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defines a formal
a
solution of the problem
D
u
xx
for x
2
.0; 1/; t > 0;
u
t
(8.77)
u
.0; t /
D
u
.1; t /
D
0
for t>0;
(8.78)
u
.x; 0/
D
f.x/
for x
2
.0; 1/:
(8.79)
a
Recall that we have not analyzed the convergence properties of this series
properly. Under what circumstances will the Fourier series of a function con-
verge toward the correct limit? Can we differentiate term-wise? Will the limit,
if it exists, satisfy the diffusion equation? Thus we refer to (8.76)asa“formal
solution”.
Note that, for a given positive integer N , we can approximate the formal solution
(8.76)bytheN th partial sum
u
N
of the Fourier series, i.e.,
N
X
c
k
e
k
2
2
t
sin.kx/:
u
.x; t /
u
N
.x; t /
D
(8.80)
k
D
1
This is important from a practical point of view. By hand, or on a computer, we can,
in general, not compute an infinite sum of sine functions!
8.2.8
More Examples
We will now consider some slightly more advanced examples illuminating the
theory developed above.
Example 8.5.
We want to determine the Fourier sine series of the constant function
f.x/
D
10
for x
2
.0; 1/:
From formula (8.74)wefindthat
D
2
Z
1
0
f.x/sin.kx/
dx
D
2
Z
1
0
c
k
10 sin.kx/
dx
D
20
k
cos.kx/
1
0
1
D
20
k
.cos.k/
1/
D
0
if k is even;
40
k
if k is odd:
