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218 Polynomial Approximation of Differential Equations
Then, one has
(9.9.3)
⎡
⎣ Mn
−µIn
⎤⎡
µIn
⎦⎣
¯pn
⎤
⎦ =
Mn
¯qn
⎡
⎣
⎤
¯fn
⎦,
where In denotes the (n − 1) × (n − 1) identity matrix and Mn is the reduced
(n − 1) × (n − 1) matrix corresponding to problem (7.4.9) (see (7.4.11) for the case
n = 3). In the ultraspherical case with −1 < α = β ≤ 1, one shows that the matrix
in (9.9.3) has eigenvalues with positive real part (see section 8.2). In addition, we can
invert the (2n − 2) × (2n − 2) linear system (9.9.3) by blocks, obtaining
(9.9.4)
⎡
⎣ ¯pn
⎤
⎦ =
¯qn
⎡
⎣ (M2 n + µ 2 In) −1 0
0 (M 2 n + µ 2 In) −1
Therefore, we only need to compute the inverse of M 2 n + µ 2 In.
9.10 Integral equations
¯gn
⎤⎡
⎦⎣
Mn
⎤⎡
⎤
−µIn
¯fn
⎦⎣
⎦.
In concluding this chapter, we say a few words about the possibility of using polynomials
in the approximation of equations in integral form. A standard example is given by the
following singular integral equation:
(9.10.1) A U(x) + B
π
1
−1
U(s)
s − x
µIn
ds = f(x), ∀x ∈] − 1,1[,
where A,B ∈ R are constants and f :] − 1,1[→ R is a given function. The integral
in (9.10.1) is a principal value integral. This is given by the limit
(9.10.2) lim
ǫ→0 +
x−ǫ
−1
U(s)
s − x
ds +
1
x+ǫ
Mn
U(s)
s − x ds
, ∀x ∈] − 1,1[.
¯gn