we get the desired result using again (4.5). □

We are now in a position allowing to obtain the behavior of $B_{m} (y)$ for large $| y |$ .
Lemma 4.2.
Let $m \in Z$ , $p \in N^{}$ . Then* $B_{m}$ can be written as $\begin{matrix} B_{m} = P_{m - 2 n, 2 n + 1, b^{(m)}} + r_{m, p}, \end{matrix}$ where the vector $b^{(m)} = (b_{1}^{(m)}, \dots, b_{p}^{(m)})$ has its entries defined recursively as follows: $b_{1}^{(m)} = \frac{1}{α_{0}}$ , and, for any $l = 2, 3, \dots, p$ , $b_{l}^{(m)} = \frac{(l - 1) (2 n + 1) - m - 1}{α_{0}} b_{l - 1}^{(m)}$ .

Moreover, the function $r_{m, p} : R^{} \to R$ satisfies the following property: there exist* $c_{m, p}, d_{m, p} > 0$ such that $\begin{matrix} | r_{m, p} (y) | ⩽ c_{m, p} | y |^{m + 1 - (p + 1) (2 n + 1)}, \forall y \in R with | y | ⩾ d_{m, p} . \end{matrix}$
Proof.
Let first $y < 0$ .

Using (4.6) one gets: $\begin{matrix} B_{m} (y) = \frac{y^{m - 2 n}}{α_{0}} + \frac{(2 n + 1) - m - 1}{α_{0}} B_{m - (2 n + 1)} (y) . \end{matrix}$

Next, $\begin{matrix} (4.8) & B_{m - (2 n + 1)} (y) = \frac{y^{m + 1 - 2 (2 n + 1)}}{α_{0}} + \frac{2 (2 n + 1) - m - 1}{α_{0}} B_{m - 2 (2 n + 1)} (y), \end{matrix}$ and in general, for $l = 1, 2, \dots, p - 1$ , one has $\begin{matrix} (4.9) & B_{m - l (2 n + 1)} (y) = \frac{y^{m + 1 - (l + 1) (2 n + 1)}}{α_{0}} + \frac{(l + 1) (2 n + 1) - m - 1}{α_{0}} B_{m - (l + 1) (2 n + 1)} (y) . \end{matrix}$

The last one reads: $\begin{matrix} (4.10) & B_{m - (p - 1) (2 n + 1)} (y) = \frac{y^{m + 1 - p (2 n + 1)}}{α_{0}} + \frac{p (2 n + 1) - m - 1}{α_{0}} B_{m - p (2 n + 1)} (y) . \end{matrix}$

Let the following numbers be defined recursively as: $\begin{matrix} {\hat{b}}_{1} : = 1, \dots, {\hat{b}}_{l} : = \frac{(l - 1) (2 n + 1) - m - 1}{α_{0}} {\hat{b}}_{l - 1}, \forall l = 2, 3, \dots . \end{matrix}$

Next, multiply equation (4.9) with $l = 1$ by ${\hat{b}}_{2}$ , (4.9) with $l = 2$ by ${\hat{b}}_{3}, \dots,$ (4.9) with $l = p - 1$ by ${\hat{b}}_{p}$ and sum-up all terms. Upon canceling equal terms, the result reads: $\begin{array}{rcl} B_{m} (y) & = & \frac{1}{α_{0}} [{\hat{b}}_{1} y^{m + 1 - (2 n + 1)} + {\hat{b}}_{2} y^{m + 1 - 2 (2 n + 1)} + \dots + {\hat{b}}_{p} y^{m + 1 - p (2 n + 1)}] \\ (4.11) & + {\hat{b}}_{p + 1} B_{m - p (2 n + 1)} (y) . \end{array}$

Observe that $b_{l}^{(m)} = \frac{1}{α_{0}} {\hat{b}}_{l}$ , for any $\forall l = 1, 2, \dots$ , and $\begin{matrix} (4.12) & | B_{m - p (2 n + 1)} (y) | ⩽ c_{m - p (2 n + 1)} | y |^{m + 1 - (p + 1) (2 n + 1)} \end{matrix}$ for $| y |$ large enough (see Lemma 4.1) which gives the result.

Next, assume $y > 0$ . In this case, one uses (4.7) and by a chain of reasoning similar to that of case where $y < 0$ one gets $\begin{array}{rcl} B_{m} (y) & = & \frac{1}{α_{0}} [{\hat{b}}_{1} y^{m + 1 - (2 n + 1)} + {\hat{b}}_{2} y^{m + 1 - 2 (2 n + 1)} + \dots + {\hat{b}}_{p} y^{m + 1 - p (2 n + 1)}] \\ (4.13) & + const \cdot e^{- θ (y)} + {\hat{b}}_{p + 1} B_{m - p (2 n + 1)} (y) \end{array}$ and notice that, in this case, the remaining part $r_{m, p} (y)$ will contain the additional term “ $const \cdot e^{- θ (y)}$ ” that tends fast to 0 for $y \to + \infty$ . □

We now must focus on finding convenient solutions $A_{0} (y)$ , $A_{1} (y), \dots, A_{j} (y), \dots,$ of the following differential equations defined recursively right below: $\begin{array}{l} (4.14) & A_{0}^{″} + {(α_{0} y^{2 n} A_{0})}^{'} = 0, \\ (4.15) & A_{1}^{″} + {(α_{0} y^{2 n} A_{1} + α_{1} y^{2 n + 1} A_{0})}^{'} = 0 \end{array}$ and in general, for $j \in N^{}$ , $\begin{matrix} (4.16) & A_{j}^{″} + {(α_{0} y^{2 n} A_{j} + α_{1} y^{2 n + 1} A_{j - 1} + \dots + α_{j} y^{2 n + j} A_{0})}^{'} = 0 . \end{matrix}$

We choose the solution of (4.14) given by $\begin{matrix} (4.17) & A_{0} (y) = e^{- θ (y)} \int_{- \infty}^{y} e^{θ (s)} d s . \end{matrix}$

Hence $A_{0} = B_{0}$ . Next for $j \in N^{}$ we choose as solution of (4.16) the expression given recursively by $\begin{matrix} (4.18) & A_{j} (y) = - e^{- θ (y)} \int_{- \infty}^{y} [α_{1} s^{2 n + 1} A_{j - 1} (s) + α_{2} s^{2 n + 2} A_{j - 2} (s) + \dots + α_{j} s^{2 n + j} A_{0} (s)] e^{θ (s)} d s . \end{matrix}$

Clearly $A_{j} \in C^{\infty} (R)$ and ${lim}_{y \to - \infty} [A_{j} (y) e^{θ (y)}] = 0$ for all $j \in R$ .

The following Lemma gives a polynomial approximation for $A_{m}$ and states the behavior of $A_{m}$ for $| y | \to \infty$ : Lemma 4.3.
Let $m \in N$ , $p \in N^{}$ . Then one has* $\begin{matrix} (4.19) & A_{m} = P_{m - 2 n, 2 n + 1, a^{(m)}} + R_{m, p}, \end{matrix}$ where the vector $a^{(m)} \in R^{p}$ has its entries defined as follows:

$a^{(0)} = b^{(0)}$ and, for any $m ⩾ 1$ , one has $\begin{matrix} a_{i}^{(m)} = - e_{1}^{(m)} b_{i}^{(m)} - e_{2}^{(m)} b_{i - 1}^{(m - (2 n + 1))} - \dots - e_{i}^{(m)} b_{1}^{(m - (i - 1) (2 n + 1))} for i = 1, 2, \dots, p, \end{matrix}$ where $e \in R^{p}$ is such that $e^{(m)} = α_{1} a^{(m - 1)} + α_{2} a^{(m - 2)} + \dots + α_{m} a^{(0)}$ .

Moreover the function $R_{m, p} : R^{} \to R$ satisfies the condition:* $\exists c_{m, p}, d_{m, p} > 0$ such that $\begin{matrix} | R_{m, p} (y) | ⩽ c_{m, p} | y |^{m + 1 - (p + 1) (2 n + 1)}, \forall y \in R with | y | ⩾ d_{m, p} . \end{matrix}$
Proof.
The proof is by recurrence w.r.t. m. The result is obvious for $m = 0$ . Now suppose that $m \in N^{}$ and assume the result is true up to $m - 1$ .

