3.30. In (a), use the property mentioned in class that the minimum
of a harmonic function must occur on the boundary. In (b), try
in the domain given by . In (c), consider the
functions and .
4.16. This is the heat equation equivalent of the uniqueness
proof of the Poisson equation. You need to use method 2 of solved
problem 4.2. Ignore the hint, which is wrong. Instead, you can
assume that , since has already been covered in
4.2. Use that to eliminate
.
4.17a. (Question (b) was done in class, and the stated condition
that only needs to be continuous is not sufficient, but integrable
and continuous would do it.)
4.18. (This is the basic solution for the temperature in a
bar of length 1 where the ends of the bar are kept at zero
temperature. Of course, the values of the constants will
normally follow from some given initial temperature, and the number
of terms in the sum will normally be infinite.)
4.19. Plane wave solutions are solutions that take the form (2)
in solved problem 4.12, with a constant vector and
a constant. This sort of solutions are a multi-dimensional
generalization of the moving “wave” solution
of the one-dimensional wave equation. In fact, if you take
to be a unit vector, it gives the oblique direction of
propagation of the wave and gives the wave propagation speed.
However, in this case you will see that the function cannot be
an arbitrary function unless . You may want to do the case
separately. And also split up the cases for .
Solve the wave equation
using a Fourier
transform in . From it, show that the solution takes the form
Hint: stick with complex exponentials in the solution of the ODE.
5.27(a). Include a sketch of the characteristic lines. Is the
solution you get valid everywhere?