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where
E
is the diagonal matrix defined by Eq. (8), and the present solution
is weakly stable since
()
|
u
(0)
≤
u t
,
|
where
C
is a positive constant depending on
N
,
n
,
α
, and
t
, such that
lim
. This is the condition of weak stability given in Eq. (1).
C
=∞
N
→∞
3.3 RESULTS
We test the numerical solution (9) in this section. To show the performance
of this procedure, we take
n
= 2, 3 and three problems with known solu-
tion. The solution for the first one is a Dirac delta function; the second
example consists of a homogeneous temperature distribution (which hap-
pens to be a nonsquare-integrable function); and for the third example,
we use a volcano-shaped radially symmetric distribution with a nontrivial
peak. In each of these examples, we use
α
= 0.0002 m
2
/s, which corre-
sponds to the thermal diffusivity of Helium. We also take the same number
of nodes
N
for all dimensions.
In the following examples, the exact values of
u
(
x
, 0) are known. Thus,
in order to measure the error of our method, we defi ne
as the maxi-
mum absolute value of the differences between the exact and approxi-
mated initial values on the plane
ζ
m
x
=
ζ
,
m
that is,
3
(
)
(11)
=
max
u
ξηζ
,
,
, 0
−
u
(0) ,
ζ
j
km
q
m
jk
,
where
q
is the index associated to (
j, k, m
) through Eq. (7) and
ξ
j
and
η
k
are
nodes on the directions
x
2
and
x
3
, respectively.
3.3.1
A DIRAC DELTA DISTRIBUTION
In this example, we take
x
∈
and a final temperature given by the fol-
2
lowing equation:
1
()
2
/4
(12)
−
x
α
t
uxt
,
=
e
,
4
πα
t
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