Environmental Engineering Reference
In-Depth Information
1
τ
2
W
ϕ
∂
1
τ
0
∂
W
ϕ
∂
+
∂
B
2
∂
∂
A
2
=
−
Δ
W
ϕ
−
t
Δ
W
ϕ
t
t
2
0
1
τ
2
W
ϕ
∂
+
∂
∂
0
∂
W
ϕ
∂
t
+
∂
B
2
∂
∂
A
2
−
Δ
W
ϕ
−
t
Δ
W
ϕ
t
2
t
1
τ
W
ϕ
1
2
W
ϕ
1
∂
0
∂
W
ϕ
1
∂
t
+
∂
B
2
∂
∂
A
2
+
−
Δ
W
ϕ
1
−
t
Δ
=
0
,
t
2
in which we have used Eqs. (6.6a) and (6.7a).
Satisfaction of Boundary Conditions
Substituting Eq. (6.5) into the boundary conditions of PDS (6.4) leads to
L
u
∂Ω
,
∂
u
∂
n
L
1
W
ϕ
+
1
W
ϕ
+
W
ϕ
1
∂Ω
τ
0
+
∂
∂
∂
τ
0
+
∂
=
W
ϕ
1
,
∂
t
n
∂
t
∂Ω
+
∂
∂Ω
+
∂Ω
=
L
W
ϕ
,
∂
t
L
W
ϕ
,
∂
L
W
ϕ
1
,
∂
1
τ
W
ϕ
∂
W
ϕ
∂
W
ϕ
1
∂
=
0
,
n
∂
n
n
0
where we have used Eqs. (6.6b) and (6.7b).
Satisfaction of Initial Conditions
By using Eqs. (6.6c) and (6.7c), we have
1
W
ϕ
(
t
=
0
τ
0
+
∂
u
(
M
,
0
)=
M
,
t
)+
W
ϕ
1
(
M
,
t
)
∂
t
t
=
0
+
1
τ
0
W
ϕ
(
)+
∂
W
ϕ
∂
=
M
,
0
W
ϕ
1
(
M
,
0
)=
ϕ
(
M
)
.
t
Also
1
W
ϕ
(
∂
u
t
=
∂
τ
0
+
∂
M
,
t
)+
W
ϕ
1
(
M
,
t
)
∂
∂
t
∂
t
2
W
ϕ
∂
τ
0
∂
1
W
ϕ
∂
t
+
∂
+
∂
W
ϕ
1
∂
=
t
2
t
B
2
∂
∂
W
ϕ
+
∂
W
ϕ
1
∂
A
2
=
Δ
W
ϕ
+
t
Δ
t
∂
W
ϕ
∂
+
∂
W
ϕ
1
∂
A
2
B
2
=
Δ
W
ϕ
+
Δ
.
t
t
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