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266 Polynomial Approximation of Differential Equations
(12.1.5) Γ(0) = 0, λ Γ ′ (t) = ∂U
(Γ(t),t), ∀t ∈]0,T].
∂x
In (12.1.5), λ is a positive given constant.
Let us provide a physical interpretation. We assume that the segment [−1,1]
represents a one-dimensional bar of some prescribed material. Initially, the interval
[−1,0[ is in the solid state, while the interval [0,1] is in the liquid state. After appropriate
scaling, the function U represents the temperature of the bar, in such a way that U = 0
corresponds to the solidification temperature of the liquid. Below this level the material
changes its physical state. The initial configuration is expressed by conditions (12.1.4).
Then, as the time passes, due to the lower temperature of the solid part, the liquid begins
a solidification process and the boundary between the solid and the liquid regions is
determined by the function Γ. For any t ∈]0,T], in the interval ] − 1,Γ(t)[ the
temperature is governed by the heat equation (12.1.1) (see also section 10.2). In the
remaining portion of the bar, the temperature is assumed to be zero (condition (12.1.3)).
Relations (12.1.2) are the boundary conditions of equation (12.1.1). In particular, at the
endpoint x = −1 the heat flux is set to zero. Thus, the bar is thermally insulated, i.e.,
there is no heat transfer across that point. Finally, the differential relation in (12.1.5)
describes the evolution of Γ. This is known as the Stefan condition and states that, for
any t ∈]0,T], the moving point Γ(t) has a velocity proportional to the heat flux at
(Γ(t),t). The parameter λ is the Stefan constant and it is proportional to the latent
heat of the material. For high values of λ solidification is slow, and vice versa. For
simplicity, we do not take into account the effects of heat conduction inside the portion
of liquid material.
According to the literature, (12.1.1)-(12.1.5) is a single-phase Stefan problem and rep-
resents an example of free boundary problem (see for instance friedman(1959), rubin-
stein(1971), fasano and primicerio(1972), elliott and ockendon(1982), hill and
dewynne(1987), chapter 7). A theoretical analysis shows that U and Γ are uniquely
determined and Γ, as expected, is an increasing function.