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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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