Let $y < 0$ first. Using (4.18) with $j = m$ leads to $\begin{array}{rcl} A_{m} (y) & = & - e^{- θ (y)} \int_{- \infty}^{y} [α_{1} s^{2 n + 1} P_{m - 1 - 2 n, 2 n + 1, a^{(m - 1)}} (s) + α_{2} s^{2 n + 2} P_{m - 2 - 2 n, 2 n + 1, a^{(m - 2)}} (s) + \dots \\ + α_{m} s^{2 n + m} P_{- 2 n, 2 n + 1, a^{(0)}} (s)] e^{θ (s)} d s + R^{(1)} (y), \end{array}$ where $\begin{array}{l} (4.20) & R^{(1)} (y) = - e^{- θ (y)} \int_{- \infty}^{y} [α_{1} s^{2 n + 1} R_{m - 1, p} (s) + α_{2} s^{2 n + 2} R_{m - 2, p} (s) + \dots + α_{m} s^{2 n + m} R_{0, p} (s)] e^{θ (s)} d s . \end{array}$

Invoking the fact that $s^{2 n + l} P_{m - l - 2 n, 2 n + 1, a^{(m - l)}} = P_{m, 2 n + 1, a^{(m - l)}}$ for any $l = 1, 2, \dots, m$ , $A_{m} (y)$ can be re-written as: $\begin{matrix} (4.21) & A_{m} (y) = - e^{- θ (y)} \int_{- \infty}^{y} P_{m, 2 n + 1, e} (s) e^{θ (s)} d s + R^{(1)} (y) \end{matrix}$ with $e = α_{1} a^{(m - 1)} + α_{2} a^{(m - 2)} + \dots + α_{m} a^{(0)}$ (hence $e_{j} = α_{1} a_{j}^{(m - 1)} + α_{2} a_{j}^{(m - 2)} + \dots + α_{m} a_{j}^{(0)}$ , $j = 1, 2, \dots, p$ ). Therefore, $\begin{matrix} (4.22) & A_{m} (y) = - e_{1} B_{m} (y) - e_{2} B_{m - (2 n + 1)} (y) - \dots - e_{p} B_{m - (p - 1) (2 n + 1)} (y) + R^{(1)} (y) . \end{matrix}$

Next, invoking the recurrence hypothesis, we obtain for $| y |$ large enough: $\begin{array}{l} | e^{- θ (y)} \int_{- \infty}^{y} s^{2 n + l} R_{m - l, p} (s) e^{θ (s)} d s | ⩽ c e^{- θ (y)} \int_{- \infty}^{y} | s |^{m - p (2 n + 1)} e^{θ s} d s \end{array}$ which gives using Lemma 4.1 $\begin{array}{l} | e^{- θ (y)} \int_{- \infty}^{y} s^{2 n + l} R_{m - l, p} (s) e^{θ (s)} d s | ⩽ c | y |^{m + 1 - (p + 1) (2 n + 1)}, \forall l = 1, \dots, m . \end{array}$

Henceforth, for $y < 0$ and $| y |$ large enough, $\begin{matrix} (4.23) & | R^{(1)} (y) | ⩽ c | y |^{m + 1 - (p + 1) (2 n + 1)} . \end{matrix}$

Using now (4.22), (4.23) and Lemma 4.2 leads to the result.

Consider next $y > 0$ . Equation (4.18) reads $\begin{array}{rcl} A_{m} (y) & = & const \cdot e^{- θ (y)} - e^{- θ (y)} \int_{1}^{y} [α_{1} s^{2 n + 1} A_{m - 1} (s) + α_{2} s^{2 n + 2} A_{m - 2} (s) + \dots \\ + α_{m} s^{2 n + m} A_{0} (s)] e^{θ (s)} d s, \end{array}$ where $\begin{matrix} const = - \int_{- \infty}^{1} [α_{1} s^{2 n + 1} A_{m - 1} (s) + α_{2} s^{2 n + 2} A_{m - 2} (s) + \dots + α_{m} s^{2 n + m} A_{0} (s)] e^{θ (s)} d s . \end{matrix}$

The rest of the proof is identical to the (precedent) case where $y < 0$ , with $\int_{1}^{y}$ replacing wherever necessary $\int_{- \infty}^{y}$ . □
Remark 4.1.
The above Lemma gives us the behavior of $R_{m, p}$ near $\pm \infty$ . But taking into account the fact that $A_{m}$ is bounded on any bounded set, we deduce immediately from (4.19) that for any $d > 0$ there exists $c > 0$ such that $\begin{array}{l} | R_{m, p} (y) | ⩽ c | y |^{min {m + 1 - p (2 n + 1), 0}}, \forall y \in R^{} with | y | ⩽ d . \end{array}$

5. The asymptotic expansion and the main result

This Section goal is to produce an approximation $u_{app}^{ϵ}$ of $u_{ϵ}$ . We shall apply Lemma 3.2 with $v_{ϵ}$ given by the sum of two functions $u_{int, k}^{ϵ}$ and $u_{ext, k, p}^{ϵ}$ denoted in Section 5.3 by $v_{app}^{ϵ}$ .

The functions introduced above have the following patterns: $u_{int, k}^{ϵ}$ has a small residue for $x \sim 0$ and its not so small residue for x away from zero will be balanced by the presence of $u_{ext, k, p}^{ϵ}$ ; it means $u_{int, k}^{ϵ} + u_{ext, k, p}^{ϵ}$ has a small residue on entire Ω. Also, these functions will have small periodicity defects.

5.1. The function $u_{int, k}^{ϵ}$

Let us set $β : = \frac{1}{2 n + 1}$ . One defines for any $k \in N^{*}$ $\begin{matrix} u_{int, k}^{ϵ} (x) = \frac{1}{ϵ^{β}} A_{k, ϵ} (\frac{x}{ϵ^{β}}), \end{matrix}$ where $A_{k, ϵ} : R \to R$ is in turn defined as: $\begin{matrix} (5.1) & A_{k, ϵ} = A_{0} + ϵ^{β} A_{1} + ϵ^{2 β} A_{2} + \dots + ϵ^{k β} A_{k} . \end{matrix}$

In the above, the $A_{j}$ functions are defined in equations (4.14)–(4.18).

We now carry out the change of variable $y = \frac{x}{ϵ^{β}}$ . For any function $v : Ω_{ϵ} \to R$ , let $\tilde{v} : Ω \to R$ , $\tilde{v} (x) : = v (\frac{x}{ϵ^{β}})$ , where $Ω_{ϵ} = \frac{1}{ϵ^{β}} Ω$ . As a matter of commodity, at instances where no reasonable confusion should arise we’ll use v instead of $\tilde{v}$ .

For any function $v \in H^{2} (Ω_{ϵ})$ we express the residue of $\tilde{v}$ as a function of y by: $\begin{matrix} (5.2) & (L_{ϵ} \tilde{v}) (y) : = ϵ \frac{1}{ϵ^{2 β}} \frac{d^{2} v}{d y^{2}} (y) + \frac{1}{ϵ^{β}} \frac{d}{d y} [a (ϵ^{β} y) v (y)] . \end{matrix}$

The above entails that $\begin{matrix} (5.3) & (L_{ϵ} u_{int, k}^{ϵ}) (y) = ϵ^{1 - 3 β} {(A_{k, ϵ})}^{″} (y) + ϵ^{- 2 β} \frac{d}{d y} [a (ϵ^{β} y) A_{k, ϵ} (y)] . \end{matrix}$

In (5.3) above, both members in the r.h.s. must be of same order in ϵ. As $a (ϵ^{β} y) \sim ϵ^{2 n β}$ , it follows that $ϵ^{1 - 3 β} \sim ϵ^{- 2 β + 2 n β}$ , from which it results $β = \frac{1}{2 n + 1}$ , thus justifying the previously introduced definition of β.

