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(f) Show that
Z 1
u .x; t / dx D Z 1
0
f.x/ dx
for t 0:
0
(g) If heat can neither leave nor enter a body ˝ ( D .0; 1/) then the temperature
throughout ˝ will approach a constant temperature C as time increases. Thus,
our conjecture is that
u .x; t / ! C as t !1 (8.104)
for all x 2 .0; 1/. Furthermore, it seems reasonable that C equals the average of
the initial temperature distribution f .
Let
f.x/ D 3:14 C cos.2x/
for x 2 .0; 1/
and define
C D Z 1
0
f.x/ dx :
Design a numerical experiment suitable for testing the hypothesis (8.104).
Perform several experiments and comment on your results.
We will now modify the method of separation of variables, presented in Sect. 8.2 ,
to handle Neumann boundary conditions. It turns out that this can be accomplished
in a rather straight forward manner.
(h) Apply the ansatz
u .x; t / D X.x/T .t/
to ( 8.99 )-( 8.100 ) and show that this leads to the two eigenvalue problems
T 0 .t / D T .t /;
(8.105)
X 00 .x/ D X.x/;
X 0 .0/ D 0 and X 0 .1/ D 0;
(8.106)
where is a constant.
(i) Show that
T.t/ D ce t
solves ( 8.105 ) for any constant c.
(j) Show that
X.x/ D cos.kx/
satisfies ( 8.106 )fork D 0; 1; 2; : : :
(k) Use the results obtained in (i) and (j) and with the super-positioning principle
to conclude that any convergent series of the form
X
c k e k 2 2 t cos.kx/
u .x; t / D c 0 C
k D 1
defines a formal solution of ( 8.99 )and( 8.100 ).
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