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()
uxt
,
()
2
n
=∇
α
uxt
, ,
x
, 0
<≤
t t
f
t
(
)
()
n
uxt
,
=
f x
,
x
,
(1)
f
where α is the thermal diffusivity or diffusion coefficient. Now, if g ( x ) is
a given bounded continuous function, the unique bounded solution of the
direct problem, i.e., of the initial value problem:
()
uxt
,
()
=∇
α
2
uxt
, ,
x
n
,
t
>
0
t
(
)
( )
n
(2)
ux
,0
=
gx
,
x
can be obtained by the using Green's functions [12, 13]
2
xy
t
1
()
(3)
uxt
,
=
e
uy
( , 0)
dy
4
α
(4
πα
t
)
n
/2
n
The partial differential equation appearing in the inverse and direct prob-
lems is the same; however, these problems have some remarkable differ-
ences. In the direct problem, Eq. (2), a small change in the data always
gives a small change in the solution. In the inverse problem, Eq. (1), this
is not true.
The integral transform in Eq. (3) is known in the literature as Gauss or
Weierstrass transform of the function u ( x, 0). Therefore, solving Eq. (1) is
equivalent to fi nd the inverse Gauss transform of Eq. (3). This can be done
by using the Fourier transform and the convolution theorem. Taking into
account that the Fourier transform of the kernel is
2
2
ik x
|| || /4
x
α
t
n
/ 2
α
|| ||
t k
ee
x
=
(4
πα
t e
)
n
the convolution of Eq. (3) yields
(
)
2
(
)
α
t
|| ||
k
(4)
Uk
,0
=
e
Ukt
,
,
f
 
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