# History of fundamental solutions

I have a few questions on the history of PDE.

1. Who first wrote down the formula for the solution of the Cauchy problem for the heat equation involving the heat kernel? I have seen it called Poisson's formula. If it is true Poisson has a formula for each of the heat, wave, and Laplace equations.
2. Who is the discoverer of the analogous formula for the wave equation in 2 and 3 dimensions? I have seen they were called both Kirchhoff's formula and Poisson's formula.
3. Is there a book to look up such questions? I have Dieudonne's History of functional analysis, but it does not have much on PDEs other than the Laplace equation.
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I would have a look at Jahnke (ed.): A history of analysis, ams.org/bookstore-getitem/item=hmath-24 The answer may not be there, but it is a good source and you will find the right references. – András Bátkai Aug 31 '11 at 7:46
For question 3: I don't know of any comprehensive book on the history of PDEs. It is such a diverse subject. The best way I think is to ask some experts about what historical texts/papers may have citations to the original papers, and start doing a literature search based on that. Another resource is the Springer EOM. It doesn't always have the primary (in terms of historical primacy) references, but it usually provides enough citation information that digging down a few levels you can probably find the "first" paper. – Willie Wong Aug 31 '11 at 12:05
For question 1, it is not inconceivable that Poisson had a formula for the solution of the heat equation. After all, one of the treatises which he didn't quit finish writing at the time of his death is a mathematical theory of heat. One should also note that he and Fourier are contemporaries. – Willie Wong Aug 31 '11 at 12:31

Question 2 is getting clearer now. My sources are Parseval's article from 1800, Poisson's memoire from 1819, Hadamard's Lectures on Cauchy's Problem in Linear Partial Differential Equations (1923), and Baker and Copson's The Mathematical Theory of Huygens' Principle (1939).

On page 133 of the afore-mentioned memoire, Poisson gives the 3-dimensional formula

$$u(x,t) = t M_{x,t}u_0 + \partial_t (t M_{x,t}u_1); \qquad u_0(x) := u(x,0), \quad u_1(x) := \partial_tu(x,0),$$

where $M_{x,t}g$ is the average of $g$ (defined in $\mathbb{R}^3$) over the sphere centred at $x$ of radius $t$. Then he goes on to prove it, and by the method of descent, derives several special cases, including the 1 and 2 dimensional formulas. So the 3D case is due to Poisson.

Later in 1882, Kirchhoff published a more general formula expressing $u(x,t)$ in terms of the values, the normal and time derivatives of $u$ over an arbitrary closed surface containing $x$, therefore mathematically justifying the Huygens principle. The analogue of Kirchhoff's 1882 formula for 2 dimensions was published by Volterra in 1894. These developments were closely related to the discoveries of fundamental solutions of the Helmholtz equation in 3 dimensions by Helmholtz in 1859, and for 2 dimensions by Weber in 1869.

As for who was the first to discover the 2 dimensional analogue of Poisson's 1819 formula, when he coins the term "method of descent", Hadamard notes

Creating a phrase for an idea which is merely childish and has been used since the very first steps of the theory is, I must confess, rather ambitious;

and cites Parseval's afore-mentioned article of 1800, Poisson's memoir of 1819, and Duhem's book from 1891. After giving the 2D formula on page 141 of his memoir, Poisson cites Parseval's article, and says something like "Parseval previously integrated this equation but in a less simple way". Parseval seems to give the formula on page 519 of his article, but I don't understand sufficiently to say the formula is complete. In particular there seem to be no explicit formulas for the quantities Q and Q'. So the 2D case can be said due to Parseval-Poisson.

