Environmental Engineering Reference
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Thus the solution for the case of
f
=
ϕ
=
0 reads, by Eq. (6.59),
⎧
⎨
+
∞
m
=
0
B
m
e
α
m
t
sin
β
m
t
·
cos
(
2
m
+
1
)
π
x
u
(
x
,
t
)=
W
ψ
(
x
,
t
)=
,
2
l
l
0
ψ
(
cos
(
+
)
π
⎩
2
2
m
1
x
B
m
=
x
)
d
x
.
l
β
m
2
l
where
4
1
τ
0
+
λ
m
B
2
1
m
B
2
2
1
2
1
2
=
−
,
=
m
A
2
−
τ
0
+
λ
.
α
β
λ
m
m
Finally, the solution of PDS (6.63) is , by the solution structure theorem,
1
W
ϕ
(
t
τ
0
+
∂
B
2
W
λ
m
ϕ
(
u
(
x
,
t
)=
x
,
t
)+
x
,
t
)+
W
ψ
(
x
,
t
)+
W
f
τ
(
x
,
t
−
τ
)
d
τ
,
∂
t
0
where
f
τ
=
f
(
x
,
τ
)
.
6.4.2 Two-Dimensional Mixed Problems
D
consists of
four boundary lines. 81 combinations of linear homogeneous boundary conditions
can be written in a general form as
Let
D
be the rectangular domain: 0
<
x
<
l
1
,
0
<
y
<
l
2
; its boundary
∂
L
(
u
,
u
n
)
|
∂
D
=
0
,
where the
u
n
, the normal derivative, are
±
u
x
or
±
u
y
. The corresponding eigenvalues
and the eigenfunction set are denoted by
λ
m
,
X
m
(
x
)
;
λ
n
,
Y
n
(
y
)
, respectively.
=
W
ψ
(
,
,
)
Theorem 2
.Let
u
x
y
t
be the solution of
⎧
⎨
B
2
∂
∂
A
2
u
t
/
τ
0
+
u
tt
=
Δ
2
u
+
t
Δ
2
u
,
D
×
(
0
,
+
∞
)
,
(6.64)
⎩
L
(
u
,
u
n
)
|
∂
D
=
0
,
u
(
x
,
y
,
0
)=
0
,
u
t
(
x
,
y
,
0
)=
ψ
(
x
,
y
)
.
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