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196 Polynomial Approximation of Differential Equations
As pointed out in section 9.2, the use of ultraspherical weight functions with ν = 0,
can lead to some trouble in the treatment of the integrals. This is why the bilinear form
Bw has been decomposed into two components. In the one containing the weight
function w, integration by parts is allowed since φ − ˜ φ vanishes at the points x = ±1.
The other component does not contain the weight function and takes into account the
boundary constraints.
Variational formulations also exist for fourth-order equations. For example, we
examine problem (9.1.6). We assume σi = 0, 1 ≤ i ≤ 4. Otherwise, we replace U by
U − s, where s ∈ P3 is given in (7.4.20). Then, for f ∈ L 2 w(I), we define
(9.3.22) Bw(ψ,φ) :=
(9.3.23) Fw(φ) :=
A natural choice for the functional space is
ψ
I
′′ (φw) ′′ dx, ∀ψ,φ ∈ X,
I
fφw dx, ∀φ ∈ X.
(9.3.24) X ≡ H 2 0,w(I) := {φ| φ ∈ H 2 w(I), φ(±1) = φ ′ (±1) = 0}.
It is left for the reader to check that Bw and Fw satisfy the hypotheses of Lax-Milgram
theorem, when w is the ultraspherical weight function with −1 < ν < 1. In particular,
coerciveness is a consequence of lemma 8.2.5. This property provides a unique weak
solution U ∈ X of the problem Bw(U,φ) = Fw(φ), ∀φ ∈ X. In the standard way, one
verifies that U is solution to (9.1.6) whenever f ∈ C0 ( Ī). More properties and results
are discussed in bernardi and maday(1990) and funaro and heinrichs(1990).
In a more advanced setting, we can construct variational formulations of the prob-
lems considered above, where two different spaces are used for the solution and the test
functions. In this case, one defines an appropriate bilinear form B : X × Y → R, and
a linear operator F : Y → R, in order that U ∈ X is solution of the weak problem
B(U,φ) = F(φ), ∀φ ∈ Y. A generalization of theorem 9.3.1 is obtained by modifying
condition (9.3.6) as follows: