Environmental Engineering Reference
In-Depth Information
functions and their normal squares. The results for the remaining six combinations
can be readily obtained using a similar approach and are summarized in Table 2.1.
It has been proven by the theory of eigenvalue problems that all eigenfunction
groups in Table 2.1 are complete and orthogonal in
. Therefore, they can be
used to expand the functions including solutions of PDS. The expanding series is
called the
generalized Fourier series
. The generalized Fourier method of expansion
solves PDS by using the generalized Fourier series to expand solutions. Using this
method, the coefficients in the series can be easily determined by the completeness
and the orthogonality of eigenfunction groups. We show this method here by solving
⎧
⎨
[
0
,
l
]
a
2
u
xx
+
u
tt
=
f
(
x
,
t
)
,
(
0
,
l
)
×
(
0
,
+
∞
)
u
x
(
0
,
t
)=
0
,
u
x
(
l
,
t
)+
hu
(
l
,
t
)=
0
,
⎩
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
ψ
(
x
)
.
Solution.
We fir s t deve l op
W
ψ
(
0. Based
on the given boundary conditions, we should use the eigenfunctions in Row 6 in
Table 2.1 to expand the solution so that
x
,
t
)
, the solution for the case of
f
=
ϕ
=
∞
m
=
1
T
m
(
t
)
cos
μ
m
x
u
=
,
l
where
T
m
(
t
)
is the function to be determined later and
μ
m
is positive zero point of
(
)=
−
/
f
x
lh
.
Substituting the above equation into the equation of the PDS leads to
cot
x
x
μ
m
a
l
2
T
m
+
T
m
=
0
,
which has the general solution
a
m
cos
μ
m
at
l
+
b
m
sin
μ
m
at
l
T
m
(
t
)=
.
Applying the initial condition
u
(
x
,
0
)=
0 yields
a
m
=
0,
m
=
1
,
2
, ···
. To satisfy the
initial condition
u
t
(
x
,
0
)=
ψ
(
x
)
,
b
m
must be determined such that
∞
m
=
1
b
m
μ
m
a
cos
μ
m
x
l
=
ψ
(
x
)
.
l
The
b
m
is thus, by the completeness and the orthogonality of the eigenfunction set,
l
0
ψ
(
l
cos
μ
m
x
l
b
m
=
x
)
d
x
,
μ
m
M
m
a
1
.
l
2
sin2
μ
m
where
M
m
=
+
2
μ
m
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