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12
EXAMPLES
In this chapter, we show how to apply spectral methods to the approximation of four
upgraded model problems, without pretending that our approach is competitive or im-
proves upon other known numerical algorithms. No theoretical convergence analysis is
provided for the schemes proposed here.
12.1 A free boundary problem
In our first example, given T > 0, we search for two functions U : [−1,1] ×[0,T] → R,
and Γ : [0,T] → [0,1], coupled by the following set of equations:
(12.1.1)
(12.1.2)
∂U
∂t (x,t) = ∂2U (x,t), ∀x ∈] − 1,Γ(t)[, ∀t ∈]0,T],
∂x2 ∂U
(−1,t) = 0, U(Γ(t),t) = 0, ∀t ∈]0,T],
∂x
(12.1.3) U(x,t) = 0, ∀x ∈]Γ(t),1], ∀t ∈]0,T],
(12.1.4) U(x,0) = −1, ∀x ∈ [−1,0[, U(x,0) = 0, ∀x ∈ [0,1],