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156 Polynomial Approximation of Differential Equations
(8.1.10) 2γ[pn,m|(η (n)
0 )] 2 ˜w (n)
0 +
= (2γ ˜w (n)
0
By taking γn := (2 ˜w (n)
0 ) −1 = 1
4
n
i=0
= 2γ[|pn,m|(η (n)
0 )] 2 ˜w (n)
0 +
d
dx (|pn,m| 2 )(η (n)
i ) ˜w (n)
i
1
−1
d
dx |pn,m| 2 dx
− 1)[|pn,m|(η (n)
0 )] 2 + [|pn,m|(η (n)
n )] 2 , 0 ≤ m ≤ n.
n(n + 1), we conclude the proof.
We note that γn must be proportional to n 2 , to ensure that the eigenvalues have
positive real part. This is also true for the Chebyshev case. In general, there is a real
eigenvalue growing linearly with γ. When γ tends to infinity, this eigenvalue diverges
and the remaining n eigenvalues approach those of problem (8.1.1) for 1 ≤ m ≤ n.
Heuristically, for γ = +∞, we are forcing the eigenfunctions to satisfy a vanishing
boundary condition at the point x = −1.
Setting q = λp in (7.4.5) and arguing as in theorem 8.1.1, we obtain, for any γ ∈ R
n
j d
(8.1.11) λ n+1
n,m + γλ n n,m + γ
j=1
dx j ˜ l (n)
0 (−1)
λ n−j
n,m = 0, 0 ≤ m ≤ n.
For boundary conditions at x = 1, similar statements hold. Now, the counterparts
of (8.1.1) and (8.1.6) are respectively the following ones:
⎧
⎨
(8.1.12)
⎩
(8.1.13)
⎧
⎨
⎩
−p ′ n,m(η (n)
i ) = λn,m pn,m(η (n)
i ), 1 ≤ i ≤ n,
pn,m(η (n)
n ) = λn,m pn,m(η (n)
n ),
−p ′ n,m(η (n)
i ) = λn,m pn,m(η (n)
i ), 1 ≤ i ≤ n,
−p ′ n,m(η (n)
n ) + γpn,m(η (n)
n ) = λn,m pn,m(η (n)
n ),
0 ≤ m ≤ n.
The change in the sign ensures that the eigenvalues verify Reλn,m > 0, 0 ≤ m ≤ n,
provided that −1 < α ≤ 0, β > −1, and γ > γn > 0.