Environmental Engineering Reference
In-Depth Information
wherewehaveused
F
−
1
e
−
(
a
ω
)
−
τ
)
x
2
1
2
a
√
t
e
−
2
(
t
4
a
2
=
.
(
t
−
τ
)
−
τ
7.3 Methods for Solving Nonhomogeneous Potential Equations
We have discussed several methods for obtaining solutions of homogeneous poten-
tial equations (i.e. the Laplace equations) in Section 7.1 and Section 7.2. In this
section, we discuss the two methods for solving nonhomogeneous potential equa-
tions, i.e. the Poisson equations.
7.3.1 Equation Homogenization by Function Transformation
Some problems of Poisson equations can be transformed into those of Laplace equa-
tions by some proper function transformations. We demonstrate this with two exam-
ples.
Example 1.
Find the solution of
⎧
⎨
Δ
u
=
py
+
q
,
0
<
x
<
a
,
0
<
y
<
b
,
u
x
(
0
,
y
)=
u
(
a
,
y
)=
0
,
(7.42)
⎩
u
(
x
,
0
)=
u
(
x
,
b
)=
0
.
where
p
and
q
are constants.
Solution.
Consider a function transformation
x
2
a
2
v
(
x
,
y
)=
u
(
x
,
y
)+(
−
)(
c
1
y
+
c
2
)
.
Such a transformation preserves the homogeneity of boundary conditions with re-
spect to
x
in PDS (7.42), regardless of the values of constants
c
1
and
c
2
,i.e.
v
x
(
0
,
y
)=
v
(
a
,
y
)=
0
.
To homogenize the equation, substitute
v
(
x
,
y
)
into the Poisson equation in PDS (7.42)
yields
c
1
=
−
p
/
2
,
c
2
=
−
q
/
2
.
Search WWH ::
Custom Search