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Ordinary Differential Equations 211
(9.6.2)
n
k=0
1
ck T
−1
′ k φ ′ dx =
1
−1
fφ dx, ∀φ ∈ P 0 n.
The coefficients ck, 0 ≤ k ≤ n, are determined from the boundary conditions (these
imply n k=0 ck = n k=0 (−1)kck = 0) and testing (9.6.2) on the set of polynomials
φ(x) := Tj(x) − 1
(1 + x), x ∈ Ī, 2 ≤ j ≤ n. The matrix corresponding
2 (−1)j (1 − x) − 1
2
to the linear system is full and its entries are obtained by computing the integrals
Ikj := 1
−1 T ′ kT ′ jdx. We can provide an explicit expression of these quantities by recalling
(1.5.7) and arguing as in section 2.6. In this approach, we can combine the benefits of
working with Chebyshev polynomials, with a variational formulation which does not
involve singular weight functions. The collocation scheme is instead obtained by taking
pn = n−1 (n)
j=1 pn(η j ) ˜l (n)
j
9.7 Boundary layers
in (9.4.14), and testing on φ := ˜ l (n)
j , 1 ≤ j ≤ n − 1.
Small perturbations of certain differential operators can force the solution to degen-
erate in a singular behavior near the boundary points, which is often recognized as a
boundary layer. The terminology was first introduced in prandtl(1905). General mo-
tivations and theoretical results are given in eckhaus(1979), chang and howes(1984),
lagerstrom(1988). A typical example is given by the equation
(9.7.1)
⎧
⎨
−ǫ U ′′
ǫ + Uǫ = 0 in I :=] − 1,1[,
⎩
Uǫ(−1) = 0, Uǫ(1) = 1,
where ǫ > 0 is a given parameter. The solution of (9.7.1) is
(9.7.2) Uǫ(x) = e(1+x)/√ ǫ − e −(x+1)/ √ ǫ
e 2/√ ǫ − e −2/ √ ǫ
, x ∈ Ī.