Environmental Engineering Reference
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or, by separation of variables,
1
τ
0
T
(
T
(
t
)+
t
)
X
(
x
)
)
=
,
A
2
T
B
2
T
(
(
t
)+
t
X
(
x
)
where the primes on the function
X
and
T
represent differentiation with respect
to the only variable present. Therefore, we obtain the separation equation for the
temporal part
T
(
t
)
with
−
λ
as the separation constant,
1
τ
0
+
λ
B
2
T
(
T
(
A
2
T
t
)+
t
)+
λ
(
t
)=
0
(6.41)
and the homogeneous system for the spatial part
X
(
x
)
X
(
x
)+
λ
X
(
x
)=
0
,
(6.42)
b
1
X
(
b
2
X
(
−
0
)+
k
1
X
(
0
)=
0
,
l
)+
k
2
X
(
l
)=
0
.
The problem (6.42) is called an
eigenvalue problem
because it has solutions only
for certain values of the separation constant
λ
=
λ
k
,
k
=
1
,
2
,
3
, ···
, which are called
the
eigenvalues
; the corresponding solutions
X
k
(
are called the
eigenfunctions
of
the problem. The eigenvalue problem (6.42) is a special case of a more general
eigenvalue problem called the Sturm-Liouville problem, which is discussed in Ap-
pendix D.
The equation in the eigenvalue problem generally depends on the equation of
the original PDS. If after substituting
u
x
)
, it is impossible to have an
equation with terms involving
x
and
t
on two separate sides, then we cannot have an
eigenvalue problem.
(
x
,
t
)=
X
(
x
)
T
(
t
)
6.3.2 Eigenvalues and Eigenfunctions
The boundary condition at
x
=
0 in Eq. (6.42) contains three cases:
b
1
=
0,
k
1
=
0;
b
1
=
0,
k
1
=
0;
k
1
=
0,
b
1
=
0 . Similarly, there also exist three kinds of bound-
ary condition at
x
l
. Hence the eigenvalue problem (6.42) is a general form en-
compassing nine problems, each corresponding to nine combinations of boundary
conditions. We consider the case of all nonzero
b
1
,
k
1
,
b
2
and
k
2
, i.e. the problem
⎨
=
X
(
x
)+
λ
X
(
x
)=
0
,
X
(
X
(
0
)
−
h
1
X
(
0
)=
0
,
l
)+
h
2
X
(
l
)=
0
,
(6.43)
⎩
h
1
=
k
1
/
b
1
,
h
2
=
k
2
/
b
2
.
If
λ
=
0, its general solution is
X
(
x
)=
c
1
x
+
c
2
. Applying the boundary conditions
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