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(
)
(
)
n
−
1
qj
=+ −
j
1
N
+…+ −
j
1
N
,
j
1,2,
= …
,
N
.
(7)
1
2
n
m
Here,
e
αt
||
a κ
||
2
stands for the
N
n
×
N
n
diagonal matrix whose diagonal ele-
ments are given by the following equation:
22
2
2
e
ακκ
ta
(
++…+
κ
)
(8)
j
j
j
n
1
2
For simplicity, we have taken the same number of nodes
N
for all di-
mensions. Taking this and Eq. (6) into account, the numerical solution of
the BHCP Eq. (1) is given by the following equation:
()
()
(9)
n
⎡
α
t
2
⎤
u
0
=
a ixft
e
a
κ
xft
⎡
u t
⎤
,
⎣
⎦
⎣
⎦
q
n
n
q
xft
[]
⋅
ixft
[]
⋅
stand for the
n
-dimensional version of the
XFT algorithm and its inverse, respectively.
Where
and
n
n
3.2.2 STABILITY
Note that the ill-posedness nature of the discrete BHCP comes from the
exponential term in Eq. (9), which becomes greater as the backward time
t
,
N
, or
n
are increased. Therefore, to compute a large time
t
, the number of
nodes should be relatively small, according to the given number of dimen-
sions
n
. In addition, note that for a given
α
, the product
αt
and the machine
numerical precision define a characteristic backward time beyond which
the method is unable to give a result.
Let
F
be the matrix associated to the
n
-dimensional XFT algorithm and
F
−1
its inverse. Then,
F
and
F
−1
can be written as the Kronecker product
of the one-dimensional XFT matrices
F
k
:
Therefore, the solution (9) can be written in matrix form as follows:
()
()
n
−
1
u
0
=
a F
u t
,
(10)
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