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222 Polynomial Approximation of Differential Equations
Proof - Using (10.1.1), we can easily check the inequality G ′ (t) ≤ 0, ∀t ∈ [0,T], where
we defined
G(t) := e −Ct
g(0) +
t
0
[Cg(s) + h(s)]ds
−
t
0
h(s)e −Cs ds, t ∈ [0,T].
Therefore, the function G is not increasing. Finally, by (10.1.1), we get
(10.1.3) g(t)e −Ct −
This implies (10.1.2).
t
10.2 Approximation of the heat equation
0
h(s)e −Cs ds ≤ G(t) ≤ G(0) = g(0), ∀t ∈ [0,T].
In the field of linear partial differential equations, the heat equation is a classical example.
The solution U ≡ U(x,t), physically interpreted as a temperature, depends on the space
variable x and the time variable t. In the case of a heated bar I =]−1,1[ of homogeneous
material, the temperature evolves in the time interval [0,T], T > 0, according to the
equation
(10.2.1)
∂U
∂t (x,t) = ζ ∂2U (x,t), ∀x ∈ I, ∀t ∈]0,T],
∂x2 where ζ > 0 is a constant (thermal diffusivity).
To pose the problem properly, we need an initial condition:
(10.2.2) U(x,0) ≡ U0(x), ∀x ∈ Ī,
where U0 : Ī → R is a given continuous function.