Some solutions of the heat equation:
- A solution of the heat equation on 0 < x < 1 with Dirichlet
boundary condition u(0, t) = u(1,t) = 0.
Notice as t -> infinity, the solution -> 0 (heat is lost at the endpoints of
the rod).
- A solution of the heat equation on 0 < x < 1 with
Neumann boundary conditions u_x(0,t) = u_x(1,t) = 0
Notice as t-> infinity, the solution goes to the constant temperature
2 (no heat is lost).
- For Homework #5, problem #1, here's a plot of
the heat of the rod at various times,
and here's the plot of
the solution as a surface, where time
goes from .08 to 1.5.
(Compare this with the picture of the solution at
fixed times.)
- The error function, Erf(x/sqrt(4kt)) solves the heat equation
on the half-line with Dirichlet boundary condition 0 and initial
condition 1. Here's a plot of this function at
various times and a plot of the surface.
(See Example #1 in Section 3.1 of your book.)
This includes pictures and movies of the oscillating sine wave and of the
hammer blow (from homework problem 2.1, #5) and also of the example from class
of the wave equation on the half-line with a Dirichlet boundary condition.