MTH 453/553 – Homework 1
1. (20 points) [Solution to advection system] Consider the wave equation
utt = c2uxx
written as a first order system of two equations in the form
where
Diagonalize A and decouple the system. Write this decoupled system in terms of the variable w. Determine the characteristics and, hence, w. Knowing w, determine u(x, t) such that it satisfies the initial data
u(x, 0) = η(x), ut(x, 0) = μ(x).
2. (20 points) [Well-posedness of advection-diffusion]
Consider the advection-diffusion equation
ut + aux = νuxx
with constant coefficients ν > 0 and a.
(a) Show that the Cauchy problem is well-posed.
(b) What happens as ν → 0 (in particular, what happens to the bound on the norm of the solution)?
(c) What would happen if ν < 0?
yt + Ayx = 0, 0 −c
A= −c 0 .