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Time-Dependent Problems 223
In addition, at any t ∈]0,T], suitable boundary conditions at the endpoints of the
interval I are assumed. These are for instance
(10.2.3) U(−1,t) = σ1(t), U(1,t) = σ2(t), ∀t ∈]0,T],
where σ1 : [0,T] → R, σ2 : [0,T] → R, are given continuous functions.
Equation (10.2.1) is of parabolic type. The literature on this subject is extensive and
cannot be covered here. For a discussion of the general properties of parabolic equa-
tions, we only mention the following comprehensive books: courant and hilbert
(1953), sneddon (1957), greenspan(1961), epstein(1962), sobolev(1964), wein-
berger(1965). The same books can be used as references for the other sections in this
chapter. An approach more closely related to the heat equation is analyzed in widder
(1975), hill and dewynne(1987).
Under appropriate hypotheses on the data U0, σ1 and σ2, existence and unique-
ness of a solution satisfying (10.2.1), (10.2.2), (10.2.3) can be proven. In most cases,
this can be expressed by a suitable series expansion.
We note that the compatibility conditions
(10.2.4) σ1(0) = U0(−1), σ2(0) = U0(1),
are not required in general. However, though the solution U is very smooth for t ∈]0,T],
if we drop conditions (10.2.4), heat propagation with infinite velocity is manifested at
t = 0. This can adversely affect the numerical analysis. Thus, we assume henceforth
that relations (10.2.4) are satisfied.
The substitution V (x,t) := U(x,t) − 1
2 (1 − x)σ1(t) − 1
2 (1 + x)σ2(t), x ∈ [−1,1],
t ∈ [0,T], leads to an equivalent formulation with homogeneous boundary conditions.
Indeed, we now have
(10.2.5)
∂V
∂t (x,t) = ζ ∂2V (x,t) + f(x,t), ∀x ∈ I, ∀t ∈]0,T],
∂x2 (10.2.6) V (x,0) = U0(x) − 1
2 (1 − x)U0(−1) − 1
2 (1 + x)U0(1), ∀x ∈ Ī,