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242 Polynomial Approximation of Differential Equations
= ζ − Rp 2 n(1,t) + 2 √ RLpn(1,t)qn(1,t) + (L − 2 L/R)q 2
n(1,t)
+ ζ − Lq 2 n(−1,t) + 2 √ RLpn(−1,t)qn(−1,t) + (R − 2 R/L)p 2
n(−1,t)
The last terms in (10.5.11) are negative in view of the inequalities
2 √ RLab ≤ Ra 2 + Lb 2 ≤ Ra 2
+ 2
L/R − L b 2 ,
2 √ RLab ≤ La 2 + Rb 2 ≤ La 2 +
where a,b ∈ R, R > 0, L > 0, RL < 1.
2
R/L − R b 2 ,
By (3.8.6), we obtain a bound to the norm Rpn(·,t) 2 w + Lqn(·,t) 2 w, t ∈]0,T], in
terms of the initial data.
≤ 0.
We also outline the proof of convergence. One introduces two additional polyno-
mials ˆpn(·,t) ∈ Pn, ˆqn(·,t) ∈ Pn, n ≥ 1, t ∈]0,T]. These are solution of
⎧
(10.5.12)
with
(10.5.13)
⎪⎨
⎪⎩
∂ˆpn
∂t (η(n)
i ,t) = −ζ ∂ˆpn
∂x (η(n)
i ,t) 1 ≤ i ≤ n, t ∈]0,T],
∂ˆqn
∂t (η(n)
i ,t) = ζ ∂ˆqn
∂x (η(n)
i ,t) 0 ≤ i ≤ n − 1, t ∈]0,T],
⎧
⎪⎨
⎪⎩
∂ˆpn
∂ˆpn
(−1,t) = −ζ
∂t ∂x (−1,t) − γ[ˆpn − V ](−1,t)
∂ˆqn ∂ˆqn
(1,t) = ζ
∂t ∂x (1,t) − γ[ˆqn − W](1,t)
∀t ∈]0,T].
The functions ˆpn and ˆqn are decoupled. Therefore, we can estimate the errors
(ˆpn − V )(·,t)w and (ˆqn − W)(·,t)w, t ∈]0,T], by virtue of the results of section
10.3. On the other hand, by subtracting the equations (10.5.12) from the equations
(10.5.6), we get an equivalent collocation scheme in the unknowns rn := pn − ˆpn and
sn := qn − ˆqn , with the following boundary relations, obtained by combining (10.5.5),
(10.5.9) and (10.5.13):