Information Technology Reference
In-Depth Information
Let us consider a problem with a modified initial condition,
v
t
D
v
xx
for x
2
.0; 1/; t > 0;
(8.12)
v
.0; t /
D
v
.1; t /
D
0
for t>0;
(8.13)
v
.x; 0/
D
g.x/
for x
2
.0; 1/:
(8.14)
If g is close to f , will
v
be approximately equal to
u
? To answer this question we
will study the difference e between
u
and
v
, that is, we will analyze the function
e.x; t/
D
u
.x; t /
v
.x; t /
for x
2
Œ0; 1; t
0:
From (
8.1
)-(
8.3
)and(
8.12
)-(
8.14
), we find that
e
t
D
.
u
v
/
t
D
u
t
v
t
D
u
xx
v
xx
D
.
u
v
/
xx
D
e
xx
;
and furthermore
e.0; t/
D
u
.0; t /
v
.0; t /
D
0
0
D
0 for t>0;
e.1; t/
D
u
.1; t /
v
.1; t /
D
0
0
D
0 for t>0;
e.x; 0/
D
u
.x; 0/
v
.x; 0/
D
f.x/
g.x/
for x
2
.0; 1/:
So, interestingly, e solves the diffusion equation with homogeneous Dirichlet bound-
ary conditions at x
D
0; 1 and with initial condition
h
D
f
g;
that is, e satisfies
e
t
D
e
xx
for x
2
.0; 1/; t > 0;
e.0; t/
D
e.1; t/
D
0
for t>0;
e.x; 0/
D
f.x/
g.x/
for x
2
.0; 1/:
In particular, this means that e must satisfy the properties derived above for the
model problem (
8.1
)-(
8.3
). Hence, by inequality (8.7), we conclude that
Z
1
e
2
.x; t /
dx
Z
1
0
h
2
.x/
dx
for t>0;
0
or
Z
1
.
u
.x; t /
v
.x; t //
2
dx
Z
1
0
.f .x/
g.x//
2
dx
for t>0:
(8.15)
0