Next, making use of (5.1) and (2.1) into (5.3) gives: $\begin{array}{rcl} (L_{ϵ} u_{int, k}^{ϵ}) (y) & = & ϵ^{1 - 3 β} [A_{0}^{″} (y) + ϵ^{β} A_{1}^{″} (y) + \dots + ϵ^{k β} A_{k}^{″} (y)] \\ + ϵ^{1 - 3 β} \frac{d}{d y} {[α_{0} y^{2 n} + α_{1} ϵ^{β} y^{2 n + 1} + \dots + α_{k} ϵ^{k β} y^{2 n + k} + α_{k + 1} (ϵ^{β} y) ϵ^{(k + 1) β} y^{2 n + k + 1}] \\ \times [A_{0} (y) + ϵ^{β} A_{1} (y) + \dots + ϵ^{k β} A_{k} (y)]} . \end{array}$

From above one gathers: $\begin{array}{l} O (ϵ^{1 - 3 β}) : A_{0}^{″} + {(α_{0} y^{2 n} A_{0})}^{'}, \\ O (ϵ^{1 - 2 β}) : A_{1}^{″} + {(α_{0} y^{2 n} A_{1} + α_{1} y^{2 n + 1} A_{0})}^{'}, \\ ⋮ \\ O (ϵ^{1 + (k - 3) β}) : A_{k}^{″} + {(α_{0} y^{2 n} A_{k} + \dots + α_{k} y^{2 n + k} A_{0})}^{'} . \end{array}$

Invoking the definitions of $A_{0}, \dots, A_{k}$ in (4.14)–(4.18) one observes that the right hand sides in the above are all equal to 0. Consequently we have that $\begin{matrix} (5.4) & (L_{ϵ} u_{int, k}^{ϵ}) (y) = ϵ^{1 - 3 β} \frac{d}{d y} ω^{ϵ} (y), \end{matrix}$ where $\begin{matrix} (5.5) & ω^{ϵ} (y) = ϵ^{(k + 1) β} ω_{1} (y) + ϵ^{(k + 2) β} ω_{2} (y) + \dots + ϵ^{(2 k + 1) β} ω_{k + 1} (y) \end{matrix}$ with $\begin{array}{l} (5.6a) & \begin{matrix} ω_{1} (y) = & α_{1} y^{2 n + 1} A_{k} (y) + α_{2} y^{2 n + 2} A_{k - 1} (y) + \dots + α_{k} y^{2 n + k} A_{1} (y) \\ + α_{k + 1} (ϵ^{β} y) y^{2 n + k + 1} A_{0} (y), \end{matrix} \\ (5.6b) & \begin{matrix} ω_{2} (y) = & α_{2} y^{2 n + 2} A_{k} (y) + α_{3} y^{2 n + 3} A_{k - 1} (y) + \dots + α_{k} y^{2 n + k} A_{2} (y) \\ + α_{k + 1} (ϵ^{β} y) y^{2 n + k + 1} A_{1} (y), \end{matrix} \\ ⋮ \\ (5.6c) & ω_{k} (y) = α_{k} y^{2 n + k} A_{k} (y) + α_{k + 1} (ϵ^{β} y) y^{2 n + k + 1} A_{k - 1} (y), \\ (5.6d) & ω_{k + 1} (y) = α_{k + 1} (ϵ^{β} y) y^{2 n + k + 1} A_{k} (y) . \end{array}$

Remark 5.1.
At this stage it is important to observe that the quantity $ω^{ϵ} (y)$ giving the residue of $u_{int, k}^{ϵ}$ can be chosen as small as needed by selecting a large enough k. This is true only for y bounded and independent of ϵ, which means for $x \sim ϵ^{β}$ . For x away from 0 (e.g. $y \sim ϵ^{- β}$ ) $ω^{ϵ} (y)$ is no longer “small”. This shortcoming is removed by adding the term $u_{ext, k, p}^{ϵ}$ .

5.2. The function $u_{ext, k, p}^{ϵ}$

Consider now $k \in N^{*}$ . We use here Lemma 4.3 to approximate each function $A_{j}$ . In doing so, we approximate each function $ω_{l}$ , $l = 1, 2, \dots, k + 1$ , by a corresponding function $ω_{l, p}$ given in (5.8) below. Specifically, $\begin{array}{rcl} ω_{l} (y) & = & α_{l} y^{2 n + l} A_{k} (y) + α_{l + 1} y^{2 n + l + 1} A_{k - 1} (y) + \dots + α_{k} y^{2 n + k} A_{l} (y) \\ (5.7) & + α_{k + 1} (ϵ^{β} y) y^{2 n + k + 1} A_{l - 1} (y), l = 1, 2, \dots k . \end{array}$

Define now, for any $p \in N^{*}$ , a polynomial-like approximation of $ω_{l}$ as: $\begin{array}{rcl} ω_{l, p} (y) & = & α_{l} y^{2 n + l} P_{k - 2 n, 2 n + 1, a^{(k)}} (y) + α_{l + 1} y^{2 n + l + 1} P_{k - 1 - 2 n, 2 n + 1, a^{(k - 1)}} (y) + \dots \\ + α_{k} y^{2 n + k} P_{l - 2 n, 2 n + 1, a^{(l)}} (y) \\ (5.8) & + α_{k + 1} (ϵ^{β} y) y^{2 n + k + 1} P_{l - 2 n - 1, 2 n + 1, a^{(l - 1)}} (y), l = 1, 2, \dots, k . \end{array}$

We also define: $\begin{matrix} ω_{k + 1, p} (y) = α_{k + 1} (ϵ^{β} y) y^{2 n + k + 1} P_{k - 2 n, 2 n + 1, a^{(k)}} (y) . \end{matrix}$

Next, thanks to the equality $\begin{matrix} y^{2 n + l + j} P_{k - 2 n - j, 2 n + 1, a^{(k - j)}} (y) = P_{l + k, 2 n + 1, a^{(k - j)}} (y) \end{matrix}$ we obtain, for any $j = 0, 1, \dots, k + 1 - l$ , and any $l = 1, 2, \dots, k$ , that $\begin{matrix} (5.9) & ω_{l, p} (y) = P_{l + k, 2 n + 1, g^{(l)}} (y) + α_{k + 1} (ϵ^{β} y) P_{l + k, 2 n + 1, a^{(l - 1)}} (y), \end{matrix}$ where $g^{(l)} \in R^{p}$ is a vector the entries of which are related to those of the vectors $a^{(l)}, a^{(l + 1)}, \dots, a^{(k)} \in R^{p}$ via $\begin{matrix} (5.10) & g^{(l)} = α_{l} a^{(k)} + α_{l + 1} a^{(k - 1)} + \dots + α_{k} a^{(l)}, l = 1, 2, \dots, k . \end{matrix}$

Moreover one has that $\begin{matrix} (5.11) & ω_{k + 1, p} (y) = α_{k + 1} (ϵ^{β} y) P_{2 k + 1, 2 n + 1, a^{(k)}} (y) . \end{matrix}$

We now introduce the approximation $ω_{p}^{ϵ}$ of $ω^{ϵ}$ : $\begin{matrix} (5.12) & ω_{p}^{ϵ} (y) = ϵ^{(k + 1) β} ω_{1, p} (y) + ϵ^{(k + 2) β} ω_{2, p} (y) + \dots + ϵ^{(2 k + 1) β} ω_{k + 1, p} (y) . \end{matrix}$

Next, we introduce the error functions related to approximating $ω_{l}$ by $ω_{l, p}$ , and to approximating $ω^{ϵ}$ by $ω_{p}^{ϵ}$ . They are defined, respectively, as follows (and in doing so we capitalize on (5.7) and Lemma 4.3): $\begin{array}{rcl} ξ_{l, p} (y) & = & α_{l} y^{2 n + l} R_{k, p} (y) + α_{l + 1} y^{2 n + l + 1} R_{k - 1, p} (y) + \dots + α_{k} y^{2 n + k} R_{l, p} (y) \\ (5.13) & + α_{k + 1} (ϵ^{β} y) y^{2 n + k + 1} R_{l - 1, p} (y), \forall l = 1, 2, \dots, k \end{array}$ and $\begin{matrix} ξ_{k + 1, p} (y) = α_{k + 1} (ϵ^{β} y) y^{2 n + k + 1} R_{k, p} (y) . \end{matrix}$

Next, $\begin{matrix} (5.14) & ξ_{p}^{ϵ} (y) = ϵ^{(k + 1) β} ξ_{1, p} (y) + ϵ^{(k + 2) β} ξ_{2, p} (y) + \dots + ϵ^{(2 k + 1) β} ξ_{k + 1, p} (y) . \end{matrix}$

Clearly $ω_{l} = ω_{l, p} + ξ_{l, p}$ , and $ω^{ϵ} = ω_{p}^{ϵ} + ξ_{p}^{ϵ}$ .

