Environmental Engineering Reference
In-Depth Information
2.2 Fourier Method for One-Dimensional Mixed Problems
In this section, we discuss the Fourier method for finding solutions of
⎧
⎨
a
2
u
xx
+
u
tt
=
f
(
x
,
t
)
,
(
0
,
l
)
×
(
0
,
+
∞
)
L
(
u
,
u
x
)
|
x
=
0
,
l
=
0
,
(2.5)
⎩
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
ψ
(
x
)
.
By the solution structure theorem in Sect. 2.1, the solution of PDS (2.5) is read-
ilyavailable if we have
u
=
W
ψ
(
x
,
t
)
solving
⎧
⎨
⎩
a
2
u
xx
,
u
tt
=
(
0
,
l
)
×
(
0
,
+
∞
)
L
(
u
,
u
x
)
|
x
=
0
,
l
=
0
,
(2.6)
u
(
x
,
0
)=
0
,
u
t
(
x
,
0
)=
ψ
(
x
)
.
depends on the boundary conditions at two
ends. We apply the Fourier method here to find solutions of (2.6) with boundary
conditions of the first kind and of the second kind at the ends.
Clearly, the structure of
u
=
W
ψ
(
x
,
t
)
2.2.1 Boundary Condition of the First Kind
When
u
(
0
,
t
)=
u
(
l
,
t
)=
0, by the theory of Fourier series, PDS (2.6) has a solution
of type
∞
k
=
1
T
k
(
t
)
sin
k
π
x
u
(
x
,
t
)=
,
(2.7)
l
where
T
k
(
is undetermined function.
Substituting Eq. (2.7) into the equation of (2.6) leads to
t
)
k
2
π
a
T
k
(
t
)+
T
k
(
t
)=
0
,
l
which has the solution
a
k
cos
k
π
at
l
+
b
k
sin
k
π
at
T
k
(
t
)=
,
(2.8)
l
where
a
k
and
b
k
are undetermined coefficients.
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