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and our goal is to show that
u
.x; t /
D
X
k
2
S
c
k
e
k
2
2
t
sin.kx/
solves this problem.
Since the sum is finite we can differentiate term-wise, that is,
c
k
e
k
2
2
t
sin.kx/
D
X
k
2
S
u
t
.x; t /
D
X
k
2
S
@
@t
k
2
2
c
k
e
k
2
2
t
sin.kx/;
and
c
k
e
k
2
2
t
sin.kx/
D
X
k
2
S
u
xx
.x; t /
D
X
k
2
S
@
2
@x
2
k
2
2
c
k
e
k
2
2
t
sin.kx/;
and we see that this function satisfies the diffusion equation. Moreover, by elemen-
tary properties of the sine function, it follows that
u
.0; t /
D
X
k
2
S
c
k
e
k
2
2
t
sin.0/
D
0;
u
.1; t /
D
X
k
2
S
c
k
e
k
2
2
t
sin.k/
D
0;
and thus the boundary conditions are fulfilled. Finally, since
u
.x; 0/
D
X
k
2
S
c
k
e
0
sin.kx/
D
X
k
2
S
c
k
sin.kx/
D
f.x/;
we conclude that we have found the unique smooth solution of this problem.
Can we develop this technique one step further? What about initial conditions
defined in terms of infinite series of sine functions? This means that the initial
condition is of the form
X
f.x/
D
c
k
sin.kx/
for x
2
.0; 1/;
k
D
1
where the sum on the right-hand side is an infinite series. Assume that this series
converges for all x
2
.0; 1/ and t
0. Consider the formal solution
X
c
k
e
k
2
2
t
sin.kx/
u
.x; t /
D
(8.69)
k
D
1
