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Published in:
2009 | OriginalPaper | Chapter
Parabolic equations
Published in: Numerical Models for Differential Problems
Publisher: Springer Milan
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In this chapter, we consider parabolic equations of the form
5.1
$$ \frac{{\partial u}} {{\partial t}} + Lu = f, x \in \Omega , t > 0, $$
where Ω is a domain of ℝ
d
, d = 1, 2, 3, f = f(x, t) is a given function, L = L(x) is a generic elliptic operator acting on the unknown u = u(x,t). When solved only for a bounded temporal interval, say for 0 < t < T, the region QT = Ω × (0, T) is called cylinder in the space ℝ
d
× ℝ
+
(see Fig. 5.1). In the case where T = +∞, Q={(x,t):x ∈ Ω, t > 0} will be an infinite cylinder.