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defines a formal solution of (
8.62
)and(
8.63
), provided that the series in (8.65)
converges.
In this section we have added simple functions, given in terms of the sine func-
tion, to obtain more general solutions of our problem. This technique is often
referred to as super-positioning, and hence the title of this section.
8.2.3
Fourier Series and the Initial Condition
On the one hand, in Sect.
8.1.3
we showed that our model problem (
8.1
)-(
8.3
) can
have at most one smooth solution. On the other hand, in Sect.
8.2.2
we constructed
infinitely many functions satisfying the diffusion equation (
8.1
) and the boundary
condition (
8.2
). This means that, for a given initial condition f ,
-
There is at most one function on the form (8.65) that fulfills the initial condition
(
8.3
), or
-
None of these functions satisfies this condition.
The purpose of this section is to clarify the role of the initial condition f in
Fourier's approach to heat transfer problems. This is a difficult issue and in this
introductory text we can only scratch the surface of this field.
20
To gain further
insight, we will start our investigations with a few examples.
8.2.4
Some Simple Examples
In examples 1 and 2 we saw how the method of separation of variables can be used
to compute the analytical solution of our model problem in some relatively simple
cases. Let us now consider some examples in which it is necessary to apply the
super-positioning technique developed above.
Example 8.3.
In this example we consider a problem with an initial condition given
by two sine modes
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
(8.66)
u
.0; t /
D
u
.1; t /
D
0
for t>0;
(8.67)
u
.x; 0/
D
2:3 sin.3x/
C
10 sin.6x/
for x
2
.0; 1/:
(8.68)
Consider the general formula (8.65) for a formal solution of the diffusion equation
(
8.66
) and the boundary condition (
8.67
). Let
c
k
D
0 for k
¤
3 and k
¤
6;
c
3
D
2:3 and c
6
D
10:
20
The theory for general Fourier series and Hilbert spaces.
