Does anyone know if there is an analogous transformation for greater than two dimensions? On a related topic, is it ever valid in the separation of variables to create a product of functions, some of multiple variables? eg., if F=F(x,y,t) with a suitable transform can we use separation of var's with F=G(x,y)*H(t)? What I'm trying to work out is how to separate PDE's of arbitrary order, dimension etc into mixed systems of ODE's and PDE's. Sorry if this is silly

You may always do F=G(x,y)*H(t). You have to make sure though that you do not losesolutions by doing that. To be valid you need a uniquiness theorem for your problem.Hope this helps

z,You can't do F=G(x,y)*H(t) separations in general. It only works on certain manifolds.N

QuoteOriginally posted by: zetaDoes anyone know if there is an analogous transformation for greater than two dimensions? On a related topic, is it ever valid in the separation of variables to create a product of functions, some of multiple variables? eg., if F=F(x,y,t) with a suitable transform can we use separation of var's with F=G(x,y)*H(t)? What I'm trying to work out is how to separate PDE's of arbitrary order, dimension etc into mixed systems of ODE's and PDE's. Sorry if this is sillyZeta,One can use the separation of variables for simple equations but even in this case we must make an assumption (ansatz) that you are allowed to do this. Then of course you prove that the Fourier series converges absolutely to the true solution.

Then of course you prove that the Fourier series converges absolutely to the true solution. I hate to break the news, but harmonic functions don't span L^2.

Thanks everybody. I guess I wonder whether the assumption eg., k^2+l^2=m^2 in sep of variables is valid for where one eigen val is attributedto an equation of 2 or more indep vars; N if you don't like harmonics, is the idea just that your solution constructed from functions over Hilbert space coverges? Or is it more than that?

Sorry, that was a little cryptic. Want I am trying to do is see if there is a simple, systematic way to reduce a general, complex PDE to several simple PDE's but I'm pretty sure there's not. Besides complex analysis and other transforms for static problems in 2D there's no systematic framework for higher dimensions is there?I'm trying to avoid reinventing the wheel, that's all

N if you don't like harmonics, is the idea just that your solution constructed from functions over Hilbert space coverges? Or is it more than that? z,Functions over Hilbert space are quite restrictive and aren't useful in many practical problems.N

Quotez,Functions over Hilbert space are quite restrictive and aren't useful in many practical problems.NAgreed, I'm trying non-orthogonal functions and differential forms; N, we're back on the same page! Halleijuah !

Sorry, that was a little cryptic. Want I am trying to do is see if there is a simple, systematic way to reduce a general, complex PDE to several simple PDE's but I'm pretty sure there's not. Besides complex analysis and other transforms for static problems in 2D there's no systematic framework for higher dimensions is there?z,I'd like to tell you but I make my living from research, and the kids have to eat.

QuoteI'd like to tell you but I make my living from research,Hey, that's my line! ED: may the best man win