Information Technology Reference
In-Depth Information
where
U
(
k, t
) is the Fourier transform of
u
(
x, t
). Taking the inverse
transform, we obtain the solution:
1
(
)
()
2
−⋅
ik x
α
|| ||
tk
ux
,0
=
e
e
Uktdk
,
(5)
()
n
2
π
n
of the final boundary problem (1), and the problem now is to obtain a reli-
able numerical approximation of
u
(
x,
0).
3.2.1 NUMERICAL SOLUTION
To do this, we use the XFT, a new algorithm [11] based on the FFT for
computing the Fourier transform:
∞
()
()
∫
ik x
⋅
Ukt
,
=
e uxtdx
,
−∞
where
t
is a parameter (time in this case). According to this,
1.
π
(
)
of the real line measured
in any system of units and compute the vector
v
according to the
following relation:
consider the points
x
=
2
N
2
j
−
N
−
1
j
2
1
−−
iN
N
π
()
()
ν
t
=
e
j
ut
,
j
=
1, 2,
…
,
N
.
i
j
Here, the vector
u
(
t
)=(
u
1
(
t
),
u
2
(
t
), … ,
u
N
(
t
))
T
is considered to be
formed with the values of the temperature
u
(
x, t
) at the nodes
x
j
,
that is,
u
j
(
t
)=
u
(
x
j
, t
).
(
)
2.
Consider the points
measured in the sys-
tem of units such that the product
x
j
κ
m
is dimensionless.
κπ
=
(/22 )2
NmN
−
−
1
m
3.
Then, the XFT gives an approximation [A scaled function by
a
=4/
π
is denoted by
()
() ( )
()
i.e.,
.Therefore,
denotes
fyt
,
f
y t
,
=
f
ay t
,
J
Ut
(
)
to the
Fourier transform
U
(
k, t
) evaluated at the points
aκ
j
(
a=4/π
) ac-
cording to the following relation:
the approximation to
()
()
()
()
Ua
κ
, ]
tUt
=
(
U t U t
,
,
…
,
U t
)
T
j
1
2
x
Search WWH ::
Custom Search