Environmental Engineering Reference
In-Depth Information
are the first-order, the second-order and the third-order PDE, respectively. Here,
subscripts on dependent variables denote differentiations, e.g.
2 u
u x =
u
u xy =
x ,
y .
x
In general, the first- and the second-order PDE of u
(
x
,
y
)
may be written in the form
F
(
u
,
x
,
y
,
u x ,
u y )=
0
,
F
(
u
,
x
,
y
,
u x ,
u y ,
u xx ,
u xy ,
u yy )=
0
,
respectively. The k -th order PDE of an unknown function u
(
x 1 ,
x 2 , ··· ,
x n )
of n in-
dependent variables x 1 ,
x 2 , ··· ,
x n can be written in a general form
F u
2 u
2 u
2 u
k u
u
x n ,
u
x 2 , ··· ,
x n , ··· ,
,
x 1
,
x 2
, ··· ,
x n
,
x 1 , ··· ,
x 1 ,
=
0
.
x 1
x n
For example, the three-dimensional wave equation
2
2
2
Δ =
x 2 +
y 2 +
a 2
a 2
u tt =
Δ
u
+
f
(
x
,
y
,
z
,
t
) ,
z 2 ,
=
constant
is a second-order PDE of u
(
x
,
y
,
z
,
t
)
;the two-dimensional heat-conduction equation
2
2
Δ =
x 2 +
a 2
a 2
u t =
Δ
u
+
f
(
x
,
y
,
t
) ,
y 2 ,
=
constant
is a second-order PDE of u
(
x
,
y
,
t
)
. The second-order PDE
Δ
u
(
x
,
y
)=
0
,
Δ
u
(
x
,
y
,
z
)=
0
are called the two-dimensional and the three-dimensional Laplace equations ,re-
spectively. Equations of mathematical physics are the PDE that comes from phys-
ical laws and describe physical processes or physical states. The wave equation,
the heat-conduction equation and the Laplace equation are three typical equations
of mathematical physics. They can be viewed as three special cases of the dual-
phase-lagging heat-conduction equation . The latter has been recently developed in
examining energy transport involving high-rate heating.
1.1.2 Linear, Nonlinear and Quasi-Linear Equations
A PDE can be linear or nonlinear, as is the case for an ordinary differential equation.
The linear PDE has many good properties and it frequently arises in problems of
mathematical physics. We shall primarily consider linear PDE in this topic.
 
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