Environmental Engineering Reference
In-Depth Information
m
n
1
2
3
4
5
. . .
1
11
B
12
B
13
B
14
B
15
...
2
21
B
22
B
23
B
24
B
25
...
3
31
B
32
B
33
B
34
B
35
...
4
41
B
42
B
43
B
44
B
45
...
5
51
B
52
B
53
B
54
B
55
...
...
...
...
...
...
...
which is a matrix.
B
=(
B
mn
)
∞
×
∞
.
If the first 3, 6 or 10 terms are used, we normally use those in the upper triangular
matrix.
m
,
n
,
k
is a triple summation at the three-dimensional case. For any fixed value of
m
, we have a coefficient matrix
B
mnk
)
∞
×
∞
. Take the starting values of
m
,
n
and
k
as equal to 1 as an example. Calculating the summation, the first eight terms
consists of the four terms
(
(
B
1
nk
)
2
×
2
at
m
=
1 and the four terms
(
B
2
mk
)
2
×
2
at
m
=
2. If taking
m
,
n
,
k
=
1
,
2
,
3, we have a total of 27 terms.
4. If
β
mnk
is purely imaginary for some
m
,
n
and
k
, we can change sin
β
mnk
t
into
e
iβ
mnk
t
e
−
iβ
mnk
t
2i
e
i
z
e
−
i
z
2i
−
−
by using the formula sin
z
=
for any imaginary vari-
able
z
.If
β
132
=
i
r
132
with
r
132
taking a positive real value, for example, sin
β
132
t
can be changed to an exponential function such that
e
α
132
t
i
r
132
e
−
r
132
t
e
r
132
t
e
α
132
t
2
r
132
(
1
β
132
−
e
α
132
t
sin
e
r
132
t
e
−
r
132
t
β
132
t
=
=
−
)
.
2i
0, the term
1
β
132
β
132
t
decays as
t
e
α
132
t
sin
Since
|
α
132
| >
r
132
and
α
132
<
→
∞
.
5. If the
satisfy consistency conditions, we can also use The-
orem 1 in Section 6.1 to express the
ϕ
(
M
)
and the
ψ
(
M
)
ϕ
-contribution to the solution by the
ψ
-contribution.
6.5 Mixed Problems in a Circular Domain
Boundary conditions of all three kinds for mixed problems in a circular domain
become separable with respect to the spatial variables in a polar coordinate system.
In this section we use separation of variables to solve mixed problems in a polar
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