INTRODUCTION OF PDE’S AND FOURIER SERIES
1
1.
INTRODUCTION
1.1.
General Form of P.D.E.
F
(
x, y, z, t,
Φ
x
,
Φ
y
,
Φ
z
,
Φ
xx
,
Φ
yy
,
Φ
tt
. . .
) = 0
.
(1.1)
where
•
(
x, y, z
) : Space Coordinates
•
t
: Time
•
Φ: Vector function.
1.2.
Definitions.
1.2.1.
Order of PDE – n.
•
if Φ(
x, y, z, t
) is scalar, n is the highest order n of the derivatives;
•
if Φ(
x, y, z, t
) is vector,
n
=
m
×
n
d
, where
m
: number of the components;
n
d
: the highest order.
1.2.2.
Solution of PDE.
Φ(
x, y, z, t
) has all necessary derivatives and satisfied the equation (1.1)
and all conditions.
Remark
1.2.1
.
: Week solution; discontinuous solutions
1.3.
Linear and Non-linear Equations. Examples
:
(1) Linear
A
(
x, y, z
)
u
x
+
B
(
x, y, z
)
u
y
=
C
(
x, y, z
)
(2) Quasi-linear
A
(
x, u
)
u
x
+
B
(
x, u
)
u
y
=
C
(
x, u
)
(3) Semi-linear
A
(
x, y, z
)
u
x
+
B
(
x, y, z
)
u
y
=
C
(
x, y, z, u
)
(4) Non-linear
A
(
x, y, z
)
u
2
x
+
B
(
x, y, z
)
u
1
2
y
=
W.
1.4.
General Solution.
Some PDE’s have the general solution.
Examples
:
(1)
Φ
xy
(
x, y
) = 0
(
n
= 2)
⇒
Φ(
x, y
) =
f
(
x
) +
g
(
y
)
.