Environmental Engineering Reference
In-Depth Information
W
f
τ
(
M
,
t
−
τ
)
thus satisfies
⎧
⎨
2
W
f
τ
∂
∂
W
f
τ
∂
t
+
τ
0
∂
a
2
=
Δ
W
f
τ
,
Ω
,
0
<
τ
<
t
<
+
∞
,
t
2
∂Ω
=
L
W
f
τ
,
∂
W
f
τ
∂
0
,
⎩
n
t
=
τ
=
W
f
τ
t
=
τ
=
∂
∂
1
τ
0
,
t
W
f
τ
f
(
M
,
τ
)
.
0
Remark 2.
The solution structure theorem is also valid for Cauchy problems. How-
ever, the structure of
W
ψ
(
,
)
M
t
differs from that of mixed problems.
4.2 One-Dimensional Mixed Problems
In this section, we use Table 2.1, the solution structure theorem and the Fourier
method of expansion to solve
⎧
⎨
a
2
u
xx
+
u
t
+
τ
0
u
tt
=
f
(
x
,
t
)
,
(
0
,
l
)
×
(
0
,
+
∞
)
,
L
1
(
u
,
u
x
)
|
x
=
0
=
L
2
(
u
,
u
x
)
|
x
=
l
=
0
,
(4.7)
⎩
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
ψ
(
x
)
.
If all combinations of linear homogeneous boundary conditions of the first, second
and third kinds are considered at
x
l
, there exist nine combinations of
boundary conditions. We detail the process of seeking the solutions of PDS (4.7)
for two combinations of boundary conditions. The results for the remaining seven
combinations can be readily obtained by using a similar approach.
=
0and
x
=
4.2.1 Mixed Boundary Conditions of the First and the Third Kind
Consider the boundary conditions of type
(
,
)=
(
,
)+
(
,
)=
.
u
0
t
u
x
l
t
hu
l
t
0
(4.8)
By the solution structure theorem, we first develop
W
ψ
(
x
,
t
)
, the solution for the
case of
0. Based on the given boundary conditions (4.8), we should use the
eigenfunctions in Row 3 in Table 2.1 to expand the solution
ϕ
=
f
=
+
∞
m
=
1
T
m
(
t
)
sin
μ
m
x
u
(
x
,
t
)=
,
(4.9)
l
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