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268 Polynomial Approximation of Differential Equations
where the new weights are defined by ˜w (n,k)
j
(see (3.5.6)).
:= 1
2 (1 + γk) ˜w (n)
j , 0 ≤ j ≤ n, 0 ≤ k ≤ m
We are now ready to describe the algorithm. In view of (12.1.4), the first polynomial
is determined by the relations
(12.1.9) p (0)
n (θ (n,0)
j ) := −1, 0 ≤ j ≤ n − 1, p (0)
n (θ (n,0)
n ) = 0.
To proceed we need to establish the new position of the free boundary. Following
(12.1.5) we define
(12.1.10) γk+1 := γk + h
d
λ dx p(k)
n (γk), 0 ≤ k ≤ m − 1.
The next step is to adapt the polynomial p (k)
n to the stretched interval [−1,γk+1]. To
this purpose we define an auxiliary polynomial q (k)
n ∈ Pn satisfying for 0 ≤ j ≤ n
(12.1.11)
⎧
⎨ q (k)
⎩
n (θ (n,k+1)
j ) = p (k)
n (θ (n,k+1)
j ) if θ (n,k+1)
j ∈ [−1,γk[,
q (k)
n (θ (n,k+1)
j ) = 0 if θ (n,k+1)
j ∈ [γk,γk+1].
The computation of q (k)
n can be done by interpolation. In fact, in analogy with (3.2.7),
we write
(12.1.12) p (k)
n (x) =
n
j=0
n (θ (n,k)
j ) ˜l (n,k)
j (x), ∀x ∈ [−1,γk],
where ˜l (n,k)
j
θ (n,k)
i , 0 ≤ i ≤ n. It is an easy matter to check that (see also (11.2.16))
(12.1.13)
p (k)
∈ Pn, 0 ≤ j ≤ n, are the Lagrange polynomials with respect to the points
˜(n,k) l j (x) = ˜l (n)
j 2 x+1
1+γk
− 1 , ∀x ∈ [−1,γk].
Thus, by virtue of (3.2.8) with α = β = 0, we can determine the values of q (k)
n at the
points θ (n,k+1)
j , 0 ≤ j ≤ n.