HEAT AND WAVE EQUATION
Math 21b, Fall 2004
FUNCTIONS OF TWO VARIABLES. We consider functions
f
(
x, t
) which are for ±xed
t
a piecewise smooth
function in
x
.
Analogously as we studied the motion of a
vector
~v
(
t
), we are now interested in the motion
of a
function
f
(
x, t
). While the governing equation for a vector was an ordinary di²erential equation (ODE),
the describing equation will now be a
partial diFerential equation
(PDE). The function
f
(
x, t
) could denote
the
temperature of a stick
at a position
x
at time
t
or the
displacement of a string
at the position
x
at time
t
.
The motion of these dynamical systems can be understood using orthonormal Fourier basis
1
/
√
2
,
sin(
nx
)
,
cos(
nx
) treated in an earlier lecture.
The homework to this lecture is at the end of this 2 page handout.
PARTIAL DERIVATIVES. We write
f
x
(
x, t
) and
f
t
(
x, t
) for the
partial derivatives
with respect to
x
or
t
.
The notation
f
xx
(
x, t
) means that we di²erentiate twice with respect to
x
.
Example: for
f
(
x, t
) = cos(
x
+ 4
t
2
), we have
•
f
x
(
x, t
) =

sin(
x
+ 4
t
2
)
•
f
t
(
x, t
) =

8
t
sin(
x
+ 4
t
2
).
•
f
xx
(
x, t
) =

cos(
x
+ 4
t
2
).
One also uses the notation
∂f
(
x,y
)
∂x
for the partial derivative with respect to
x
. Tired of all the ”partial derivative
signs”, we always write
f
x
(
x, y
) or
f
t
(
x, y
) in this handout.
This is an o³cial abbreviation in the scienti±c
literature.
PARTIAL DIFFERENTIAL EQUATIONS. A partial di²erential equation is an equation for an unknown
function
f
(
x, t
) in which di²erent partial derivatives occur.
•
f
t
(
x, t
) +
f
x
(
x, t
) = 0 with
f
(
x,
0) = sin(
x
) has a
solution
f
(
x, t
) = sin(
x

t
).
•
f
tt
(
x, t
)

f
xx
(
x, t
) = 0 has a solution
f
(
x, t
) =
sin(
x

t
) + sin(
x
+
t
).
THE HEAT EQUATION. The temperature distribution
f
(
x, t
) in a metal bar [0
, π
] satis±es the
heat equation
f
t
(
x, t
) =
μf
xx
(
x, t
)
This partial di²erential equation tells that the rate of change of the temperature at
x
is proportional to the
second space derivative of
f
(
x, t
) at
x
. The function
f
(
x, t
) is assumed to be zero at both ends of the bar and
f
(
x
) =
f
(
x, t
) is a given initial temperature distribution.
The constant
μ
depends on the heat conductivity