Wikipedia:Reference desk/Archives/Mathematics/2007 June 11
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June 11
editLaplace transform for PDEs?
editIn class my professor mentioned that it is possible to use the Laplace transform to solve PDEs but did not go into details, and as the article doesn't mention this technique (actually I have seen it mentioned in a few sparse places but nothing really specific on how it works), I'll ask: is there another name for the technique? And how does it work? Regards. 75.58.181.204 00:06, 11 June 2007 (UTC)
- Example. (I had trouble making an ordinary wiki link.) —Bromskloss 07:05, 11 June 2007 (UTC)
- I turned the exolink into a standard wikilink. The problem was an anomaly (double space) in the section title at the linkend. --LambiamTalk 11:22, 11 June 2007 (UTC)
- Thanks. —Bromskloss 12:29, 11 June 2007 (UTC)
- I turned the exolink into a standard wikilink. The problem was an anomaly (double space) in the section title at the linkend. --LambiamTalk 11:22, 11 June 2007 (UTC)
- That's an ordinary differential equation, not a partial one. Is the Laplace transform easy to apply to PDE problems too? nadav (talk) 11:15, 11 June 2007 (UTC)
- Yes. You just transform one of the variables and treat the other variables as constants during the transform. Note that derivatives with respect to the other variables kind of "pass through" the transform unaffected (i.e. the transform of the derivative of something with respect to the another variable is just the derivative with respect to that variable of the transform of that thing.) --Spoon! 15:44, 11 June 2007 (UTC)
- This is what I was missing; thank you very much. 76.195.72.198 22:55, 12 June 2007 (UTC)
- Yes. You just transform one of the variables and treat the other variables as constants during the transform. Note that derivatives with respect to the other variables kind of "pass through" the transform unaffected (i.e. the transform of the derivative of something with respect to the another variable is just the derivative with respect to that variable of the transform of that thing.) --Spoon! 15:44, 11 June 2007 (UTC)
- For certain types of PDEs (usually ones where values at the boundaries of the desired solution are known), there is something called the multigrid method which can computationally efficient and robust. It works by starting with an estimate of the solution, and then by evaluating the error (using a Laplacian operator) and adjusting the solution locally to spread that error out across the solution, producing a new estimate that is better than the one you started with (this process is done on various frequency levels and repeated until the error is reduced to an acceptable level). I don't know if this is what you're looking for, but it sounds like the kind of thing you are talking about. I've also heard of things called "rapid Poisson solvers" which are used to a similar end, but I haven't worked with them personally so I can't tell you much about them. - Rainwarrior 04:09, 12 June 2007 (UTC)