Environmental Engineering Reference
In-Depth Information
3.4 One-Dimensional Cauchy Problems: Fundamental Solution
In this section, we apply the Fourier transformation to develop solutions of Cauchy
problems and discuss fundamental solutions. We also develop solutions of problem
in a semi-infinite domain by using the method of continuation.
3.4.1 One-Dimensional Cauchy Problems
Using the Fourier transformation, find the solution of
u
t
a
2
u
xx
=
+
f
(
x
,
t
)
, −
∞
<
x
<
+
∞
,
0
<
t
,
(3.25)
(
,
)=
ϕ
(
)
.
u
x
0
x
Solution.
We fir s t deve l op
W
ϕ
(
x
,
t
)
, the solution of PDS (3.25) with
f
=
0. View-
ing
u
as a function of
x
, apply the Fourier transformation to the equation of
PDS (3.25) to arrive at
(
x
,
t
)
2
u
u
t
(
ω
,
t
)+(
ω
a
)
(
ω
,
t
)=
0
,
2
t
. Here,
A
e
−
(
ω
a
)
which yields
u
is the function to be determined by
the initial condition. Applying the initial condition, we have
A
(
ω
,
t
)=
A
(
ω
)
(
ω
)
(
ω
)=
ϕ
(
ω
)
¯
.There-
fore,
2
t
e
−
(
ω
a
)
¯
¯
u
(
ω
,
t
)=
ϕ
(
ω
)
,
F
[
ϕ
(
x
)] =
ϕ
(
ω
)
.
(3.26)
The solution
u
(
x
,
t
)
follows from the inverse Fourier transformation of Eq. (3.26),
F
−
1
¯
2
t
1
2
a
√
π
x
2
4
a
2
t
t
e
−
e
−
(
ω
a
)
u
=
W
ϕ
(
x
,
t
)=
ϕ
(
ω
)
=
ϕ
(
x
)
×
2
e
−
(
x
−
ξ
)
+
∞
1
2
a
√
π
4
a
2
t
=
ϕ
(
ξ
)
d
ξ
.
(3.27)
t
−
∞
The solution of PDS (3.25) with
ϕ
=
0 is, by the solution structure theorem,
t
t
+
∞
e
−
(
x
−
ξ
)
2
(
ξ
,
τ
)
√
t
1
2
a
√
π
f
u
=
W
f
τ
(
x
,
t
−
τ
)
d
τ
=
d
τ
4
a
2
−
τ
)
d
ξ
.
(3.28)
(
t
−
τ
0
0
−
∞
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