Series Solution of Differential Equations - Real or Fake?

[Pejman]

Hi guys,

I was browsing in regards to differential equations, the non-linear de and came up with this site in facebook:

https://www.facebook.com/nonlinearDE

Are these people for real? Can just solve any DE like that, come up with a series? Not an expert in this area, so I do not know what if this is actually possible? If it is possible, what are the downfalls?
Thanks.

 

[Geofleur]

This paper may be relevant: http://arxiv.org/pdf/1206.2346.pdf. I'm not sure if it's the same people, but the authors seem to be claiming the same thing. I'm not sure I want analytic solutions if they have to be that complicated and ugly. Why not just have a numerical solution for the "full" problem and then compare that with simple analytic solutions that describe "parts" of whatever's going on?

 

[Pejman]

Thanks a lot for the reply and paper, I looked through it and it seems similar, however, these fb guys series does not come as power series of x, some times it eventually become a closed form solution which I found very interesting, some times as power series of tanh() for example . They also have a heat PDE in (x,y) plane; I know that I had a similar problem but I had difficulties finding a numerical solution. Is it possible to find a numerical solution to this one? What method do you use? It's a BV problem, shooting method in 2-d?

 

[Geofleur]

If you mean the heat equation Dirichlet BV problem on the Facebook page then, yes, that can be solved numerically. I would probably use the finite analytic method on it. A good book to look at is the one by Richard Bernatz, Fourier Series and Numerical Methods for Partial Differential Equations. There is a rather egregious error in the front of the book (to a physicist, anyway!) where he says that the solution to the Schrodinger equation represents a velocity, but otherwise it's very good.