Toronto Math Forum

APM346-2018S => APM346--Lectures => Topic started by: Ioana Nedelcu on January 24, 2018, 12:29:49 AM

Title: Uniqueness/ continuity of solutions
Post 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)
Title: Re: Uniqueness/ continuity of solutions
Post by: Victor Ivrii on January 24, 2018, 03:27:49 AM
Also existence problems at $(0,0)$.

Draw integral lines and consider their intersections with $\{(x,t)\colon t=0\}$.
Title: Re: Uniqueness/ continuity of solutions
Post by: Jingxuan Zhang on January 24, 2018, 06:01:01 AM
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?
Title: Re: Uniqueness/ continuity of solutions
Post by: Victor Ivrii on January 24, 2018, 06:48:43 AM
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?)
Title: Re: Uniqueness/ continuity of solutions
Post by: Jingxuan Zhang on January 24, 2018, 07:13:23 AM
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 $t-x^{2}=0$ does not require this, and so shouldn't it rather be no issue at origin?
Title: Re: Uniqueness/ continuity of solutions
Post by: Victor Ivrii on January 24, 2018, 07:43:34 AM
It means "in the arbitrarily small vicinity of the origin"
Title: Re: Uniqueness/ continuity of solutions
Post by: Ioana Nedelcu on January 24, 2018, 05:40:54 PM
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 x-axis 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?
Title: Re: Uniqueness/ continuity of solutions
Post by: Victor Ivrii on January 25, 2018, 04:22:51 AM
Ioana, read what Jingxuan posted:

1) In yellow domain characteristics do not intersect $\{t=0\}$ (non-unicity)
2) In white domain they intersect $\{t=0\}$ twice (possible non-existence)