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194 Polynomial Approximation of Differential Equations
We now propose a weak formulation of the Neumann problem (9.1.5). Let us
first discuss the Legendre case. We set X ≡ H 1 w(I), where w ≡ 1. We introduce
Bw : X × X → R and Fw : X → R such that
(9.3.15) Bw(ψ,φ) :=
(9.3.16) Fw(ψ) :=
I
ψ
I
′ φ ′ dx + µ
I
ψφ dx, ∀ψ,φ ∈ X,
fφ dx + σ2φ(1) − σ1φ(−1), ∀φ ∈ X.
According to these new definitions, we consider the problem of finding U ∈ X which
satisfies (9.3.4).
Theorem 9.3.3 - Let w ≡ 1 and X ≡ H 1 w(I). Let Bw and Fw be respectively defined
by (9.3.15) and (9.3.16). Then, problem (9.3.4) has a unique weak solution U ∈ X.
Moreover, if U ∈ C2 ( Ī) , then U is solution of (9.1.5).
Proof - It is an easy matter to verify the hypotheses of theorem 9.3.1. In particular,
to prove (9.3.7) we note that, by (5.7.4), one has |φ(±1)| ≤ φ C 0 (Ī) ≤ K2K −1
1 φX.
Then, we have existence and uniqueness of a weak solution U ∈ X. When U is more
regular, we can write
(9.3.17) U ′ (1)φ(1) − U ′ (−1)φ(−1) +
=
I
I
[−U ′′ + µU]φ dx
fφ dx + σ2φ(1) − σ1φ(−1), ∀φ ∈ X,
which leads to −U ′′ + µU = f in I, by testing on the functions φ ∈ H 1 0,w(I) ⊂ X.
We can now remove the integrals in (9.3.17) and recover the boundary conditions
U ′ (−1) = σ1, U ′ (1) = σ2.
We can also study variational formulations of the Neumann problem based on
different Jacobi weight functions. We set X ≡ H 1 w(I), where w is the ultraspherical
weight function with ν := α = β satisfying −1 < ν < 1. Then, Bw : X × X → R
and Fw : X → R are modified to get