Question

why does Picard iteration work???

I mean, I'm familiar with the process. But I cannot figure out why we are getting closer and closer to the actual answer with every iteration

Answer 1

you might want to show what a Picard interation is, and what its used for.

Answer 2

http://www.sosmath.com/diffeq/first/picard/picard.html

Answer 3

i did see that. It kind of reminded me of a Euler solution

Answer 4

This is based off of from http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem

But you start with the assumption that you have \(y'(t)=f(t,y(t))\) as a differential equation that has some continuity conditions on some intervals. Then you integrate both sides from \(t_0\) to \(t\) to get\[y(t)-y(t_0)=\int_{t_0}^tf(s,y(s))\;\text{d}s.\]Add \(y(t_0)\) to both sides, and then you have something that looks like the picard iteration theorem. So if you have a solution, then if you apply picard iteration to it, it won't do anything. Once you prove that picard iteration converges to that solution (which is not super trivial), and use the fact that there's a unique solution, you're done.

Answer 5

I'm struggling to understand why picard iteration converges to the solution

Answer 6

Like I said, that's not super trivial, and I don't really understand it, and seems to rely mostly on http://en.wikipedia.org/wiki/Banach_fixed_point_theorem. If you could fully understand that theorem, you would probably have a better grasp.

Answer 7

thanks :) I'm still a level one undergrad and this was the best help I could find. I'll try to have a go at it. Thanks for your time :)

Answer 8

You're welcome.