On one hand, from Lemma 4.3, for any $\forall l = 1, 2, \dots, k + 1$ , we observe that $\exists c_{l, p} > 0$ , $\exists d_{l, p} > 0$ , such that $\begin{array}{l} (5.15) & | ξ_{l, p} (y) | ⩽ c_{l, p} | y |^{k + l - p (2 n + 1)}, \forall y s.t. | y | ⩾ d_{l, p} . \end{array}$

We also deduce from Remark 4.1 that for any $d > 0$ there exists $c > 0$ such that for any $l = 1, 2, \dots, k + 1$ we have $\begin{matrix} (5.16) & | ξ_{l, p} (y) | ⩽ c | y |^{2 n + l + min {k + 1 - p (2 n + 1), 0}}, \forall y \in R^{*} with | y | ⩽ d . \end{matrix}$

On the other, one has $\begin{matrix} \frac{d ω^{ϵ}}{d y} (y) = ϵ^{β} \frac{d {\tilde{ω}}^{ϵ}}{d x} (x) \end{matrix}$ which leads upon using (5.4) to $\begin{array}{l} (5.17) & (L_{ϵ} u_{int, k}^{ϵ}) (x) = ϵ^{1 - 2 β} \frac{d {\tilde{ω}}^{ϵ}}{d x} (x) = ϵ^{1 - 2 β} \frac{d {\tilde{ω}}_{p}^{ϵ}}{d x} (x) + ϵ^{1 - 2 β} \frac{d {\tilde{ξ}}_{p}^{ϵ}}{d x} (x) . \end{array}$

We shall later define $u_{ext, k, p}^{ϵ}$ as an approximation of a solution of the following equation whose unknown is v: $\begin{matrix} (5.18) & (L_{ϵ} v) (x) = - ϵ^{1 - 2 β} \frac{d {\tilde{ω}}_{p}^{ϵ}}{d x} (x) . \end{matrix}$

In the following we undertake to make explicit ${\tilde{ω}}_{p}^{ϵ}$ (see (5.19) further down).

Invoking (5.12) one gets $\begin{matrix} {\tilde{ω}}_{p}^{ϵ} (x) = ϵ^{(k + 1) β} ω_{1, p} (\frac{x}{ϵ^{β}}) + ϵ^{(k + 2) β} ω_{2, p} (\frac{x}{ϵ^{β}}) + \dots + ϵ^{(2 k + 1) β} ω_{k + 1, p} (\frac{x}{ϵ^{β}}) . \end{matrix}$

Next, using (5.9), $\begin{matrix} ω_{l, p} (\frac{x}{ϵ^{β}}) = P_{l + k, 2 n + 1, g^{(l)}} (\frac{x}{ϵ^{β}}) + α_{k + 1} (x) P_{l + k, 2 n + 1, a^{(l - 1)}} (\frac{x}{ϵ^{β}}), l = 1, 2, \dots, k \end{matrix}$ and, using (5.11), $\begin{matrix} ω_{k + 1, p} (\frac{x}{ϵ^{β}}) = α_{k + 1} (x) P_{2 k + 1, 2 n + 1, a^{(k)}} (\frac{x}{ϵ^{β}}) . \end{matrix}$

We get for ${\tilde{ω}}_{p}^{ϵ}$ the following rather lengthy expression: $\begin{array}{rcl} {\tilde{ω}}_{p}^{ϵ} (x) & = & ϵ^{(k + 1) β} [g_{1}^{(1)} \frac{x^{k + 1}}{ϵ^{(k + 1) β}} + g_{2}^{(1)} \frac{x^{k + 1 - (2 n + 1)}}{ϵ^{β (k + 1 - (2 n + 1))}} + \dots + g_{p}^{(1)} \frac{x^{k + 1 - (p - 1) (2 n + 1)}}{ϵ^{β (k + 1 - (p - 1) (2 n + 1))}} \\ + α_{k + 1} (x) \frac{x^{k + 1}}{ϵ^{(k + 1) β}} a_{1}^{(0)} + α_{k + 1} (x) \frac{x^{k + 1 - (2 n + 1)}}{ϵ^{β (k + 1 - (2 n + 1))}} a_{2}^{(0)} + \dots \\ + α_{k + 1} (x) \frac{x^{k + 1 - (p - 1) (2 n + 1)}}{ϵ^{β (k + 1 - (p - 1) (2 n + 1))}} a_{p}^{(0)}] \\ + ϵ^{(k + 2) β} [g_{1}^{(2)} \frac{x^{k + 2}}{ϵ^{(k + 2) β}} + g_{2}^{(2)} \frac{x^{k + 2 - (2 n + 1)}}{ϵ^{β (k + 2 - (2 n + 1))}} + \dots + g_{p}^{(2)} \frac{x^{k + 2 - (p - 1) (2 n + 1)}}{ϵ^{β (k + 2 - (p - 1) (2 n + 1))}} \\ + α_{k + 1} (x) \frac{x^{k + 2}}{ϵ^{(k + 2) β}} a_{1}^{(1)} + α_{k + 1} (x) \frac{x^{k + 2 - (2 n + 1)}}{ϵ^{β (k + 2 - (2 n + 1))}} a_{2}^{(1)} + \dots \\ + α_{k + 1} (x) \frac{x^{k + 2 - (p - 1) (2 n + 1)}}{ϵ^{β (k + 2 - (p - 1) (2 n + 1))}} a_{p}^{(1)}] \\ + \dots + ϵ^{2 k β} [g_{1}^{(k)} \frac{x^{2 k}}{ϵ^{2 k β}} + g_{2}^{(k)} \frac{x^{2 k - (2 n + 1)}}{ϵ^{β (2 k - (2 n + 1))}} + \dots + g_{p}^{(k)} \frac{x^{2 k - (p - 1) (2 n + 1)}}{ϵ^{β (2 k - (p - 1) (2 n + 1))}} \\ + α_{k + 1} (x) \frac{x^{2 k}}{ϵ^{2 k β}} a_{1}^{(k - 1)} + α_{k + 1} (x) \frac{x^{2 k - (2 n + 1)}}{ϵ^{β (2 k - (2 n + 1))}} a_{2}^{(k - 1)} + \dots \\ + α_{k + 1} (x) \frac{x^{2 k - (p - 1) (2 n + 1)}}{ϵ^{β (2 k - (p - 1) (2 n + 1))}} a_{p}^{(k - 1)}] \\ + ϵ^{(2 k + 1) β} α_{k + 1} (x) [\frac{x^{2 k + 1}}{ϵ^{(2 k + 1) β}} a_{1}^{(k)} + \frac{x^{2 k + 1 - (2 n + 1)}}{ϵ^{β (2 k + 1 - (2 n + 1))}} a_{2}^{(k)} + \dots + \frac{x^{2 k + 1 - (p - 1) (2 n + 1)}}{ϵ^{β (2 k + 1 - (p - 1) (2 n + 1))}} a_{p}^{(k)}] . \end{array}$

The above can be re-written as $\begin{matrix} (5.19) & {\tilde{ω}}_{p}^{ϵ} (x) = ψ_{1} (x) + ϵ ψ_{2} (x) + \dots + ϵ^{p - 1} ψ_{p} (x) \end{matrix}$ where, $\begin{array}{l} ψ_{1} (x) = [g_{1}^{(1)} + a_{1}^{(0)} α_{k + 1} (x)] x^{k + 1} + [g_{1}^{(2)} + a_{1}^{(1)} α_{k + 1} (x)] x^{k + 2} \\ + \dots + [g_{1}^{(k)} + a_{1}^{(k - 1)} α_{k + 1} (x)] x^{2 k} + a_{1}^{(k)} α_{k + 1} (x) x^{2 k + 1}, \\ ψ_{2} (x) = [g_{2}^{(1)} + a_{2}^{(0)} α_{k + 1} (x)] x^{k + 1 - (2 n + 1)} + [g_{2}^{(2)} + a_{2}^{(1)} α_{k + 1} (x)] x^{k + 2 - (2 n + 1)} \\ + \dots + [g_{2}^{(k)} + a_{2}^{(k - 1)} α_{k + 1} (x)] x^{2 k - (2 n + 1)} + a_{2}^{(k)} α_{k + 1} (x) x^{2 k + 1 - (2 n + 1)}, \\ ⋮ \\ ψ_{p} (x) = [g_{p}^{(1)} + a_{p}^{(0)} α_{k + 1} (x)] x^{k + 1 - (p - 1) (2 n + 1)} + [g_{p}^{(2)} + a_{p}^{(1)} α_{k + 1} (x)] x^{k + 2 - (p - 1) (2 n + 1)} \\ (5.20) & + \dots + [g_{p}^{(k)} + a_{p}^{(k - 1)} α_{k + 1} (x)] x^{2 k - (p - 1) (2 n + 1)} + a_{p}^{(k)} α_{k + 1} (x) x^{2 k + 1 - (p - 1) (2 n + 1)} . \end{array}$

