Untitled - Cdm.unimo.it

154 Polynomial Approximation of Differential Equations
Proof - The eigenfunctions are solutions of the equation
(8.1.3) p ′ n,m = λn,mpn,m + p ′ n,m(−1) ˜l (n)
0 , in Ī = [−1,1], 1 ≤ m ≤ n.
By differentiating the above expression k times, we get by recursion
(8.1.4)
n,mpn,m + p ′ ⎡
n,m(−1) ⎣λ k n,m ˜l (n)
0 +
k
j d
d k+1
dx k+1 pn,m = λ k+1
j=1
dxj ˜l (n)
0
λ k−j⎦,
n,m
1 ≤ m ≤ n, k ≥ 1.
Recalling that the derivative of order n + 1 of a polynomial of degree n is zero, and
pn,m(−1) = 0, 1 ≤ m ≤ n, we take k = n in (8.1.4) and evaluate at the point x = −1.
This yields (8.1.2) (it is evident from (8.1.3) that p ′ n,m(−1) = 0, 1 ≤ m ≤ n).
In other words, the λn,m’s are the roots of the characteristic polynomial of degree n
associated with our eigenvalue problem. By directly solving the differential equation in
(8.1.3), we obtain the following explicit expression for the eigenfunctions:
(8.1.5) pn,m(x) = p ′ n,m(−1) e λn,mx
x
−1
e −λn,mt ˜(n) l 0 (t) dt, x ∈ Ī, 1 ≤ m ≤ n.
Some properties can be deduced from formulas (8.1.2) and (8.1.5), but the most inter-
esting conjectures follow from numerical experiments. We will give other specifications
in section 8.3. We must pay attention when we carry out tests on problem (8.1.1). In-
deed, in trefethen and trummer(1987), the authors note that disagreeable rounding
errors occur when computing the eigenvalues without appropriate machine accuracy.
Conclusions from careless computations could sometimes be misleading.
(7.4.4) is
(8.1.6)
Let us proceed with our investigation. The eigenvalue problem associated with
⎧
⎨
⎩
p ′ n,m(η (n)
i ) = λn,m pn,m(η (n)
i ), 1 ≤ i ≤ n,
p ′ n,m(η (n)
0 ) + γpn,m(η (n)
0 ) = λn,m pn,m(η (n)
0 ),
⎤
0 ≤ m ≤ n.