Information Technology Reference
In-Depth Information
In the days of Fourier there were of course no computers available. So he devel-
oped a technique for computing the solution of the diffusion equation using only
paper and pencil. We will now go through the three basic steps in this classic
methodology.
8.2.1
Separation of Variables
To begin with, we will focus on the PDE (
8.1
) and the boundary condition (
8.2
)
separately and try to determine functions that satisfy these two equations. That is,
we are trying to find functions satisfying
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
(8.44)
u
.0; t /
D
u
.1; t /
D
0
for t>0:
(8.45)
We will show below how to take care of the initial condition (
8.3
). Note that the
zero function,
u
.x; t /
D
0 for all x
2
Œ0; 1 and t>0, satisfies these two equations.
However, we want to find non-trivial solutions of this problem.
The basic idea is to seek a solution of (
8.44
)and(
8.45
) that can be expressed as
a product of two univariate functions. Thus, we make the ansatz
15
u
.x; t /
D
X.x/T .t/;
(8.46)
where X
D
X.x/ and T
D
T.t/ only depend on x and t , respectively.
16
If a function of the form (8.46) is supposed to solve (
8.44
)and(
8.45
), then
u
.0; t /
D
0
D
X.0/T .t/
and
u
.1; t /
D
0
D
X.1/T .t/:
Consequently, X.x/ must satisfy
X.0/
D
0
and
X.1/
D
0;
(8.47)
provided that T.t/
¤
0 for t>0.
Next, by plugging the ansatz (8.46)for
u
.x; t / into the diffusion equation (
8.44
),
we find that X and T must satisfy the relation
X.x/T
0
.t /
D
X
00
.x/T .t /
for all x
2
.0; 1/; t > 0:
15
Ansatz
is the German word for
beginning
. It is frequently used in the mathematical literature. By
an ansatz, we mean a “guess”. More precisely, we guess that the solution of a particular problem
is in a specific form. Thereafter, we investigate whether or not, and under what circumstances, this
guess actually forms a solution of the problem.
16
We are trying to split
u
into a product of two univariate functions and the technique is therefore
referred to as “separation of variables”.