Remark 5.2.
As we need all functions $ψ_{1}, ψ_{2}, \dots, ψ_{p}$ to belong to $H^{1} (Ω)$ , it is necessary to assume $\begin{matrix} (5.21) & k + 1 ⩾ (p - 1) (2 n + 1) . \end{matrix}$

Next, we search for $u_{ext, k, p}^{ϵ}$ in the form $\begin{matrix} (5.22) & u_{ext, k, p}^{ϵ} (x) = ϵ^{1 - 2 β} [q_{1} (x) + ϵ q_{2} (x) + \dots + ϵ^{p - 1} q_{p} (x)], \end{matrix}$ where the above functions $q_{1}, \dots, q_{p}$ are chosen in such manner that $\begin{array}{l} (5.23) & (L_{ϵ} u_{ext, k, p}^{ϵ}) (x) “is close to” - ϵ^{1 - 2 β} \frac{d {\tilde{ω}}_{p}^{ϵ}}{d x} (x) \end{array}$ that is $\begin{array}{l} ϵ^{2 - 2 β} \frac{d^{2}}{d x^{2}} [q_{1} (x) + ϵ q_{2} (x) + \dots + ϵ^{p - 1} q_{p} (x)] \\ + ϵ^{1 - 2 β} \frac{d}{d x} [a (x) (q_{1} (x) + ϵ q_{2} (x) + \dots + ϵ^{p - 1} q_{p} (x))] \\ “is close to” - ϵ^{1 - 2 β} \frac{d}{d x} [ψ_{1} (x) + ϵ ψ_{2} (x) + \dots + ϵ^{p - 1} ψ_{p} (x)] . \end{array}$

Equating same order terms, it follows that: $\begin{array}{l} (5.24) & O (ϵ^{1 - 2 β}) : {[a (x) q_{1} (x)]}^{'} = - ψ_{1}^{'} (x), \\ (5.25) & O (ϵ^{2 - 2 β}) : {[a (x) q_{2} (x)]}^{'} = - {[ψ_{2} (x) + q_{1}^{'} (x)]}^{'} \end{array}$ and, in general, $\begin{matrix} (5.26) & O (ϵ^{j - 2 β}) : {[a (x) q_{j} (x)]}^{'} = - {[ψ_{j} (x) + q_{j - 1}^{'} (x)]}^{'}, \forall j = 2, 3, \dots, p . \end{matrix}$

In solving (5.24)–(5.26) above, we shall set all integration constants equal to 0. Next, we shall recursively construct a sequence $q_{1}, \dots, q_{p}$ such that: $\begin{array}{l} (5.27) & a q_{1} = - ψ_{1}, \\ (5.28) & a q_{j + 1} = - (ψ_{j + 1} + q_{j}^{'}), j = 1, 2, \dots, p - 1 . \end{array}$
Definition 5.1.
For any $\forall f \in C^{\infty} (Ω)$ , we shall say the degree $deg (f)$ of f is equal to $l \in N$ if there exists $\exists \hat{f} \in C^{\infty} (Ω)$ , with $\hat{f} (0) \neq 0$ , such that $f (x) = x^{l} \hat{f} (x)$ , for all $x \in Ω$ .
Remark 5.3.
We easily see that the degree is unique. Moreover, if $deg (f) = l \in N^{}$ , then $deg (f^{'}) = l - 1$ .

We now prove the following result:
Proposition 5.1.
Assume that* $k + 2 ⩾ p (2 n + 1)$ . Then the functions $q_{1}, q_{2}, \dots, q_{p}$ defined by ( 5.27 )–( 5.28 ) belong to $C^{\infty} (\overline{Ω})$ and satisfy $deg (q_{j}) ⩾ k + 2 - j (2 n + 1), \forall j = 1, 2, \dots, p$ .
Proof.
Observe, for $j = 1$ , that $\begin{matrix} q_{1} (x) = - \frac{ψ_{1}}{a} (x) = - x^{k + 1 - 2 n} \frac{ψ_{1} (x) / x^{k + 1}}{a (x) / x^{2 n}} . \end{matrix}$

However, $a (x) / x^{2 n} > 0$ on $\overline{Ω}$ , $a (x) / x^{2 n} \in C^{\infty} (\overline{Ω})$ and $ψ_{1} (x) / x^{k + 1} \in C^{\infty} (\overline{Ω})$ . Consequently, $deg (q_{1}) ⩾ k + 1 - 2 n$ and $q_{1} \in C^{\infty} (\overline{Ω})$ .

Next, we proceed recursively on j; we assume that $q_{j}$ exists for $j = 1, 2, \dots, p - 1$ , it is an element of $C^{\infty} (\overline{Ω})$ , and $deg (q_{j}) ⩾ k + 2 - j (2 n + 1)$ . From (5.27) one gets $q_{j + 1} = - \frac{ψ_{j + 1} + q_{j}^{'}}{a}$ . However, $deg (ψ_{j + 1}) ⩾ k + 1 - j (2 n + 1)$ , $deg (q_{j}^{'}) ⩾ k + 1 - j (2 n + 1)$ . Henceforth $deg (q_{j + 1}) ⩾ k + 1 - j (2 n + 1) - 2 n = k + 2 - (j + 1) (2 n + 1)$ which end the proof. □

We then have $\begin{array}{rcl} (L_{ϵ} u_{ext, k, p}^{ϵ}) (x) + ϵ^{1 - 2 β} \frac{d {\tilde{ω}}_{p}^{ϵ}}{d x} (x) & = & ϵ^{2 - 2 β} \frac{d^{2}}{d x^{2}} [q_{1} (x) + ϵ q_{2} (x) + \dots + ϵ^{p - 1} q_{p} (x)] \\ + ϵ^{1 - 2 β} \frac{d}{d x} {a (x) [q_{1} (x) + ϵ q_{2} (x) + \dots + ϵ^{p - 1} q_{p} (x)]} \\ + ϵ^{1 - 2 β} \frac{d}{d x} [ψ_{1} (x) + ϵ ψ_{2} (x) + \dots + ϵ^{p - 1} ψ_{p} (x)] \end{array}$ so we obtain, in view of the choice of $q_{1}, q_{2}, \dots, q_{p}$ , as solutions of (5.27): $\begin{matrix} (5.29) & (L_{ϵ} u_{ext, k, p}^{ϵ}) (x) = - ϵ^{1 - 2 β} \frac{d {\tilde{ω}}_{p}^{ϵ}}{d x} (x) + ϵ^{p + 1 - 2 β} q_{p}^{″} (x) . \end{matrix}$

Summing (5.29) and (5.17) and capitalizing on the linearity of $L_{ϵ}$ leads to $\begin{matrix} (5.30) & (L_{ϵ} (u_{int, k}^{ϵ} + u_{ext, k, p}^{ϵ})) (x) = \frac{d}{d x} ({Res}^{ϵ} (x)), \end{matrix}$ where the above ${Res}^{ϵ}$ denotes the “residue” $\begin{matrix} (5.31) & {Res}^{ϵ} (x) : = ϵ^{p + 1 - 2 β} q_{p}^{'} (x) + ϵ^{1 - 2 β} {\tilde{ξ}}_{p}^{ϵ} (x) . \end{matrix}$

We next undertake to proving the following Lemma: Lemma 5.1.
Assume that $k + 1 ⩾ p (2 n + 1) + n$ . Then $| {Res}^{ϵ} (x) | ⩽ c ϵ^{p + 1 - 2 β} | x |^{n}, \forall x \in Ω$ , where $c ⩾ 0$ is a constant independent of ϵ.
Proof.
From (5.14) we get $\begin{matrix} (5.32) & {\tilde{ξ}}_{p}^{ϵ} (x) = ϵ^{(k + 1) β} ξ_{1, p} (\frac{x}{ϵ^{β}}) + ϵ^{(k + 2) β} ξ_{2, p} (\frac{x}{ϵ^{β}}) + \dots + ϵ^{(2 k + 1) β} ξ_{k + 1, p} (\frac{x}{ϵ^{β}}) . \end{matrix}$

Next, from (5.15) and for any $l = 1, 2, \dots, k + 1$ , $\begin{matrix} | ξ_{l, p} (\frac{x}{ϵ^{β}}) | ⩽ c_{l, p} {| \frac{x}{ϵ^{β}} |}^{k + l - p (2 n + 1)} for | x | ⩾ d_{l, p} ϵ^{β} \end{matrix}$ which entails that $\begin{matrix} | {\tilde{ξ}}_{p}^{ϵ} (x) | ⩽ \sum_{l = 1}^{k + 1} ϵ^{(k + l) β} c_{l, p} ϵ^{p β (2 n + 1) - β (k + l)} | x |^{k + l - p (2 n + 1)} . \end{matrix}$

Consequently there exist $c_{p}, d_{p} > 0$ independent of ϵ such that $\begin{matrix} (5.33) & | {\tilde{ξ}}_{p}^{ϵ} (x) | ⩽ c ϵ^{p} | x |^{k + 1 - p (2 n + 1)} ⩽ c_{p} ϵ^{p} | x |^{n} for | x | ⩾ d_{p} ϵ^{β} \end{matrix}$ (since $| x | ⩽ 1$ ).

