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Derivative Matrices 143
This is equivalent to
(7.4.17) −p ′′ + µp − γ[p ′ (η (n)
0 ) − σ1] ˜ l (n)
0 + γ[p′ (η (n)
n ) − σ2] ˜ l (n)
n = q.
This approach was introduced in funaro(1986) for Legendre nodes, and in funaro
(1988) for Chebyshev nodes. For n = 3, we present the relative linear system
(7.4.18)
⎡
⎢
⎣
− ˜ d (2)
00 − γ ˜ d (1)
00 + µ − ˜ d (2)
01 − γ ˜ d (1)
01
− ˜ d (2)
10
− ˜ d (2)
20
− ˜ d (2)
30 + γ ˜ d (1)
30
⎡
p(η
⎢
× ⎢
⎣
(n)
0 )
p(η (n)
1 )
p(η (n)
⎤
⎥
2 ) ⎥
⎦
− ˜ d (2)
02 − γ ˜ d (1)
02
− ˜ d (2)
11 + µ − ˜ d (2)
12
− ˜ d (2)
21
− ˜ d (2)
31 + γ ˜ d (1)
31
p(η (n)
3 )
=
⎡
⎢
⎣
− ˜ d (2)
03 − γ ˜ d (1)
03
− ˜ d (2)
13
− ˜ d (2)
22 + µ − ˜ d (2)
23
− ˜ d (2)
32 + γ ˜ d (1)
32 − ˜ d (2)
33 + γ ˜ d (1)
33
q(η (n)
0 ) − γσ1
q(η (n)
1 )
q(η (n)
2 )
q(η (n)
3 ) + γσ2
⎤
⎥
⎥.
⎥
⎦
⎤
⎥
+ µ ⎦
The reason for using these type of boundary conditions will be explained in section 9.4.
Alternative techniques are presented in canuto(1986).
We now investigate fourth-order problems. A classical example is to find p ∈ Pn+2,
n ≥ 2, such that
(7.4.19)
⎧
⎪⎨
⎪⎩
p IV (η (n)
i ) = q(η (n)
i ) 1 ≤ i ≤ n − 1,
p(η (n)
0 ) = σ1, p(η (n)
n ) = σ2,
p ′ (η (n)
0 ) = σ3, p ′ (η (n)
n ) = σ4,
where q ∈ Pn−2 and σk ∈ R, 1 ≤ k ≤ 4. For the purpose of writing the associated
linear system, we construct the unique polynomial s ∈ P3 that satisfies the same
boundary conditions as p: