1
Definition of PDE
A partial differential equation (PDE) is an equation in which there appear partial derivatives
of some unknown function with respect to two or more independent variables. It’s two
or more since if there were only one independent variable we would have an ordinary
differential equation or ODE.
As an example consider the PDE
∂u
∂x
+
∂u
∂y

σu
= 0
.
Here,
u
is a function of two variables
x
and
y
and, for the purpose of this example, we take
σ
as a constant.
As a short hand notation we often write the partial derivatives using subscripts. So in
the example above, we could define
u
x
≡
∂u
∂x
u
y
≡
∂u
∂y
,
and the PDE would then be written
u
x
+
u
y

σu
= 0
.
The solution of this PDE is some function
u
(
x, y
)
that exists in some region
R
of the
x

y
plane, for which the partial derivatives of
u
(
x, y
)
, namely
u
x
and
u
y
, are defined (meaning
they exist) for each point
(
x, y
)
of the region
R
, and for which the equation becomes an
identity, meaning that at each point
(
x, y
)
, if we add
u
x
and
u
y
, and then subtract
σu
we
get the right hand side 0. If this happens, we say
u
(
x, y
)
satisfies the PDE in the region
R
.
We can extend the subscript notation to include multiple partial derivatives. For exam
ple
u
x
≡
∂u
∂x
u
y
≡
∂u
∂y
u
xx
≡
∂
2
u
∂x
2
u
yy
≡
∂
2
u
∂y
2
u
xy
≡
∂
2
u
∂x∂y
etc. Note that in using this notation we’re assuming you can take the partial derivatives
in any order which is not always true at every point. As you recall from calculus, it’s not
always true that
∂
2
u
∂x∂y
=
∂
2
u
∂y∂x
,
so where order matters we will be careful to indicate that
u
xy
is not
u
yx
. Otherwise you can
assume when we use this notation that it doesn’t matter in which order you take the partial
derivatives.
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PDE terminology
There are several descriptive terms we apply to PDE’s so as to try to characterize them:
Order
The order of a PDE is the order of the highest partial derivative appearing in the
expression for the PDE.
Linearity
A PDE is called linear if the equation is a linear relation in the dependent vari
able and its derivatives. A linear PDE with 2 independent variables
x
and
y
can be
generically written in the form
N
X
n
=0
M
X
m
=0
a
nm
(
x, y
)
∂
n
+
m
u
∂
n
x∂
m
y
=
g
(
x, y
)
,
where the term with
n
=
m
= 0
represents
u
(
x, y
)
.
Nonlinearity
A PDE is nonlinear if it isn’t linear. For example if you have a PDE of the
form
F
(
u, u
x
, u
y
, x, y
) =
g
(
x, y
)
,
and
F
is not linear in
u
or all the derivatives of
u
, then the PDE is nonlinear. In order
for a PDE to be linear
F
must be linear in
u
and all the derivatives of
u
.
Homogeneity
You can generally write any PDE in the form
F
(
u, u
x
, u
y
, x, y
) =
g
(
x, y
)
If
g
≡
0
then the PDE is said to be homogeneous. Otherwise it is inhomogeneous.
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 Spring '09
 NilesA.Pierce
 Boundary value problem, Partial differential equation, Boundary conditions

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