History of fundamental solutions I have a few questions on the history of PDE.


*

*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.

*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.

*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.

 A: 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.
A: 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. 
A: 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!
