Math 212
Introductory Partial Differential Equations
Course notes , Fall 2007
Instructor:
Dr. Friedemann Brock
Chapter 1: Introduction
1.1. Notation
IN
natural numbers
IR
real numbers
IC
complex numbers
IR
n
Euclidean space of dimension
n
,
=
{
x
= (
x
1
, . . . , x
n
) :
x
i
∈
IR
, i
= 1
, . . . , n
}
,
if
n
= 2
,
3 we often write (
x, y
), (
x, y, z
) for points in IR
n
, respectively,
A
×
B
:=
{
(
a, b
) :
a
∈
A, b
∈
B
}
A set
D
⊂
IR
n
is called a domain if it is open and connected. By
∂D
we denote the
boundary of
D
.
f
:
D
→
IR denotes a function from
D
into IR. If
u
:
D
→
IR then
∂
k
u
∂x
i
1
1
· · ·
∂x
i
n
n
,
(
k, i
1
, . . . , i
n
∈
IN
,
n
j
=1
i
j
=
k
)
,
denotes a
partial derivative of order
k
of
u
. We often abbreviate partial derivatives
using subscripts. For instance, if
u
=
u
(
x, y, t
), then (
∂u/∂x
) =
u
x
, (
∂u/∂t
) =
u
t
are partial derivatives of first order, and (
∂
2
u/∂y∂t
) =
u
yt
, (
∂
2
u/∂x∂y
) =
u
xy
are
partial derivatives of second order.
If
k
∈
IN, we set
C
(
D
)
:=
{
u
:
D
→
IR :
u
is continuous on
D
}
,
C
k
(
D
)
:=
{
u
:
D
→
IR :
u
together with its partial derivatives
of order
≤
k
are continuous
}
.
If
u
∈
C
2
(
D
), then
u
x
i
x
j
=
u
x
j
x
i
, (
i, j
= 1
, . . . n
).
1.2. Some linear PDE
(PDE =
P
artial
D
ifferential
E
quation ).
Assume that
D
is a domain in IR
n
, and
u
:
D
→
IR. We set
Lu
:=
au
+
n
i
=1
b
i
u
x
i
+
n
i,j
=1
c
ij
u
x
i
x
j
+
· · ·
,
(1)
1
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where the
a, b
i
, c
ij
’s are given functions on
D
, and the dots
· · ·
in (1) stand for further
terms involving higherorder derivatives of
u
which enter
linearly
. The operator
L
is then a
linear DOP
(DOP=
D
ifferential
Op
erator), since we have for any functions
u
i
:
D
→
IR, and numbers
c
i
∈
IR, (
i
= 1
, . . . , k
),
L
k
i
=1
c
i
u
i
=
k
i
=1
c
i
Lu
i
.
(2)
An equation of the form
Lu
= 0 is called a linear
homogeneous
PDE. If
F
:
D
→
IR,
then an equation of the form
Lu
=
F
is called a linear
nonhomogeneous
PDE.
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 Fall '08
 FriedmannBrock
 Differential Equations, Real Numbers, Equations, Derivative, Partial Differential Equations, Natural Numbers, Complex Numbers, Boundary value problem, Partial differential equation, Linear PDE

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