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Examples 271
(12.1.16)
=
=
=
d
dx p(k)
n (γk) −
d
λ Γ(t) −
dt
Γ(t)
−1
γk
U(x,t) dx
≈ λ γk+1 − γk
−
h
1
[p
h −1
(k)
d
dx p(k)
n (γk) − 1
n
[p
h
(k)
d
dx p(k)
n (γk) −
γk
−1
d 2
n
2 d
j=0
p(k)
dx2 n
j=0
p(k)
dx2 n
(x) dx −
t=tk
n − q (k−1)
n
](x) dx
n − q (k−1)
n ](θ (n,k)
j ) ˜w (n,k)
j
(θ (n,k)
j ) ˜w (n,k)
j
−
d
dx p(k)
n (−1)
d
dx p(k)
n (−1) = 0, 1 ≤ k ≤ m − 1.
A version of the scheme similar to the one proposed here has been tried for the two-phase
Stefan problem in rønquist and patera(1987). Different discretizations with finite-
differences for both the space and the time variable are compared in furzeland(1980).
For a survey of numerical methods for free boundary problems we refer the reader to
nochetto(1990).
12.2 An example in an unbounded domain
We study in this section the approximation of a second-order differential equation de-
fined in I ≡]0,+∞[. Let f : I → R be a given function. We are concerned with
finding the solution U : Ī → R to the problem
⎧
⎨
(12.2.1)
⎩U(0)
= 0, lim U(x) = 0.
x→+∞
−(e x U ′ (x)) ′ = f(x) x ∈ I,
We assume for example that f is such that the unique solution of (12.2.1) is U(x) =
e−x sin 7x
1+x2 , x ∈ Ī, which is shown in figure 12.2.1 for x ∈ [0,4].