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u
k
.x; t /
D
c
k
e
k
2
2
t
sin.kx/
for k
D
:::;
2;
1; 0; 1; 2; : : :
(8.58)
satisfy both the diffusion equation (
8.44
) and the boundary condition (
8.45
). Here,
c
k
represents an arbitrary constant, see (8.51). At this point we recommend that you
do Exercise
8.1
.
Example 8.1.
Consider the problem
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
(8.59)
u
.0; t /
D
u
.1; t /
D
0
for t>0;
(8.60)
u
.x; 0/
D
sin.x/
for x
2
.0; 1/:
(8.61)
In view of the theory developed in Sect.
8.2.1
, it follows that
u
.x; t /
D
e
2
t
sin.x/
satisfies (
8.59
)and(
8.60
). Furthermore,
u
.x; 0/
D
e
2
0
sin.x/
D
sin.x/;
and thus this is the unique smooth solution of this problem.
Example 8.2.
Our second example is
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
u
.0; t /
D
u
.1; t /
D
0
for t>0;
u
.x; 0/
D
7 sin.5x/
for x
2
.0; 1/:
Setting k
D
5 and c
5
D
7 in (8.58), we find that
u
.x; t /
D
7e
25
2
t
sin.5x/
satisfies the diffusion equation and the boundary condition of this problem. Further-
more,
u
.x; 0/
D
7e
25
2
0
sin.5x/
D
7 sin.5x/
and hence the initial condition is also satisfied.
8.2.2
Super-Positioning
Above we saw how we could use the method of separation of variables to construct
infinitely many solutions of the two equations