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Ah, so the method of spherical means is due to Poisson! That is going into my reference database. Awesome detective work there. – Willie Wong Sep 3 '11 at 15:21
Your answer probably also answers the case for the heat equation! On page 143 he starts to treat the Heat equation. The equation on the bottom of page 145 is the heat kernel! – Willie Wong Sep 3 '11 at 15:35
One last remark: I just skimmed through Poisson's memoire. There is no mention in the text of any inhomogeneous equations. He only considered, as far as I can tell, homogeneous second order PDEs with constant coefficients. In particular there is no mention of Duhamel's/Huygens' principles. Is Kirchhoff actually the first one to deal with the inhomogeneous terms? (For the inhomogeneous heat equation, I think Duhamel was the first to extend the solution from the homogeneous case.) – Willie Wong Sep 3 '11 at 15:44
Your link to Parseval's "article" looks like it is actually a book by Lacroix? (Though it is stated that the method is due to Parseval.) On page 528 of the book, amazingly we also can find the solution to the homogeneous wave equation via Fourier transform. – Willie Wong Sep 3 '11 at 16:00
About inhomogeneous case, it appears Kirchhoff was the first. And about the Lacroix's book, my understanding was that the book is essentially a collection of material written by many different people on Lacroix's request. – timur Sep 3 '11 at 22:32

Concerning question 2: Lars Garding credits G.Tedone, in a 1898 paper in the first volume of (the third series of) Annali di Matematica, for the general solution formula for the wave equation. Also Hadamard calls it Tedone's formula.

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I don't read German well, and Italian not at all, but I think depending on what exact form of the formula one is interested in, a case can be made for either Kirchoff or Tedone. Kirchoff's paper is at least as early as 1882: emis.de/cgi-bin/JFM-item?14.0829.02 While I haven't been able to find Tedone's 1889 paper (it is not in ZBMath or MathSciNet), but I could find a 1896 paper of Tedone's, which the Reviewer claims to provide a "more general formula then the Huygen's principle of Kirchhoff" emis.de/cgi-bin/JFM-item?27.0702.02 – Willie Wong Aug 31 '11 at 10:52
The association with Poisson, however, I am pretty sure is because that the time-independent case of the wave equation reduces to the Poisson equation, and formally the time-independent reduction of the Kirchhoff formula gives the same as the Poisson integral. Insofar as the name "Kirchhoff" is concerned, there may also be a citation to the correct paper of Kirchhoff in Sobolev's paper of 1933, but I can't find a copy handy to check. Sobolev's paper is zentralblatt-math.org/zmath/en/advanced/?q=an:0008.20805 – Willie Wong Aug 31 '11 at 11:15
Ah, the Annali paper is actually from 1898. @Piero: I hope you don't mind me correcting that and giving the ZBMath link. – Willie Wong Aug 31 '11 at 12:01
@timur: of course it depens what you mean by 'derive'. You can write the solution using Fourier transform, so the fundamental solution is just the Fourier transform of say $cos(t∣\xi∣∣)$, which can be computed via complex analytic methods. By the way, I believed you were asking about the higher dimensional case; I was so convinced that Kirchhoff solved the 3D case that it did not occur to me you might be asking for that case. – Piero D'Ancona Aug 31 '11 at 17:28
On the other hand, ... Hadamard attributes the 3D case to Poisson 1819, and it seems what Kirchhoff did was the inhomogeneous case. Now I am in the the same state I started. – timur Sep 2 '11 at 12:09

This is an update on Question 1. As Willie observed, in his 1819 memoir Poisson studies not only the wave equation but also the heat equation from page 143 on, and reaches the heat kernel on page 145. However, amazingly, in Fourier's original memoir where he derived the heat equation and gave a convincing case for the importance of trigonometric series, the heat kernel appears on page 454 for 1D, on page 475 for 1D in the usual form as presented today, and on page 479 for 3D. Fourier's memoir was published in 1822 after a long delay, and it is said that the memoir is essentially Fourier's 1811 work that won a mathematical prize, which was in turn a continuation of his work presented in 1807, and summarized by Poisson in 1808. That said, even more amazingly, a new player appears in the story. After giving the 1D heat kernel on page 454, Fourier says something like

This integral, which contains an arbitrary function, was not known when we started our research on the theory of heat, which were presented at the Institute of France in December 1807. It was given by Mr. Laplace, in volume VI of des Mémoires de l'école polytechnique, and we have only applied his results here.

Poisson also mentions Laplace on page 148, and says that his 3D result was a straightforward extension of Laplace's formula. I found volume 6 of Journal de l'école polytechnique but there is nothing by Laplace, and moreover the journal is from 1806. I wondered if des Mémoires is different than Journal, but skimmed through Fourier's book to find that on page 513 he cites Laplace again, but now says volume 8. Then volume 8 it is! It is published in 1809, and the heat kernel appears on its page 241!

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