Toronto Math Forum
APM3462018S => APM346Lectures => Topic started by: Ioana Nedelcu on January 24, 2018, 12:29:49 AM

I'm still trying to understand how the initial and boundary conditions affect the existence and types of solutions.
For example, in one of the tutorials the solution to $$ xu_{t} + u_{x} = 0 $$ $$ u(x, 0) = f(x) $$ is not uniquely determined. What exactly determines this?
(It's also possible I understood the question completely wrong but I'm generally confused about uniqueness)

Also existence problems at $(0,0)$.
Draw integral lines and consider their intersections with $\{(x,t)\colon t=0\}$.

But should there be issue with existence? I think there at the origin we have this parabolic characteristic tangent to it and what's wrong with that?

Look, at the picture. In the "yellow" domain initial condition $u(x,0)=f(x)$ does not define solution (why?).
In the white domain initial condition could be impossible to satisfy (why?)

Look, at the picture. In the "yellow" domain initial condition $u(x,0)=f(x)$ does not define solution (why?).
In the white domain initial condition could be impossible to satisfy (why?)
But you mentioned in another post that existence problem is at $(0,0)$. For the white domain indeed you have to have even initial function to meet the symmetrical curves, but $tx^{2}=0$ does not require this, and so shouldn't it rather be no issue at origin?

It means "in the arbitrarily small vicinity of the origin"

For the yellow domain, is it because when t=0 there are infinite possibilities for f(x)? So then there is no unique solution there
Impossible solution in the white domain because the characteristic curve doesn't cross the xaxis so there is no function of x, ie solution doesn't exist
Because the solution u(x, t) is constant on/ determined by the characteristic curves, do we essentially just look at how those curves behave at the initial and boundary conditions?

Ioana, read what Jingxuan posted:
1) In yellow domain characteristics do not intersect $\{t=0\}$ (nonunicity)
2) In white domain they intersect $\{t=0\}$ twice (possible nonexistence)