# PDE - Characteristics

1. Oct 1, 2015

### Nicolaus

1. The problem statement, all variables and given/known data
1. ut +ux = 0 subject to u(t,x) = x on x^2 + y^2 = 1

Is this a well-posed PDE BVP?

2. Relevant equations

3. The attempt at a solution
This is an easy one to solve: u(t,x) = f(x-t)
I let t(0) = 0 as an initial condition, and so t=s => x= ts + xo, where x(0) = xo
s is the variable such that ∂(u(t(s), x(s))/∂s = 0
If I let u(t,x) = x = f(x-t), would this not be well-posed since f must be a function of (x-t)?

2. Oct 2, 2015

### Geofleur

In the problem setup, is the condition really $x^2 + y^2 = 1$, or did you mean $x^2 + t^2 = 1$?