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Time-Dependent Problems 225
The next step is to discretize (10.2.11) with respect to the variable t. This will be
investigated in section 10.6. Let us first analyze the convergence of the solution of the
semi-discrete problem (10.2.8), (10.2.9), (10.2.10) to the solution of problem (10.2.5),
(10.2.6), (10.2.7). Arguing as in section 9.3, we can propose a variational formulation
for equation (10.2.5) as follows:
(10.2.12)
1
−1
1
∂V
∂V ∂(φw)
φw dx = −ζ
∂t −1 ∂x ∂x
dx +
1
−1
fφw dx,
∀φ ∈ X, ∀t ∈]0,T],
where the solution and the test functions both belong to the space X ≡ H 1 0,w(I) (see
(5.7.6)). Existence and uniqueness of weak solutions of (10.2.12) are discussed in lions
and magenes(1972) for the case w ≡ 1.
With the help of the quadrature formula (3.5.1), the approximating polynomials satisfy
(10.2.13)
n
j=0
+
∂pn
∂t φ
(η (n)
j ,t) ˜w (n)
j
n
(fφ)(η (n)
j=0
j ,t) ˜w (n)
1
∂pn ∂(φw)
= −ζ
−1 ∂x ∂x
j , ∀φ ∈ P 0 n, ∀t ∈]0,T].
Interesting properties are obtained from this equation. Consider ν := α = β with
−1 < ν ≤ 1. For any t ∈]0,T], setting φ(x) := pn(x,t), x ∈ I, one gets
(10.2.14)
=
n
(fpn)(η (n)
j=0
⎛
1 d
⎝
2 dt
n
[pn(η (n)
j=0
j ,t) ˜w (n)
j
1
≤
2
j ,t)] 2 ˜w (n)
j
⎞
n
[pn(η (n)
j=0
⎠ + ζ
1
−1
j ,t)] 2 ˜w (n)
j
∂pn
∂x
1
+
2
∂(pnw)
∂x
dx
dx
n
[f(η (n)
j=0
j ,t)] 2 ˜w (n)
The last inequality in (10.2.14) is derived from the relation ab ≤ 1
2 (a2 +b 2 ), a,b ∈ R.
j .