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132 Polynomial Approximation of Differential Equations
(7.2.12) ˜ d (1)
ij =
⎧
⎪⎨
⎪⎩
α − n(n + α + β + 1)
2(β + 2)
(−1) n n! Γ(β + 2) un(η (n)
i )
Γ(n + β + 1) (1 + η (n)
i )
(−1) n Γ(β + 2) Γ(n + α + 1)
2 Γ(α + 2) Γ(n + β + 1)
−(−1) n Γ(n + β + 1)
n! Γ(β + 2) un(η (n)
j )(1 + η (n)
j )
un(η (n)
i )
un(η (n)
j )
η (n)
i
1
− η(n)
j
+ α − β
(α + β)η (n)
i
2 (1 − (η (n)
i ) 2 )
Γ(n + α + 1)
n! Γ(α + 2) un(η (n)
j )(1 − η (n)
j )
−(−1) n Γ(α + 2) Γ(n + β + 1)
2 Γ(β + 2) Γ(n + α + 1)
−n! Γ(α + 2) un(η (n)
i )
Γ(n + α + 1) (1 − η (n)
i )
n(n + α + β + 1) − β
2(α + 2)
i = j = 0,
1 ≤ i ≤ n − 1, j = 0,
i = n, j = 0,
i = 0, 1 ≤ j ≤ n − 1;
i = j, 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n − 1,
1 ≤ i = j ≤ n − 1,
i = n, 1 ≤ j ≤ n − 1,
i = 0, j = n,
1 ≤ i ≤ n − 1, j = n,
i = j = n.
To check (7.2.12), the reader must remember that u ′ n(η (n)
j ) = 0, 1 ≤ j ≤ n − 1, while
u ′ n(1) and u ′ n(−1) are respectively given by (3.1.17) and (3.1.18). Further relations,
useful in this computation, are obtained from the differential equation (1.3.1). This can
be used for instance to obtain the values of u ′′ n at the nodes.