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3.3.4 BOUNDED DOMAINS
In this subsection, we illustrate with an example how a slight modifica-
tion of the present method can help in solving inverse heat conduction
problems in bounded domains. Considerthefollowingone-dimensional
problem in (0,
ℓ
),
ℓ
> 0:
• Find the heat flux
(
t
) at the boundary
x
= 0, where there is a heat
source, when the temperature
u
ℓ
(
t
) at
x
=
ℓ
is given and the initial
condition is zero.
• Boundary element methods have been used in [14] for solving this
problem with a discontinuous function
ϕ
(
t
); thus, this example can
also be used for benchmarking the performance of the present meth-
od against others. The temperature
ϕ
()
ut
and the flux
ϕ
(
t
) are related
by (see [14] for details)
t
2
/4
(
)
∫
−
t
−
τ
e
∞
()
()
(
) ()
(15)
∫
0
ut
=
φτ
d
τ
=
gt
−
τ
q d
τ
τ
,
(
)
πτ
t
−
−∞
()
2
/4
−
t
(
t
)
H
(
t
), and
H
(
t
) is the Heaviside
function. Therefore, convolving Eq. (15), we obtain the analog of Eq. (5):
where
gt
=
e
Ht
()/
π
t
,
q
(
t
) =
ϕ
∞
1
()
()
∫
−
it
ω
qt
=
e
U
()/
ωωω
G
d
,
a
2
π
−∞
()
u
and
g
(
t
), re-
spectively. Therefore, the numerical solution of this problem is
where
U
a
(
ω
) and
G
(
ω
) are the Fourier transforms of
q
=
a ixft
[
xft
[
u
( )] /
t
xft
[
g t
( )]]
(16)
k
1
1
1
k
which is the analog of Eq. (9). In this example, we take as data the tem-
perature
()
ut
obtained by a numerical integration of Eq. (15) where the
function
ϕ
(
t
) is known a priori and given by
1,
t
≤≤
t
t
,
t
≤≤
t
t
=
⎩
()
1
2
3
4
φ
t
,
0,
otherwise
where
t
1
= 0.2,
t
2
=0.7,
t
3
= 1, and
t
4
= 1.5. We take
ℓ
= 1/3. In Figure (
3.6),
the numerical results are compared with the exact solution and a plot of
the error |
q
(
t
k
)−
q
k
| is given.
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