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Z
1
k.x/
u
x
.x; t /
dx
Z
1
0
k.x/
u
x
.x; 0/
dx
D
Z
1
0
k.x/f
x
.x/
dx
for t
0
(8.119)
0
holds.
Design a series of numerical experiments suitable for testing this conjecture.
Apply the definitions of f , k and T given in (c). According to your experiments,
is (8.119)true?
(f) Recapture the mathematical derivation of inequality (8.11), adapt that argument
to the present case and prove that (8.119) must hold.
(g) Use the result obtained in (e) to show that
Z
1
Z
1
Z
1
M
m
M
m
u
x
.x; t /
dx
u
x
.x; 0/
dx
D
f
x
.x/
dx
for t
0:
0
0
0
