Untitled - Cdm.unimo.it

10
TIME-DEPENDENT PROBLEMS
The purpose of this chapter is to analyze the approximation, using spectral methods,
of differential problems where the solution is time-dependent. We consider parabolic
and hyperbolic partial differential equations in one space variable. Finite-differences are
usually employed for the treatment of the time variable, hence part of the analysis will
be devoted to this topic.
10.1 The Gronwall inequality
A basic result, known as Gronwall lemma, will be used in the following. Several versions
exist, but we recall only one of them here.
Theorem 10.1.1 - Let T > 0 and C ≥ 0. Let g and h be two continuous functions
in the interval [0,T], such that
(10.1.1) g(t) ≤ g(0) +
Then, we have
(10.1.2) g(t) ≤ e Ct
t
0
g(0) +
[Cg(s) + h(s)]ds, ∀t ∈ [0,T].
t
0
h(s)e −Cs
ds , ∀t ∈ [0,T].