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and use this inequality to conclude that
u
`
C
1
max
u
max
for `
D
1;:::;m:
(d) Show that
u
max
max
max
x
f.x/;0
:
(e) Prove in a similar manner that
u
min
min
min
x
f.x/;0
:
From the inequalities derived in (d) and (e) we conclude that
min
min
x
f.x/;0
for i
D
1;:::;n and `
D
0;:::;m;
(8.98)
which is a discrete analog to the maximum principle (8.41)-(
8.43
) valid for the
continuous problem (
8.1
)-(
8.3
).
f.x/;0
u
i
max
max
x
(f) Above we mentioned that we would derive a stronger bound for the numerical
approximations of the solution of the diffusion equation than inequality (8.96).
Explain why (8.98) is a stronger result than (8.96), i.e., use (8.98)toderive
(8.96).
˘
Exercise 8.10.
The purpose of this exercise is to give a slightly different argument
for the property (8.7) than in the text above. Recall that the starting point of our
analysis leading to this inequality was to multiply the left- and right-hand sides of
the diffusion equation by
u
. In this exercise we ask you to start off by differentiating
the function E
1
.t /,definedin(8.4), with respect to t and apply the differential
equation (
8.1
), integration by parts, the boundary condition (
8.2
), and so on, in order
to derive (8.7).
˘
Exercise 8.11.
Assume that g.t/ is a function of t>0and that f.x/ is a function
of x
2
.0; 1/. Prove that if
g.t/
D
f.x/
for all x
2
.0; 1/ and t>0;
then there must exist a constant such that
g.t/
D
for all t>0;
and
f.x/
D
for all x
2
.0; 1/:
˘
