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139
where
u
j
is the coeﬃcient of
φ
j
in the representation of
u
h
and
f
j
=
Z
1
0
f
(
t
)
φ
j
(
t
)d
t
=
Z
(
j
+1)
h
(
j
−
1)
h
f
(
t
)
φ
j
(
t
)d
t,
which is
h
times a weighted average of
f
over the
j
th interval. The only diFerence
between the ±nite diFerence system and this system is that we have replaced point
samples of
f
by average values. Note that if
a
(
t
) is not constant, then the systems
look even more diFerent.
CHALLENGE 23.5.
See the program posted on the website.
CHALLENGE 23.6.
See the program posted on the website.
CHALLENGE 23.7.
Here are the results, in dull tables with interesting entries:
PROBLEM 1
Using coeﬃcient functions
a
(1) and
c
(1) with true solution
u
(1)
In±nity norm of the error at the grid points
for various methods and numbers of interior grid points
M
M
=
9
99
999
1st order ±nite diFerence
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 Fall '11
 Dr.Robin

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