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u
t
D
u
xx
for x
2
.0; 1/; t > 0;
(8.16)
u
.0; t /
D
g
1
.t / and
u
.1; t /
D
g
2
.t /
for t>0;
(8.17)
u
.x; 0/
D
f.x/
for x
2
.0; 1/;
(8.18)
for an appropriate choice of scales.
8.1.6
Physical Considerations
Let us first, for the sake of simplicity, consider the case of g
1
.t /
D
g
2
.t /
D
0 for all
t>0, i.e., the temperature at the end points of the rod is kept at zero. In this case,
we know from physical experience that the temperature at every point of the rod will
approach zero as time increases. Furthermore, heat will not accumulate at any point.
On the contrary, this is a diffusion process and the heat will thus be smoothed out
in time. Consequently, it seems reasonable that the temperature at any point x and
time t cannot exceed the maximum value of the initial temperature distribution f
and the zero temperature kept at the endpoints of the rod. In mathematical symbols
this reads
u
.x; t /
max.max
x
for all x
2
Œ0; 1; t > 0:
f.x/;0/
In a similar manner we can argue that the temperature throughout the rod at any
time cannot be less than the minimum value of the initial temperature distribution f
and the temperature at the end points of the rod, i.e.,
u
.x; t /
min.min
x
f.x/;0/
for all x
2
Œ0; 1; t > 0:
We can carry this discussion one step further by allowing non-zero temperatures
on the boundary. In such more general cases we would also, due to the physical inter-
pretation of heat conduction, expect the highest and lowest temperatures to appear
either initially or at the boundary of the rod. For example, assume that f.x/
D
0 for
x
2
.0; 1/ and that g
1
.t /
D
g
2
.t /
D
c,wherec is a positive constant, for all t>0.
Then the temperature throughout the rod will increase with time from zero to c,and
the temperature cannot, at any time or position, be higher than c or lower than zero!
8.1.7
Analytical Considerations
Above we argued from a “physical” point of view that the maximum and minimum
temperature of the rod is obtained either at the boundaries or initially. What about
the mathematical model (
8.16
)-(
8.18
)? Will a solution of this problem satisfy this
