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142 Polynomial Approximation of Differential Equations
Different boundary conditions are possible. A classical problem is to find p ∈ Pn
such that
(7.4.13)
⎧
⎪⎨
⎪⎩
−p ′′ (η (n)
i ) + µp(η (n)
i ) = q(η (n)
i ) 1 ≤ i ≤ n − 1,
−p ′ (η (n)
0 ) = −σ1,
p ′ (η (n)
n ) = σ2,
where q ∈ Pn−2, σ1,σ2 ∈ R, and µ > 0.
We changed the sign at relation p ′ (η (n)
0 ) = σ1 for a reason that will be understood in
section 8.2. The matrix associated to (7.4.13), for n = 3, takes the form
(7.4.14)
⎡
⎢
⎣
− ˜ d (1)
00
− ˜ d (1)
01
− ˜ d (1)
02
− ˜ d (2)
10 − ˜ d (2)
11 + µ − ˜ d (2)
12
− ˜ d (2)
20
˜d (1)
30
− ˜ d (2)
21
˜d (1)
31
− ˜ d (1)
03
− ˜ d (2)
13
− ˜ d (2)
22 + µ − ˜ d (2)
23
˜d (1)
32
˜d (1)
33
⎤⎡
p(η
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎦⎣
(n)
0 )
p(η (n)
1 )
p(η (n)
⎤
⎥
2 ) ⎥
⎦
p(η (n)
3 )
=
⎡
−σ1
⎢
⎢q(η
⎢
⎣
(n)
1 )
q(η (n)
⎤
⎥
⎥.
⎥
2 ) ⎥
⎦
This situation can be easily generalized to mixed type boundary conditions, such as
(7.4.15)
⎧
⎨τ1p
′ (−1) + τ2p(−1) = σ1
⎩
τ3p ′ (1) + τ4p(1) = σ2
τi ∈ R, 1 ≤ i ≤ 4.
Here τ1 and τ2 (as well as τ3 and τ4) do not vanish simultaneously.
A suitable modification furnishes another formulation. Let q ∈ Pn, σ1,σ2 ∈ R, µ > 0
and γ ∈ R with γ = 0. Then, we seek p ∈ Pn such that
(7.4.16)
⎧
⎪⎨
⎪⎩
−p ′′ (η (n)
i ) + µp(η (n)
i ) = q(η (n)
i ) 1 ≤ i ≤ n − 1,
−p ′′ (η (n)
0 ) + µp(η (n)
0 ) − γp ′ (η (n)
0 ) = q(η (n)
0 ) − γσ1,
−p ′′ (η (n)
n ) + µp(η (n)
n ) + γp ′ (η (n)
n ) = q(η (n)
n ) + γσ2.
σ2