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240 Polynomial Approximation of Differential Equations
(10.5.8)
d
dt
⎡
pn(η
⎢
⎣
(n)
1 ,t)
pn(η (n)
2 ,t)
qn(η (n)
⎤
⎥
0 ,t) ⎥
⎦
qn(η (n)
1 ,t)
⎡
⎢
= ζ ⎢
⎣
− ˜ d (1)
11 − ˜ d (1)
12 −L ˜ d (1)
10
− ˜ d (1)
21 − ˜ d (1)
22 −L ˜ d (1)
20
0 R ˜ d (1)
02
0 R ˜ d (1)
12
˜d (1)
00
˜d (1)
10
0
0
˜d (1)
01
˜d (1)
11
⎤⎡
pn(η
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎦⎣
(n)
1 ,t)
pn(η (n)
2 ,t)
qn(η (n)
⎤
⎥
⎥,
0 ,t) ⎥
⎦
qn(η (n)
1 ,t)
t ∈]0,T].
We note that the determinant of the matrix of the system vanishes when RL = 1.
In fact, polynomials of degree zero satisfying (10.5.7) are eigenfunctions relative to the
eigenvalue zero. Unfortunately, we are not aware of stability and convergence results
for this approximation scheme.
Approximations of equation (10.5.4) by polynomials in Pn, using the collocation
method at the points η (n+1)
i , 1 ≤ i ≤ n are considered in gottlieb, lustman
and tadmor(1987a) and gottlieb, lustman and tadmor(1987b). The boundary
conditions are treated as in (10.5.7), where |RL| < 1. In this case, a theoretical
analysis is given for the ultraspherical case when ν := α = β satisfies −1 < ν ≤ 0.
In funaro and gottlieb(1989), the boundary relations (10.5.7) are modified as
(10.5.9)
⎧
⎪⎨
⎪⎩
∂pn
∂pn
(−1,t) = −ζ
∂t ∂x (−1,t) − γ[pn − L qn](−1,t)
∂qn ∂qn
(1,t) = ζ
∂t ∂x (1,t) − γ[qn − R pn](1,t)
∀t ∈]0,T],
where γ > 0 is a constant. As in (10.3.9) the differential equation is also considered at
the points x = ±1. The new collocation scheme is now equivalent to a (2n+2)×(2n+2)
differential system. For n = 2, we have ∀t ∈]0,T]