Next, assume $| x | ⩽ d_{p} ϵ^{β}$ (i.e. $| y | ⩽ d_{p}$ ). From (5.32), (5.16) and the hypothesis of this lemma, one gets $\begin{matrix} | {\tilde{ξ}}_{p}^{ϵ} (x) | ⩽ c_{1} \sum_{l = 1}^{k + 1} ϵ^{(k + l) β} {(\frac{| x |}{ϵ^{β}})}^{2 n + l} \end{matrix}$ (remark that ${\tilde{ξ}}_{p}^{ϵ}$ can be extended by 0 in $x = 0$ ), so $\begin{matrix} | {\tilde{ξ}}_{p}^{ϵ} (x) | ⩽ c_{1} ϵ^{(k + 1) β} {(\frac{| x |}{ϵ^{β}})}^{n} d_{p}^{n + l} = c_{2} ϵ^{(k + 1 - n) β} | x |^{n} . \end{matrix}$

It follows that $\begin{matrix} (5.34) & | {\tilde{ξ}}_{p}^{ϵ} (x) | ⩽ c ϵ^{p} | x |^{n} for | x | ⩽ d_{p} ϵ^{β} . \end{matrix}$

Invoking (5.33) and (5.34) it now results that $\begin{matrix} (5.35) & | {\tilde{ξ}}_{p}^{ϵ} (x) | ⩽ c ϵ^{p} | x |^{n}, \forall x \in Ω, \end{matrix}$ where c is independent of ϵ. Moreover, from Proposition 5.17 it follows $deg (q_{p}^{'}) ⩾ k + 1 - p (2 n + 1)$ . Therefore, $\begin{matrix} (5.36) & | q_{p}^{'} (x) | ⩽ c | x |^{k + 1 - p (2 n + 1)} ⩽ c | x |^{n} . \end{matrix}$

Equations (5.35) and (5.36) eventually give the stated result. □

5.3. The function $v_{app}^{ϵ}$ and the convergence of its periodicity defect and integral

Out of a concern for clarity and to ease the reading we recall here that, for any $k, p \in N^{*}$ , $\begin{matrix} u_{int, k}^{ϵ} (x) : = \frac{1}{ϵ^{β}} \sum_{j = 0}^{k} ϵ^{j β} A_{j} (\frac{x}{ϵ^{β}}) \end{matrix}$ with $A_{j}$ being defined in (4.17)–(4.18), and that $\begin{matrix} u_{ext, k, p} (x) : = ϵ^{1 - 2 β} \sum_{j = 1}^{p} ϵ^{j - 1} q_{j} (x), \end{matrix}$ where the $q_{j}$ functions are defined in (5.27)–(5.28); in passing one observes each $q_{j}$ also depends on k.

We now define $\begin{matrix} (5.37) & v_{app}^{ϵ} = u_{int, k}^{ϵ} + u_{ext, k, p}^{ϵ} . \end{matrix}$ Remark that $v_{app}^{ϵ}$ depends also on k and p, but to avoid cumbersome notations (and keep them tractable) dependence upon these indices is dropped.

Now from (5.30) we have $\begin{matrix} (L_{ϵ} v_{app}^{ϵ}) (x) = \frac{d}{d x} ({Res}^{ϵ} (x)) \end{matrix}$ with ${Res}^{ϵ}$ being defined in (5.31).

We have the following results:

Lemma 5.2.
The periodicity defect of $v_{app}^{ϵ}$ tends to 0 for $ϵ \to 0$ . More precisely, we have the estimate: $\begin{matrix} | v_{app}^{ϵ} (- 1) - v_{app}^{ϵ} (1) | ⩽ C ϵ^{(2 n - 1) β}, C ⩾ 0 . \end{matrix}$
Proof.
With (5.37) in mind, we actually have to estimate: $\begin{matrix} | u_{int, k}^{ϵ} (- 1) + u_{ext, k, q}^{ϵ} (- 1) - u_{int, k}^{ϵ} (1) - u_{ext, k, q}^{ϵ} (1) | . \end{matrix}$

Recall that $u_{int, k}^{ϵ} (x) = \frac{1}{ϵ^{β}} A_{k, ϵ} (\frac{x}{ϵ^{β}})$ . From Lemma 4.3 and (4.1) one infers: $\begin{matrix} u_{int, k}^{ϵ} (- 1) - u_{int, k}^{ϵ} (1) = \sum_{l = 0}^{k} ϵ^{(l - 1) β} [A_{l} (- \frac{1}{ϵ^{β}}) - A_{l} (\frac{1}{ϵ^{β}})] . \end{matrix}$

From Lemma 4.3 it is clear that $A_{l} (\pm ϵ^{- β}) \sim ϵ^{(2 n - l) β}$ for $ϵ \to 0$ , implying $\begin{matrix} | u_{int, k}^{ϵ} (- 1) - u_{int, k}^{ϵ} (1) | ⩽ C ϵ^{(2 n - 1) β} . \end{matrix}$

On the other hand, from (5.22) we have $\begin{matrix} | u_{ext, k, p}^{ϵ} (x) | ⩽ C ϵ^{1 - 2 β}, \forall x \in \overline{Ω} \end{matrix}$ which gives the result. □
Lemma 5.3.
We have $\begin{matrix} (5.38) & \int_{Ω} v_{app}^{ϵ} (x) d x \to \int_{- \infty}^{\infty} A_{0} (y) d y as ϵ \to 0 . \end{matrix}$

More precisely, we have the following estimate: $\begin{matrix} (5.39) & | \int_{Ω} v_{app}^{ϵ} (x) d x - \int_{- \infty}^{\infty} A_{0} (y) d y | ⩽ C ϵ^{(2 n - 1) β} log (ϵ^{- 1}) . \end{matrix}$
Proof.
We have $\begin{array}{rcl} \int_{Ω} u_{int, k}^{ϵ} (x) d x & = & \frac{1}{ϵ^{β}} \int_{- 1}^{1} A_{k, ϵ} (\frac{x}{ϵ^{β}}) d x = \int_{- ϵ^{- β}}^{ϵ^{- β}} A_{k, ϵ} (y) d y \\ (5.40) & = & I_{0, ϵ} + I_{1, ϵ} + \dots + \dots + I_{k, ϵ}, \end{array}$ where for any $j = 0, 1, \dots, k$ we denote $\begin{matrix} I_{j, ϵ} = ϵ^{j β} [\int_{- ϵ^{- β}}^{ϵ^{- β}} A_{j} (y) d y] . \end{matrix}$

From Lemma 4.3 we deduce that for large enough $| y |$ we have $| A_{j} (y) | \sim | y |^{j - 2 n}$ ; several cases listed below are possible:
for $0 ⩽ j ⩽ 2 n - 2$ we have that $\int_{- \infty}^{+ \infty} A_{j} (y) d y < + \infty$ , which allows us to write $\begin{matrix} I_{j, ϵ} = ϵ^{j β} [\int_{- \infty}^{+ \infty} A_{j} (y) d y - \int_{- \infty}^{- ϵ^{- β}} A_{j} (y) d y - \int_{ϵ^{- β}}^{+ \infty} A_{j} (y) d y] . \end{matrix}$

