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u
t
D
u
xx
for x
2
.0; 1/; t > 0;
(8.62)
u
.0; t /
D
u
.1; t /
D
0
for t>0:
(8.63)
Our goal now is to show that we can use the solutions given in (8.58) to generate
even further solutions of this problem. More precisely, it turns out that any linear
combination of the functions in (8.58) will satisfy (
8.62
)and(
8.63
). This is a con-
sequence of the linearity of the diffusion equation. Let us have a closer look at this
matter.
Assume that both
v
1
and
v
2
satisfy (
8.62
)and(
8.63
). Let
w
D
v
1
C
v
2
;
and observe that
w
t
D
.
v
1
C
v
2
/
t
D
.
v
1
/
t
C
.
v
2
/
t
D
.
v
1
/
xx
C
.
v
2
/
xx
D
.
v
1
C
v
2
/
xx
D
w
xx
:
Furthermore,
w
.0; t /
D
v
1
.0; t /
C
v
2
.0; t /
D
0
C
0
D
0
for t>0;
and
w
.1; t /
D
v
1
.1; t /
C
v
2
.1; t /
D
0
C
0
D
0
for t>0;
and hence
w
is also a solution of these two equations.
Next, for an arbitrary constant a
1
consider the function
p.x; t/
D
a
1
v
1
.x; t /:
Since a
1
is a constant and
v
1
satisfy the diffusion equation it follows that
p
t
D
.a
1
v
1
/
t
D
a
1
.
v
1
/
t
D
a
1
.
v
1
/
xx
D
.a
1
v
1
/
xx
D
p
xx
;
and furthermore
p.0; t/
D
a
1
v
1
.0; t /
D
0
for t>0;
p.1; t/
D
a
1
v
1
.1; t /
D
0
for t>0:
Hence, we conclude that p satisfies both (
8.62
)and(
8.63
).
These properties of
w
and p show that any linear combination of solutions of
(
8.62
)and(
8.63
) will also satisfy these equations. That is, if
v
1
and
v
2
satisfy (
8.62
)
and (
8.63
), then any function of the form
a
1
v
1
C
a
2
v
2
;
