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.



History of fundamental solutions
Question 2 is getting clearer now. My sources are Parseval's [article](http://books.google.ca/books?id=SDMVAAAAQAAJ&pg=PA515) from 1800, Poisson's [memoire](http://books.google.com/books?id=TZ8AAAAAYAAJ&pg=PA121) from 1819, Hadamard's [Lectures on Cauchy's Problem in Linear Partial Differential Equations](http://books.google.com/books?id=B25O-x21uqkC%20%22Lectures%22) (1923), and Baker and Copson's [The Mathematical Theory of Huygens' Principle](http://books.google.com/books?id=d7Xb1bKLnQwC%20%22The%20%22) (1939).


On [page 133](http://books.google.com/books?id=TZ8AAAAAYAAJ&pg=PA133) 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](http://books.google.com/books?id=TZ8AAAAAYAAJ&pg=PA141) 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](http://books.google.com/books?id=SDMVAAAAQAAJ&pg=PA519) 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*. 


