introduction to partial differential equations
219
Unless otherwise stated, we will
always
assume that our functions
u
are suf
ficiently smooth to ensure that mixed partial derivatives are equal.
That is,
u
xt
=
u
tx
and
u
xxt
=
u
xtx
=
u
txx
, and so on.
Roughly speaking, a partial differential equation is any equation involving
partial derivatives of some function
u
. With the above notational conventions in
mind, we state a more precise definition.
Definition
8
.
0
.
2
.
Suppose
u
=
u
(
x
,
t
)
is a function of two variables. An equation
of the form
F
(
x
,
t
,
u
,
u
x
,
u
t
,
u
xx
,
u
xt
,
u
tt
, . . .
) =
0
is called a
partial differential equation (
pde
)
.
In this definition, it is understood that the function
F
has only finitely many
arguments. The definition is easily extended to allow for more than two in
dependent variables. The
order
of a
pde
is the order of the highest derivative
present in the equation. For example, the equation
u
t
+
u
x
=
0
is a firstorder
pde
, and the equation
u
xxt

(
u
x
)
8
=
0
is a thirdorder
pde
. The most general
form of a firstorder
pde
with three independent variables
t
,
x
, and
y
would be
F
(
t
,
x
,
y
,
u
,
u
t
,
u
x
,
u
y
) =
0. Here are some wellknown examples of
pde
s
.
The transport or advection equation:
Let
u
=
u
(
x
,
t
)
. Then the equation
u
t
+
cu
x
=
0
where
c
is a constant is called the simple
advection equation
. It can be
used to model the transport of a pollutant being carried (but not diffusing) in a
long, straight river with velocity
c
.