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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).

  1. 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


  2. Does what you see agree with your intuition?

  3. Select another pair of boundary conditions. Now what is the evolution of the temperature over time?

  4. 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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