Information Technology Reference
In-Depth Information
and consequently
w
satisfies the diffusion equation. Moreover,
w
.0; t /
D
g
1
.t / and
w
.1; t /
D
g
2
.t /
for t>0;
w
.x; 0/
D
f.x/
for x
2
.0; 1/;
and thus, from the analysis presented above, we find that
w
.x; t /
max
max
t
0
.
f.x//
.
g
1
.t //; max
t
0
.
g
2
.t //;
max
x
2
.0;1/
D
min
min
t
0
f.x/
g
1
.t /; min
t
0
g
2
.t /;
min
x
2
.0;1/
for all x
2
Œ0; 1; t > 0;
or
u
.x; t /
min
min
t
0
f.x/
g
1
.t /; min
t
0
g
2
.t /;
min
x
2
.0;1/
for all x
2
Œ0; 1; t > 0:
8.1.9
Summary
Above we have derived maximum and minimum principles for the solution
u
of the
problem
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
(8.38)
u
.0; t /
D
g
1
.t / and
u
.1; t /
D
g
2
.t /
for t>0;
(8.39)
u
.x; 0/
D
f.x/
for x
2
.0; 1/:
(8.40)
More precisely, we have shown that the maximum and minimum value of
u
is either
obtained initially or at the boundary of the solution domain.
A smooth solution of the problem (
8.38
)-(
8.40
) must satisfy the
bound
m
u
.x; t /
M
for all x
2
Œ0; 1; t > 0;
(8.41)
where
m
D
min
min
t
0
f.x/
;
g
1
.t /; min
t
0
g
2
.t /;
min
x
2
.0;1/
(8.42)
M
D
max
max
t
0
f.x/
:
g
1
.t /; max
t
0
g
2
.t /;
max
x
2
.0;1/
(8.43)
