Environmental Engineering Reference
In-Depth Information
Method of Solving (3.46)
n = 1 a n ( s ) sin n π x
Expand G
(
x
,
s
)=
. Substitute it into the integral equation to obtain
l
l
n = 1 X n sin n π x
(
)= λ
,
= λ
(
)
(
)
X
x
X n
X
s
a n
s
d s
(3.47)
l
0
Multiplying the equation of (3.47) by a m
(
x
)
and then integrating over
[
0
,
l
]
yields
l
n = 1 a mn X n ,
sin n
π
x
= λ
=
(
)
,
=
,
, ···
X m
a mn
a m
x
d x
m
1
2
(3.48)
l
0
T . Eq. (3.48) can then be written as an
Let A
=(
a mn ) × , X
=(
X 1 ,
X 2 , ··· ,
X n , ··· )
eigenvalue problem of A
X
= λ
AX
.
(3.49)
We denote its eigenvalues and the corresponding eigenvectors by
λ 1 , λ 2 , ··· , λ n , ··· ,
X 1 ,
X 2 , ··· ,
X n , ··· .
The eigenfunction set associated with the eigenvalues
λ n (
n
=
1
,
2
, ··· )
X 1 (
x
) ,
X 2 (
x
) , ··· ,
X n (
x
) , ···
can thus be obtained by an substitution into Eq. (3.47). The T
(
t
)
associated with
λ = λ
n can also be obtained by Eq. (3.43),
e ( λ n t / ρ c ) ,
(
)=
=
,
, ··· .
T n
t
n
1
2
Therefore
n = 1 C n X n ( x ) e ( λ n t / ρ c ) ,
which satisfies the equation and the boundary conditions of (3.41). Here C n ( n
n = 1 C n T n ( t ) X n ( x )=
u
(
x
,
t
)=
=
1
,
2
, ···
) are constants and can be determined by applying the initial condition
) } n = 1 in
u
(
x
,
0
)=
f
(
x
)
and using the completeness and orthogonality of
{
X n (
x
[
0
,
l
]
.
Analytical solution of PDS (3.41)
The analytical solution of PDS (3.41) follows from the above discussion,
n = 1 C n X n (
e ( λ n t / ρ c ) ,
u
(
x
,
t
)=
x
)
d x 0 X n (
C n = 0 X n (
x
)
f
(
x
)
x
)
d x
.
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