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18.152  Introduction to PDEs
, Fall 2004
Prof. Gigliola Staﬃlani
Lecture 2  First order linear PDEs and PDEs from physics
I mentioned in the ﬁrst class some basic PDEs of ﬁrst and second order. Today we illustrate
how they come naturally as a model of some basic phenomena.
1.
u
t
+
cu
x
= 0
Transport equation (simple transport)
2.
u
tt
−
c
2
u
xx
= 0
Wave equation (vibrating string)
3.
u
t
=
k
Δ
u
Parabolic equation (heat equation and diﬀusion)
4. Δ
u
= 0
Elliptic equation (stationary wave and diﬀusion)
5.
−
iu
t
= Δ
u
Schr¨
odinger equation (Hydrogen atom)
Derivation of (1):
•
Imagine putting a drop of ink in the pipe. Let
u
(
x, t
) be the concentration or density of ink
at point
x
at time
t
.
But how do we describe it?
Fix an interval [0
, b
].
ink in [0, b]
=
M
=
±
b
u
(
x, t
)
dx
time=
t
0
Suppose the water moves at speed
c
, so at time
h
+
t
the same quantity of ink will be
±
ch
+
b
M
=
u
(
x, t
+
h
)
dx
ch
±
b
±
ch
+
b
u
(
x, t
)
dx
=
u
(
x, t
+
h
)
dx
0
ch
Taking
∂
b
u
(
b, t
) =
u
(
b
+
ch, t
+
h
)
⇒
Taking
∂
h h
=0
0 =
cu
x
(
b, t
) +
u
t
(
b, t
)

⇒
1
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View Full Document Derivation of (2):
•
A typical example of wave motion in a plane is the motion of a string with ﬁxed end points.
Why does it have this shape? We will see later.
Let’s take a piece of it:
We will ignore all the forces on the string except for its tension. We will consider a perfect
string in which for all
x
we have
T
(
x, t
) =
T
v
(
x, t
)
(1)
for constant
T
and
v
(
x, t
) a unit tangent vector to the string at (
x, t
).
(1
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This note was uploaded on 10/02/2010 for the course MAT 18.152 taught by Professor Gigliolastaffilani during the Fall '04 term at MIT.
 Fall '04
 GigliolaStaffilani

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