Ex 7:
Note that there are two typos in the formula, as we discussed in the tutorials.
Ex 8-11:
Here is a brief writeup about the Method of Characteristics,
which motivates and derives the method for general nonlinear first-order PDE. The method is
then applied to the special case of quasilinear systems. The cooking recipee how to apply the
method step by step is given along an example.
Ex 11:
As seen in the tutorial, the method of characteristics yielded two solutions. Here is the
plot of the one solution, which is continuous w.r.t. the initial data
(ps-file (4 MB)).
Ex 33:
About the discussion in the tutorial, if H1-functions can have poles.
The example in this exercise shows that this can very well happen (here in 2d).
A question to you: Can 1d H1-functions have poles?
Ex 44:
As discussed in the tutorial, care has to be taken, whenever discontinuous functions
are approximated by a fourier series in practice. As the below example shows, strong
oscillations will show up at the discontinuities.
