Environmental Engineering Reference
In-Depth Information
3.2 Solutions of Mixed Problems
In this section, we follow Remark 2 in Section 3.1 to write out solutions of mixed
problems of heat-conduction equations directly from those in Section 2.2 to Sec-
tion 2.4. This is the Fourier method based on Table 2.1.
3.2.1 One-Dimensional Mixed Problems
Boundary Condition of the First Kind
Find the solution of PDS
⎧
⎨
a
2
u
xx
+
u
t
=
f
(
x
,
t
)
,
(
0
,
l
)
×
(
0
,
+
∞
)
,
u
(
0
,
t
)=
u
(
l
,
t
)=
0
,
(3.4)
⎩
u
(
x
,
0
)=
ϕ
(
x
)
.
Solution.
It follows from Section 2.2.1 that for the case
f
(
x
,
t
)=
0, the solution of
PDS (3.4) is
⎧
⎨
+
∞
k
=
1
b
k
e
−
(
2
t
sin
k
π
x
k
π
a
)
u
=
W
ϕ
(
x
,
t
)=
,
l
l
(3.5)
l
0
ϕ
(
⎩
2
l
sin
k
π
x
b
k
=
x
)
d
x
.
l
Therefore, the solution of PDS (3.4) for the case
ϕ
(
x
)=
0 would be, by the solution
structure theorem,
t
t
+
∞
k
=
1
b
k
e
−
(
(
t
−
τ
)
sin
k
π
x
2
k
π
a
l
)
u
=
W
f
τ
(
x
,
t
−
τ
)
d
τ
=
d
τ
l
0
0
t
l
=
G
(
x
,
ξ
,
t
−
τ
)
f
(
ξ
,
τ
)
d
ξ
d
τ
.
(3.6)
0
0
Here,
∞
k
=
1
2
−
τ
)
sin
k
πξ
l
sin
k
π
x
2
l
e
−
(
k
π
a
l
)
(
t
G
(
x
,
ξ
,
t
−
τ
)=
l
is called the
Green function
of the mixed problem of the one-dimensional heat-
conduction equation under boundary conditions of the first kind.
Search WWH ::
Custom Search