Week 1 notes: Math 716
1
PDE: Order, Linear & Nonlinear
A differential equation that involves more than one independent variable is called a
partial
differential equation
, abbreviated as PDE. The order of the highest derivative occurring in the
PDE is defined as the
order
of the PDE. For instance, the following
∂
2
u
∂x
2
+
∂
2
u
∂y
2
= 0
(1)
is a second order PDE for
u
(
x,y
), which is usually called the
Laplace’s equation
in two variables.
Another example of a second order linear differential equation is the socalled
heat
equation for
u
(
x,t
) in one space variable with a source:
∂u
∂t
=
κ
∂
2
u
∂x
2
+
g
(
x,t
)
for some constant
κ
(2)
As with ODEs, if the differential operator
L
is linear,
i.e.
for constant
c
1
and
c
2
,
L
(
c
1
u
1
+
c
2
u
2
) =
c
1
L
u
1
+
c
2
L
u
2
(3)
then
L
u
=
g
is called a linear differential differential equation. For instance, equations (1) and
(2) above are linear differential equations since in those cases we can check easily the linearity of
the corresponding differential operators
L
=
∂
2
∂x
2
+
∂
2
∂y
2
or
L
=
∂
∂t
−
∂
2
∂x
2
, acting on appropriate
class of functions
1
Note, that in (1),
g
= 0. In that case, the PDE is linear and homogeneous.
Partial differential equations that are not linear are called
nonlinear
; the following
semilinear
heat equation is an example:
u
t
−
u
xx
=
u
2
,
(4)
where for notational brevity, the partial derivatives are denoted by subscripts.
As for ODEs, linear PDEs are usually simpler to analyze/solve than nonlinear PDEs. Non
linear PDEs arise in many applications; but general theory rarely exists.
2
Some physical applications where PDEs arise
PDEs abound in physical sciences and engineering. Here we give some examples.
2.1
Dispersion of pollutants in a river stream
Consider predicting concentration
ρ
(measured in some units, say
Kg/m
3
) of some pollutant as
a function of position
x
and time
t
, in a river. We denote the
x
domain by Ω
⊂
R
3
. Let the fluid
velocity field in the river (domain Ω) be given by
u
(
x
,t
) (measured in some units, say
m/sec
).
We will assume that the pollutant is passive, meaning it’s inertia is neglected. Consider a small
but fixed volume
V
centered around at a point
x
(See Figure 1), entirely contained in Ω. The
1
Classically, this would be
C
2
(
R
2
) in the first case and
C
2
in
x
and
C
1
in time for the second. However, this
set can be extended further by introducing the concept of weak solutions.
1
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. x
V
δ
V
n
F
Figure 1: Control Volume V for pollutant
rate of change of mass of fluid inside that volume equals
d
dt
integraltext
V
ρ
(
x
,t
)
dV
. Assuming no pollutant
is created or destroyed within
V
, this must equal the net inward flow of pollutants into
V
:
−
integraldisplay
∂V
F
·
n
dA
where
F
(
x
,t
) is the flux (measured in units like
Kg/m
2
/sec
) of pollutants at any point
x
and
n
is the outward normal to the boundary of volume
V
. A reasonable expression for flux is:
F
=
u
ρ
−
κ
∇
ρ
where the first term is due to fluid motion carrying pollutants, usually called
advection
, while
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 Spring '08
 Tanveer,S
 Derivative, Laplace, Partial differential equation, Boundary conditions, Linear & Nonlinear

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