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232 Polynomial Approximation of Differential Equations
According to theorem 6.2.2, we have
(10.3.11)
1
−1
∂pn
∂t
− Πw,n−1
∂pn
φw dx = 0 =
∂t
d
1
(pn − Πw,n−1pn)φw dx,
dt −1
∀φ ∈ Pn−1, ∀t ∈]0,T],
which shows that the operator ∂/∂t commutes with the operator Πw,n−1. Therefore,
we can rewrite (10.3.10) as
(10.3.12)
∂
∂t (Πw,n−1pn)
(x,t) = −ζ ∂pn
(x,t),
∂x
∀x ∈] − 1,1], ∀t ∈]0,T].
In addition, we require that pn(−1,t) = 0, ∀t ∈]0,T], and we impose the initial
condition
(10.3.13) pn(·,0) = rΠw,n−1(U0r −1 ), ∀x ∈ [−1,1],
where r(x) := (1 + x), x ∈ [−1,1]. In this way, we also get pn(−1,0) = 0.
Let ck(t), 0 ≤ k ≤ n, t ∈ [0,T], be the Fourier coefficients of pn(·,t) with respect
to the basis uk ≡ P (α,β)
k , k ∈ N, α > −1, β > −1. Therefore, pn = n k=0 ckuk.
Moreover, by (10.3.12), pn satisfies the set of equations (see section 7.1):
(10.3.14)
d
dt ck(t) = −ζ c (1)
k (t), 0 ≤ k ≤ n − 1, ∀t ∈]0,T].
This is equivalent to a n × n linear system of ordinary differential equations in the
unknowns ck, 0 ≤ k ≤ n−1, where the initial values ck(0), 0 ≤ k ≤ n−1, are obtained
from (10.3.13) by virtue of the results of section 2.3. We eliminate cn by expressing it
as a linear combination of the other coefficients with the help of the relation
(10.3.15) pn(−1,t) =
n
k=0
ck(t)uk(−1) = 0, ∀t ∈ [0,T].
Another characterization is obtained from (10.3.10) by noting that pn satisfies
(10.3.16)
∂pn ∂pn
(x,t) = −ζ
∂t ∂x (x,t) + c′ n(t)un(x), ∀x ∈] − 1,1], ∀t ∈]0,T].
In fact, the last term in (10.3.16) disappears when we apply the projector Πw,n−1.