We have $\begin{matrix} | \int_{- \infty}^{- ϵ^{- β}} A_{j} (y) d y | ⩽ C \int_{- \infty}^{- ϵ^{- β}} | y |^{j - 2 n} d y ⩽ \frac{C}{2 n - j - 1} ϵ^{(2 n - j - 1) β} \end{matrix}$ and in the same manner we obtain $\begin{matrix} | \int_{ϵ^{- β}}^{\infty} A_{j} (y) d y | ⩽ \frac{C}{2 n - j - 1} ϵ^{(2 n - j - 1) β} \end{matrix}$ which finally gives $\begin{matrix} (5.41) & | I_{j, ϵ} - ϵ^{j β} \int_{- \infty}^{+ \infty} A_{j} (y) d y | ⩽ C ϵ^{(2 n - 1) β} . \end{matrix}$

for $j = 2 n - 1 ⩾ 1$ , $\begin{array}{l} (5.42) & I_{2 n - 1, ϵ} = ϵ^{β (2 n - 1)} (\int_{- ϵ^{- β}}^{- 1} + \int_{- 1}^{1} + \int_{1}^{ϵ^{- β}}) A_{2 n - 1} (y) d y . \end{array}$

Since $A_{2 n - 1} (y) \sim \frac{1}{| y |}$ for $| y |$ large enough we deduce $\begin{matrix} (5.43) & | I_{2 n - 1, ϵ} | ⩽ C ϵ^{(2 n - 1) β} log (ϵ^{- 1}) . \end{matrix}$

for $j ⩾ 2 n$ , $\begin{matrix} (5.44) & | I_{j, ϵ} | ⩽ C ϵ^{j β} \int_{- ϵ^{- β}}^{ϵ^{- β}} | y |^{j - 2 n} d y ⩽ 2 C ϵ^{j β} ϵ^{- β} ϵ^{- β (j - 2 n)} = 2 C ϵ^{β (2 n - 1)} . \end{matrix}$

We then deduce from (5.40), (5.41), (5.43) and (5.44) that $\begin{matrix} (5.45) & | \int_{Ω} u_{int, k}^{ϵ} (x) d x - \int_{- \infty}^{\infty} A_{0} (y) d y | ⩽ C ϵ^{(2 n - 1) β} log (ϵ^{- 1}) . \end{matrix}$

On the other hand from the definition (5.22) of $u_{ext, k, p}^{ϵ}$ it is clear that $\begin{matrix} | \int_{Ω} u_{ext, k, p}^{ϵ} (x) d x | ⩽ C ϵ^{(2 n - 1) β} \end{matrix}$ and from (5.45) we obtain the result. □

5.4. The main result

Now we are able to give an approximation $u_{app}^{ϵ}$ of $u_{ϵ}$ and to prove error estimates.

We define for any $k, p, q \in N^{*}$ : $\begin{matrix} u_{app}^{ϵ} (x) : = \frac{1}{λ_{app, ϵ}} [v_{app}^{ϵ} (x) - s_{app, ϵ} u_{bd, q}^{ϵ} (x)], \end{matrix}$ where $\begin{matrix} λ_{app, ϵ} = \int_{Ω} v_{app}^{ϵ} (x) d x - s_{app, ϵ} \int_{Ω} u_{bd, q}^{ϵ} (x) d x \end{matrix}$ and $\begin{matrix} s_{app, ϵ} = v_{app}^{ϵ} (- 1) - v_{app}^{ϵ} (1) \end{matrix}$ with $v_{app}^{ϵ}$ defined in (5.37).

We now give the main result of this paper which allows to obtain an approximation of the solution up to any prescribed (desired) error:

Theorem 5.1 (Main Result).

For any $k, p, q \in N^{*}$ such that $\begin{matrix} k + 1 ⩾ p (2 n + 1) + 2 n \end{matrix}$ we have $\begin{matrix} ‖ u_{ϵ} - u_{app}^{ϵ} ‖_{H^{1} (Ω)} ⩽ c (ϵ^{p - 2 β} + ϵ^{q + (2 n - 1) β}) . \end{matrix}$

Proof.
We use Lemma 3.2 with $v_{ϵ} = v_{app}^{ϵ}$ defined in (5.37), $s_{ϵ} = s_{app, ϵ}$ , $λ_{ϵ} = λ_{app, ϵ}$ and $f_{ϵ} = {Res}^{ϵ}$ with ${Res}^{ϵ}$ defined in (5.31). From Lemma 5.1 we deduce that ${Res}^{ϵ} \in L^{2} (a^{- 1} d x)$ with $\begin{matrix} {‖ {Res}^{ϵ} ‖}_{L^{2} (a^{- 1} d x)} ⩽ c ϵ^{p + 1 - 2 β} . \end{matrix}$ From Lemma 5.2 and Lemma 5.3 we deduce that $\begin{matrix} (5.46) & λ_{ϵ} \to \int_{- \infty}^{+ \infty} A_{0} (y) d y > 0 as ϵ \to 0 \end{matrix}$ and this allows to obtain the result. □

We deduce from the above theorem the following consequence: Corollary 5.1.
Under the hypothesis of Theorem 5.1 we have $\begin{matrix} {‖ u_{ϵ} - \frac{1}{ϵ^{β} {\bar{A}}_{0}} A_{0} (\frac{x}{ϵ^{β}}) - \frac{1}{{\bar{A}}_{0}} A_{1} (\frac{x}{ϵ^{β}}) ‖}_{L^{\infty} (Ω)} ⟶ 0 as ϵ \to 0, \end{matrix}$ where $\begin{matrix} {\bar{A}}_{0} = \int_{- \infty}^{\infty} A_{0} (y) d y . \end{matrix}$
Proof.
It suffices to observe, using Lemma 4.3, that for any $j \in N^{*}$ we have that $\begin{matrix} ϵ^{j} A_{j + 1} (\frac{x}{ϵ^{β}}) \to 0 for ϵ \to 0 in the norm L^{\infty} (Ω) \end{matrix}$ and also to use (5.46). □
Remark 5.4.
It is well known that $\begin{matrix} \frac{1}{ϵ^{β} {\bar{A}}_{0}} A_{0} (\frac{x}{ϵ^{β}}) \to_{ϵ \to 0}^{D^{'} (Ω)} δ_{0} \end{matrix}$ and $\begin{matrix} A_{1} (\frac{x}{ϵ^{β}}) \to_{ϵ \to 0}^{D^{'} (Ω)} 0 . \end{matrix}$ We then deduce from Corollary 5.1 the following convergence in the sense of distributions: $\begin{matrix} u_{ϵ} \to_{ϵ \to 0}^{D^{'} (Ω)} δ_{0} \end{matrix}$ which comforts the heuristic assumption that has been made in [13].

6. Particular cases

6.1. A simple theoretical case

We begin with a very simple case for this problem, for which $\begin{matrix} a (x) = x^{2 n}, \forall x \in Ω . \end{matrix}$

One has $\begin{matrix} α_{0} = 1 and α_{j} = 0, \forall j \in N^{*} . \end{matrix}$

Notice one actually controls the approximation error for $u_{ϵ}$ : using Theorem 5.1 take for instance $p = 2$ and $q = 1$ . As a consequence, one gets for the approximation of the solution an error of order $ϵ^{4 n / (2 n + 1)}$ . Next, capitalizing on the assumption made in Theorem 5.1 we can take $k = 6 n + 1$ .

We easily find that $A_{j} = 0, \forall j \in N^{*}$ , and $ψ_{1} = ψ_{2} = 0$ , which gives $\begin{matrix} u_{int, k}^{ϵ} = \frac{1}{ϵ^{β}} A_{0} (\frac{x}{ϵ^{β}}) \end{matrix}$ and $\begin{matrix} u_{ext, k, p}^{ϵ} = 0 . \end{matrix}$ Moreover, we also have (see (3.22) and (3.18)) $\begin{matrix} u_{bd, q}^{ϵ} = ζ φ_{1, ϵ} \end{matrix}$ with ζ the cut-off function introduced in Section 3 and $\begin{matrix} φ_{1, ϵ} (x) = φ_{0} (\frac{x + 1}{ϵ}) + ϵ φ_{1} (\frac{x + 1}{ϵ}), \forall x \in \overline{Ω}, \end{matrix}$ where, for any $z ⩾ 0$ , one defines $\begin{matrix} φ_{0} (z) = e^{- z} and φ_{1} (z) = n z^{2} e^{- z} \end{matrix}$ (since $γ_{0} = 1$ and $γ_{1} = - 2 n$ ).

6.2. An important case of practical relevance

As announced in Section 1, an important case for the applications is when $\begin{matrix} a (x) = {sin}^{2} (\frac{π}{2} x), \forall x \in Ω . \end{matrix}$ With $n = 1$ it gives $β = 1 / 3$ .

Take again $p = 2$ and $q = 1$ ; then $k = 7$ , which gives an error of approximation of order $ϵ^{4 / 3}$ .

Observe that $\begin{matrix} a (x) = \frac{1}{2} [1 - cos (π x)] . \end{matrix}$

Next, $\begin{matrix} cos (π x) = \sum_{k = 0}^{4} {(- 1)}^{k} \frac{π^{2 k}}{(2 k)!} x^{2 k} + x^{10} Q (x) \end{matrix}$ with $Q \in C^{\infty} (\overline{Ω})$ given by $\begin{matrix} (6.1) & Q (x) = \frac{1}{x^{10}} (cos (π x) - \sum_{k = 0}^{4} {(- 1)}^{k} \frac{π^{2 k}}{(2 k)!} x^{2 k}) = \sum_{k = 0}^{+ \infty} {(- 1)}^{k + 1} \frac{π^{2 k + 10}}{(2 k + 10)!} x^{2 k} . \end{matrix}$

Therefore, $\begin{matrix} a (x) = α_{0} x^{2} + α_{1} x^{3} + \dots + α_{7} x^{9} + α_{8} (x) x^{10} \end{matrix}$ with $\begin{array}{l} α_{0} = \frac{π^{2}}{4}, α_{2} = - \frac{π^{4}}{2 \cdot 4!}, α_{4} = \frac{π^{6}}{2 \cdot 6!}, α_{6} = - \frac{π^{8}}{2 \cdot 8!}, \\ α_{1} = α_{3} = α_{5} = α_{7} = 0 \end{array}$ and $\begin{matrix} α_{8} (x) = - \frac{1}{2} Q (x) \end{matrix}$ with Q given by (6.1).

We now focus on $u_{int, k}^{ϵ}$ , $u_{ext, k, p}^{ϵ} (x)$ , $u_{bd, q}^{ϵ} (x)$ seriatim.

Thus: $\begin{array}{rcl} u_{int, k}^{ϵ} (x) & = & ϵ^{- 1 / 3} A_{7, ϵ} (ϵ^{- 1 / 3} x) \\ = & ϵ^{- 1 / 3} [A_{0} (ϵ^{- 1 / 3} x) + ϵ^{1 / 3} A_{1} (ϵ^{- 1 / 3} x) + ϵ^{2 / 3} A_{2} (ϵ^{- 1 / 3} x) \\ (6.2) & + \dots + ϵ^{7 / 3} A_{7} (ϵ^{- 1 / 3} x)] \end{array}$ with $A_{0}$ being given in (4.17) and $A_{1}$ to $A_{7}$ in (4.18). They can be numerically calculated right away from (4.18); however, one also easily sees that $A_{1}$ to $A_{7}$ can all be expressed in terms of $A_{0}$ if so desired (though some rather tedious calculations are involved).

It is important, at this stage, to notice that our $A_{0}$ function is actually, up to a multiplicative constant, the g function given in (1.4), thus bridging and relating our work to the results of [13].

Next, $\begin{matrix} (6.3) & u_{ext, k, p}^{ϵ} (x) = ϵ^{1 / 3} [q_{1} (x) + ϵ q_{2} (x)], \end{matrix}$ where $\begin{matrix} (6.4) & q_{1} (x) = - \frac{ψ_{1} (x)}{a (x)} = - \frac{ψ_{1} (x)}{{sin}^{2} (π x / 2)} . \end{matrix}$

From (5.20) we deduce $\begin{array}{rcl} ψ_{1} (x) & = & x^{8} [g_{1}^{(1)} + a_{1}^{(0)} α_{8} (x)] + x^{9} [g_{1}^{(2)} + a_{1}^{(1)} α_{8} (x)] + x^{10} [g_{1}^{(3)} + a_{1}^{(2)} α_{8} (x)] + \dots \\ (6.5) & + x^{14} [g_{1}^{(7)} + a_{1}^{(6)} α_{8} (x)] + x^{15} a_{1}^{(7)} α_{8} (x), ψ_{1} \in C^{\infty} . \end{array}$

Remark 6.1.
It is not necessary to compute $q_{2}$ since $ϵ^{4 / 3} q_{2}$ is of the same order as the error of approximation.

Given that (see (5.10)) $\begin{matrix} g^{(l)} = α_{l} a^{(7)} + α_{l + 1} a^{(6)} + \dots + α_{7} a^{(l)} \in R^{2}, \forall l = 1, 2, \dots, 7 \end{matrix}$ we now need to calculate the vectors $a^{(0)}, \dots, a^{(7)} \in R^{2}$ . By (somewhat tedious) elementary calculations, invoking Lemma 4.2 and Lemma 4.3, we obtain $\begin{array}{l} a^{(0)} = (\frac{4}{π^{2}}, \frac{32}{π^{4}}), \\ a^{(2)} = (\frac{1}{3}, \frac{8}{3 π^{2}}), \\ a^{(4)} = (\frac{π^{2}}{60}, 0), \\ a^{(6)} = (\frac{π^{4}}{1512}, - \frac{31 π^{2}}{1890}) \end{array}$ and $\begin{matrix} a^{(1)} = a^{(3)} = a^{(5)} = a^{(7)} = (0, 0) . \end{matrix}$

Now we can calculate the $g^{(l)}$ vectors. It turns out that $g^{(2)} = g^{(4)} = g^{(6)} = g^{(7)} = (0, 0)$ , and the non-zero ones are: $\begin{array}{l} g^{(1)} = α_{2} a^{(6)} + α_{4} a^{(4)} + α_{6} a^{(2)} = (- \frac{23 π^{8}}{3628800}, \frac{π^{6}}{3240}), \\ g^{(3)} = α_{4} a^{(6)} + α_{6} a^{(4)} = (- \frac{π^{10}}{1741824}, - \frac{31 π^{8}}{2721600}), \\ g^{(5)} = α_{6} a^{(6)} = (- \frac{π^{12}}{121927680}, \frac{31 π^{10}}{152409600}) . \end{array}$

Consequently $ψ_{1} (x)$ now follows right away.

The last term in need be calculated is $u_{bd, q}^{ϵ} (x)$ , with $u_{bd, q}^{ϵ} (x) = ζ φ_{1, ϵ} (x)$ . Since in this case $γ_{0} = 1$ and $γ_{1} = 0$ one obtains $\begin{matrix} φ_{1, ϵ} (x) = e^{- (x + 1) / ϵ}, \forall x \in Ω \end{matrix}$ from which $u_{bd, q}^{ϵ} (x)$ follows on the spot.
7. Conclusions

The problem of non-Newtonian fluids fast flows is both an utterly important one in view of the large number of industrial applications and a challenging one, in particular when the fluid constitutive equation derives from a kinetical (molecular description) theory. In the later case, one first has to solve the configurational probability diffusion equation for large velocity gradients, i.e. to obtain an asymptotic solution valid for this particular flows. This probability is the key ingredient to calculating the corresponding stress tensor as (generally) a second order moment. For the Rigid Dumbbell Model (RDM) type of polymer chains, in [13] Öttinger used educated, skillful intuition to conjecture the solution of the configurational probability equation must exhibit a particular pattern: it has to be “concentrated” in the neighborhood of a preferred angle. Consequently he derived an analytical expression assigned to describe the solution.

In this paper we rigorously solved the problem by obtaining an asymptotic solution to the corresponding equation that is of a turning-point type. The techniques used in doing so are expected to apply to a wide variety of turning point type mathematical physics problems, in particular to those where asymptotic solutions are the main focus.

Footnotes

Acknowledgements

The authors are grateful to the Referee for useful comments on the originally submitted manuscript.

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A turning point asymptotic expansion for a rigid-dumbbell polymer fluid probability configurational equation for fast shear flows

Abstract

Keywords

1. Introduction

2. The problem under scrutiny

5.1. The function u int , k ϵ

Theorem 5.1 (Main Result).

6.1. A simple theoretical case

6.2. An important case of practical relevance

Footnotes

Acknowledgements

References

5.1. The function $u_{int, k}^{ϵ}$