Information Technology Reference
In-Depth Information
where a
1
and a
2
are arbitrary constants, also solves
18
this problem for all constants
a
1
and a
2
.
What does this mean for the set of solutions given in (8.58)? Well, it means that
any linear combination of these functions will form a solution of the problem. In
mathematical terms this is expressed as follows. For any, possibly infinite, sequence
of numbers
:::;c
2
;c
1
;c
0
;c
1
;c
2
;:::
such that the series of functions
X
1
X
c
k
e
k
2
2
t
sin.kx/
C
c
k
e
k
2
2
t
sin.kx/
k
D1
k
D
0
converges,
19
the function
1
X
X
c
k
e
k
2
2
t
sin.kx/
C
c
k
e
k
2
2
t
sin.kx/
u
.x; t /
D
(8.64)
k
D1
k
D
1
defines a formal solution of (
8.62
)and(
8.63
).
We close this discussion with a simplifying observation. It turns out that, without
any loss of generality, we can remove the first sum from the formula given in (8.64).
This is so because
sin.
y/
D
sin.y/
for all y
2
IR
and, consequently, for any positive integer k,
c
k
e
k
2
2
t
sin.
kx/
C
c
k
e
k
2
2
t
sin.kx/
D
.c
k
c
k
/e
k
2
2
t
sin.kx/:
Thus, the set of solutions of the diffusion equation (
8.62
) and the boundary condition
(
8.63
) that we have constructed in this section can be written in the following form:
For any sequence of numbers
c
1
;c
2
;:::;
the function
X
c
k
e
k
2
2
t
sin.kx/
u
.x; t /
D
(8.65)
k
D
1
18
This is a typical feature of linear problems.
19
From a rigorous mathematical point of view there are several open questions that we have not
addressed here. What does it mean that a series of functions converges? Can we differentiate term-
wise? And so forth? Prior to answering these questions we don't know whether or not (8.64)forms
a solution of (
8.62
)and(
8.63
). These questions are beyond the scope of this introductory text and
we will not dwell upon them. Instead we will treat (8.64) as a “formal solution” of the problem.
Further details can be found in, e.g., [28].
