Talk:Fundamental solution

An issue with the example

The convolution integral of the example doesn't exist. You certainly can't "easily find it" as is stated directly above. —Preceding unsigned comment added by 207.237.8.64 (talk) 16:40, 9 November 2008 (UTC)

The notion of fundamental solution predates distribution theory

See this 1940 paper by Dressel [1], which references a 1911 paper by Hadamard. The concept of fundamental solution for the heat equation at least was in use long before Schwartz invented distribution theory in the 1940s. Certainly the discovery of distributions shed a lot of light on the subject of fundamental solutions, but it may be worthwhile to convey the older intuition as well. Perturbationist (talk) 03:47, 21 March 2008 (UTC)

Difference between Green's function and Fundamental solution?

This article does not mark the difference between the Green's function and the fundamental solution! —Preceding unsigned comment added by 62.227.248.103 (talk) 08:26, 22 July 2008 (UTC)

I agree: I would like the distinction to be stated explicitly. Lavaka (talk) 22:53, 14 February 2011 (UTC)

(talk)

I also agree, no difference whatsoever for practical matters between widely known Green's functions and alledged 'Fundamental Solution Theory'. Definitively all results here are also illustrated in the Green's function article. The alternative insight if any is not described here at all.Polilogaritmo (talk)

Is there a difference? Mike Stone (talk) —Preceding undated comment added 23:49, 14 February 2017 (UTC)

Barely: A suitably smooth solution of the homogeneous equation is added to the fundamental solution to ensure it satisfies the boundary conditions posited. Cuzkatzimhut (talk) 16:42, 3 January 2018 (UTC)

Heh. Per my comment at bottom, I'm flabbergasted as to how anyone could say "Green's function" without blurting out Fredholm alternative in the very same sentence. Roughly speaking, you can't "do anything" with a Greens function, unless you know how to integrate over it. In physics, you write greens functions aka "propagators" as expansions over some Hilbert space: $G(x,y)=\sum _{n}\lambda _{n}^{-1}\langle x|n\rangle \langle n|y\rangle$ where $L|n\rangle =\lambda _{n}|n\rangle$ and Dirac delta $\delta (x-y)=\sum _{n}\langle x|n\rangle \langle n|y\rangle$ so that $LG=\delta$ but doing this requires you have to deal with convergence, boundaries, orthonormality, perturbative expansions, etc. I guess you don't have to actually expand over a basis like this, but ... well ... this is the lock-stock-and-barrel trade of textbooks on classical electrodynamics... and quantum ... and qft ... 67.198.37.16 (talk) 23:45, 18 November 2020 (UTC)

clean up

Someone had edited the text such that f was used both as the solution to the differential equation as well as the fundamental solution (which was elsewhere referred to as F). I tried to clear this up everywhere. Further clean-up is certainly possible! One major thing that should be fixed is the example: If the convolution doesn't cleanly produce the correct result in this case, why are we using it as an example? Might a polynomial work better? Someone else can consider it and hopefully fix it! digfarenough (talk) 21:32, 17 December 2009 (UTC)

missing information

in the "Screened Poisson equation" section there is a function K0 but no explanation nor reference is given. Is K0 a kernel? — Preceding unsigned comment added by 2001:6B0:E:4A59:A4DB:5213:66B0:501 (talk) 10:33, 17 April 2013 (UTC)

Fredholm alternative

So where I grew up, the Fredholm alternative was the right way to describe this ... 67.198.37.16 (talk) 23:19, 18 November 2020 (UTC)