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18.152  Introduction to PDEs
, Fall
2004
Prof.
Gigliola
Staﬃlani
Lecture 1  Introduction
and
Basic Facts
about
PDEs
The Content of
the
Course
•
Definition of Partial
Differential
Equation (PDE)
±
Linear
PDEs
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V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
homomgeneous
nonhomogeneous
(no forcing
term)
(with forcing
term)
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
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with variable and constant coeﬃcients
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
parabolic
example
±
*
hyperbolic example
elliptic example
(diffusion equation)
(wave equation)
(Laplace equation)
(heat equation)
2
u
t
=
κu
xx
, κ >
0
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
u
tt
=
c u
xx
u
xx
= 0
)
±
�
and nonhomogeneous case
In
studying
these
examples
of
PDEs we will
learn how
to
“impose conditions”
to
make the
problem
“wellposed”, we
will introduce fundamental
mathematical
concepts like “distribu
tions”,
”Fourier
Transform”, and “Fourier
Series”.
These tools are by now
“classical”, but
still heavily
used in the
study
of
more complex PDEs,
in particular,
the nonlinear
ones.
What
is
a
partial differential equation?
•
This is
an equation involving
a
function
u
(
x
1
, . . . , x
n
) of
n
variables and its partial
derivatives
up
to order
m
:
F
(
u, u
x
1
, . . . , u
x
n
, . . . , u
x
i
1
x
i
2
, . . . , u
x
i
1
x
i
2
...x
i
m
) = 0
, i
j
∈ {
1
, . . . , n
}
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 Fall '04
 GigliolaStaffilani
 Derivative, Partial differential equation, PDEs, VVVV VVVV VVVV

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