System of piecewise differential equations - Where to start?

[urbanist]

Hello all,

I am dealing with a rather complicated problem (as we all do), without much knowledge in differential equations.
I have coded a numerical solution, taking a straightforward approach: constructing the functions step by step, but I would like to try to achieve better precision.

Any hint about the type of problem I am facing, about relevant solution methods, etc., so that I can start exploring the literature, will be greatly appreciated!

https://dl.dropbox.com/u/17308924/problem.GIF

Thank you!

Urbanist.

 

[Simon Bridge]

Welcome to PF;
I'd normally treat piecewise defined DEs as two coupled DEs, with extra boundary conditions at the place where the defined regeons join.

Looks like you have dw/dr = f(H,w) and dH/dr = g(H,w,z) too ... fun fun fun.

 

[urbanist]

Thank you so much for your help!

I can treat them as coupled. The problem is knowing when to switch from one set of coupled DEs to the other.

If I start with w<wm, I'm in the w<wm and H>0 region. This region ends when w>=wm (because H continues to rise). How do I know when this condition is fulfilled in order to switch to the other set of DEs, which holds when w>wm and H>0? Looking at boundary value problems I see only cases when the boundary condition on y(x)=0 is on a given x, which in my case is unkown.

 

[Simon Bridge]

You'd have to solve each of the DEs uncoupled and then apply the boundary conditions. This will give you a lot of candidates - at a glance: you'll have 4 cases in which to solve simultanious DEs - hopefully the boundaries will appear once you've done the spadework ... bear in mind I'm not going to actually do the problem just to help you. Looks too much like hard work. Instead I will point in likely looking directions.

The big advantage here is that you know the problem is set for you as an example so you know it can be solved. I know we all want some sort of shortcut but none occur to me just looking at it. Sometimes (often IRL) the shortcut does not exist and you just have to push through.

 

[haruspex]

Some simplifications are available. Write w(r) = exp(v(r)), H(r) = G(r)/t. Or maybe w(r) = exp(κv(r)), H(r) = κG(r)/t.