Information Technology Reference
In-Depth Information
and
D f .0; t / j t 0 g [ f .x; 0/ j 0 x 1 g [ f .1; t / j t 0 g :
,(8.35) and the definition (8.33)of v imply that
Then, since T
v .x; t /
v .y; s/ M C
max
.y;s/ 2
for all .x; t / 2 ˝ T :
Next, again by (8.33), it follows that
u .x; t / v
for all .x; t / 2 ˝ T ;
and consequently
u .x; t / M C
for all .x; t / 2 ˝ T
and all >0:
This inequality is valid for all >0, and, hence, it follows that
u .x; t / M
for all .x; t / 2 ˝ T :
(8.36)
In the argument leading to (8.36), the end “point” T>0of the time interval
Œ0; T , in which we studied the behavior of the solution of the diffusion equation,
was arbitrary, cf. also definitions ( 8.25 )-(8.27). Thus, we can choose T as large as
we need! This means that this inequality must hold for any t>0, i.e.,
u .x; t / M
for all x 2 Œ0; 1; t > 0:
(8.37)
8.1.8
The Minimum Principle
We have now shown that the solution u of our model problem ( 8.16 )-( 8.18 )is
bounded by the maximum value M attained by the boundary conditions g 1 and g 2
and the initial temperature distribution f . Can we derive a similar property for the
minimum value achieved by u ? That is, can we prove that the temperature u .x; t /,at
any point .x; t /, is larger than the minimum value achieved by the boundary and ini-
tial conditions? From a physical point of view, this property seems to be reasonable.
Let us now have a look at the mathematics. Consider the function w defined by
w .x; t / D u .x; t /
for all x 2 Œ0; 1; t > 0:
Note that
w t
D . u / t
D u t
D u xx D . u / xx D w xx ;
 
Search WWH ::




Custom Search