Introduction
to the One-Dimensional Heat Equation
Part 3: Unequal Boundary
Conditions
Now we consider a different experiment. Again the
rod is given an initial temperature distribution. Then at the start of
the experiment, the ends are placed in baths that keep them at different
temperatures, Tl on the left and Tr
on the right. The new problem has the form
u(0,t) = Tl, u(L,t) = Tr for all t > 0,
u(x,0) = f(x).
-
We'll start with Tl = -2 and Tr
= 5. Look at the first initial temperature distribution again. What do
you expect to happen this time for small t? For larger t? In the steady-state?
Temperature of the Rod
- Does what you see agree with your intuition?
- Select another pair of boundary conditions. Now what is the evolution of
the temperature over time?
- Describe what you think will happen in general to the time evolution of
the temperature distribution for unequal boundary conditions. Check this
with the other initial conditions and various pairs of boundary conditions.
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