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Ordinary Differential Equations 195
(9.3.18) Bw(ψ,φ) :=
(9.3.19) Fw(φ) :=
(ψ −
I
˜ ψ) ′ [(φ − ˜ φ)w] ′ dx +
+ µ ψ(φ −
I
˜
φ)w dx + µ
I
f(φ − ˜ φ)w dx +
I
I
I
˜ψ ′ ˜ φ ′ dx
ψ ˜ φ dx, ∀ψ,φ ∈ X,
f ˜ φ dx + σ2φ(1) − σ1φ(−1), ∀φ ∈ X.
In the above expressions, for a given function χ ∈ X, we have defined ˜χ ∈ X to be
the function ˜χ(x) := 1
1
2 (1 − x)χ(−1) + 2 (1 + x)χ(1), x ∈ Ī. We note that ˜χ ∈ P1
and χ − ˜χ ∈ H 1 0,w(I). For ν = 0, (9.3.18) and (9.3.19) give respectively (9.3.15) and
(9.3.16). As usual, we are interested in finding the solution U ∈ X of (9.3.4).
Without entering into details of the proof, we claim that it is possible to find a constant
µ ∗ > 0 such that, for any µ ∈]0,µ ∗ ], the bilinear form Bw in (9.3.18) fulfills the
requirements (9.3.5) and (9.3.6). This result is given in funaro(1988) for the Chebyshev
case and can be easily generalized to other Jacobi weights. This implies existence and
uniqueness of a weak solution U. When f ∈ C0 ( Ī), we can relate U to problem (9.1.5)
by arguing as follows. Considering that ˜ φ ′ ∈ P0 is a constant function, we write
(9.3.20)
=
Ũ
I
′ φ ˜′ dx = (Ũ(1) − Ũ(−1))˜ φ ′ = (U(1) − U(−1)) ˜ φ ′
U
I
′ φ ˜′ ′
dx = U (1) φ(1) ˜ ′
− U (−1) φ(−1) ˜ −
Therefore, since Ũ ′′ ≡ 0 and ˜ φ(±1) = φ(±1), one has
(9.3.21) Bw(U,φ) = (−U ′′ + µU)(φ − ˜ φ)w dx +
I
I
I
U ′′ ˜ φ dx.
(−U ′′ + µU) ˜ φ dx
+ U ′ (1)φ(1) − U ′ (−1)φ(−1) = Fw(φ), ∀φ ∈ X.
One first uses test functions φ belonging to H 1 0,w(I) (therefore ˜ φ ≡ 0). This implies
the differential equation −U ′′ + µU = f in I. Finally, we eliminate the integrals and
obtain the boundary conditions.