Environmental Engineering Reference
In-Depth Information
Finally the solution of the original PDS (2.28) can be written as
u
(
x
,
t
)=
0
v
.
In practice, we can only solve eigenvalue problems where matrix
A
has a finite
order to obtain an approximate analytical solution.
(
x
,
t
,
τ
)
d
τ
2.5 Two-Dimensional Mixed Problems
Separation of variables is the main method for solving mixed problems. It is appli-
cable only for PDE with a separable equation and separable homogeneous boundary
conditions. This is the case only for some regular domains.
2.5.1 Rectangular Domain
Consider
⎧
⎨
a
2
u
tt
=
Δ
u
+
f
(
x
,
y
,
t
)
,
D
×
(
0
,
+
∞
)
u
y
)
∂
D
=
L
(
u
,
u
x
,
0
,
(2.34)
⎩
u
(
x
,
y
,
0
)=
ϕ
(
x
,
y
)
,
u
t
(
x
,
y
,
0
)=
ψ
(
x
,
y
)
,
.
If all combinations of the boundary conditions of the first, second and third kinds
are considered, for a finite rectangular domain
D
, there exist 81 combinations of
linear boundary conditions
L
where
D
is the domain 0
<
x
<
l
1
,
0
<
y
<
l
2
,
∂
D
is the boundary of
D
,and
t
∈
(
0
,
∞
)
∂
D
=
(
u
,
u
x
,
u
y
)
0. We detail the process of finding the
solution of PDS (2.34) for the combination
u
(
0
,
y
,
t
)=
u
x
(
l
1
,
y
,
t
)+
h
2
u
(
l
1
,
y
,
t
)=
0
,
(2.35)
u
y
(
x
,
0
,
t
)
−
h
1
u
(
x
,
0
,
t
)=
u
y
(
x
,
l
2
,
t
)=
0
.
The results for the remaining 80 combinations are easily obtained by using a similar
approach, Table 2.1 and the solution structure theorem.
By the solution structure theorem, we first develop
u
=
W
ψ
(
x
,
y
,
t
)
, the solution
for the case
f
0. Based on the given boundary conditions (2.35), we should
use the eigenfunctions in Rows 3 and 8 in Table 2.1 to expand the solution,
=
ϕ
=
sin
μ
n
y
n
∞
m
,
n
=
1
T
mn
(
t
)
sin
μ
m
x
(
,
,
)=
l
2
+
ϕ
,
u
x
y
t
l
1
x
l
1
h
2
μ
n
are positive zero-points of
f
where
μ
m
and
(
x
)=
tan
x
+
and
g
(
x
)=
cot
x
−
ϕ
n
=
μ
n
x
l
2
h
1
, respectively, and tan
l
2
h
1
. Substituting this equation into the wave equa-
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