Environmental Engineering Reference
In-Depth Information
Remark 1.
The
W
ψ
(
x
,
y
,
t
)
can be obtained from
G
simply by replacing the fac-
−
η
)
τ
0
in the integrand of Eq. (4.31) by
tor
δ
(
x
−
ξ
,
y
ψ
(
x
,
y
)
and letting
τ
=
0.
G
(
x
,
ξ
;
y
,
η
;
t
−
τ
)
can be obtained from
W
ψ
(
x
,
y
,
t
)
in Eq. (4.28) simply by replacing
−
η
)
τ
0
,
t
and the factor
ψ
(
x
,
y
)
of the integrand in Eq. (4.28) by
t
−
τ
and
δ
(
x
−
ξ
,
y
respectively,
+
∞
m
,
n
=
0
1
τ
0
γ
mn
M
mn
sin
t
−
τ
2
e
−
G
(
x
,
ξ
;
y
,
η
;
t
−
τ
)=
τ
0
α
m
ξ
sin
α
m
·
x
cos
β
n
η
cos
β
n
y
sin
γ
mn
(
t
−
τ
)
.
(4.36)
Remark 2.
The Green function
G
(
x
,
ξ
;
y
,
η
;
t
−
τ
)
is the solution of
⎧
⎨
a
2
u
t
+
τ
0
u
tt
=
Δ
u
+
δ
(
x
−
ξ
,
y
−
η
,
t
−
τ
)
,
D
,
0
<
τ
<
t
<
+
∞
,
∂
D
=
L
(
u
,
u
x
,
u
y
)
0
,
(4.37)
⎩
u
(
x
,
y
,
t
)
|
t
=
τ
=
u
t
(
x
,
y
,
t
)
|
t
=
τ
=
0
.
It is the temperature distribution due to a point source
δ
(
x
−
ξ
,
y
−
η
,
t
−
τ
)
at
t
=
τ
and point
(
ξ
,
η
)
that satisfies
t
δ
(
x
−
ξ
,
y
−
η
,
t
−
τ
)
d
σ
d
τ
=
1
.
0
D
When all combinations of linear homogeneous boundary conditions of the first, the
second and the third kinds are considered, on
D
, there exist 81 combinations of
boundary conditions in PDS (4.37). Clearly, the Green function
G
differs from one
combination to another.
Remark 3.
Example 2 shows that it is crucial to obtain
G
for solving
⎧
⎨
∂
a
2
u
t
+
τ
0
u
tt
=
Δ
u
+
f
(
x
,
y
,
t
)
,
D
×
(
0
,
+
∞
)
,
u
y
)
∂
D
=
L
(
u
,
u
x
,
0
,
(4.38)
⎩
u
(
x
,
y
,
0
)=
ϕ
(
x
,
y
)
,
u
t
(
x
,
y
,
0
)=
ψ
(
x
,
y
)
.
Once the
G
is available, we can readily write out the solution
u
of PDS (4.38). The
main steps consist of:
t
Find G
→
write
u
3
=
W
f
τ
(
x
,
y
,
t
−
τ
)
d
τ
.
1
W
ϕ
(
0
τ
0
+
∂
Find G
→
write
u
2
=
W
ψ
(
x
,
y
,
t
)
→
write
u
1
=
x
,
y
,
t
)
.
∂
t
The Green function (Eq. (4.32) and Eq. (4.36)) has an elegant, universal structure.
It can be written out directly from the eigenfunctions in Table 2.1. The underlying
rule will be demonstrated by examining how to construct the Green function
G
in
Eq. (4.32) and Eq. (4.36) from the corresponding eigenfunction sets in Table 2.1.
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