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Who was the first mathematician/physicist to derive the heat equation $u_t=\Delta u$ and when? Was it already known to explain most diffusion phenomena? How much time passed between the first appearance of the heat equation and Fourier's solution? What were the previous attempts?
I'm not looking for a very detailed monograph of this part of history but rather a short account that may be condensed in an answer.
Heat equation was first formulated by Fourier in a manuscript presented to Institut de France in 1807, followed by his book Theorie de la Propagation de la Chaleur dans les Solides the same year, see Narasimhan, Fourier’s heat conduction equation: History, influence, and connections. "Diffusion phenomena" were not studied until much later, when atomic theory was accepted, Fourier succeeded exactly by ignoring microscopic physics.
"Essentially, Fourier moved away from discontinuous bodies and towards continuous bodies. Instead of starting with the basic equations of action at a distance, Fourier took an empirical, observational approach to idealize how matter behaved macroscopically. In this way he also avoided discussion of the nature of heat... In formulating heat conduction in terms of a partial differential equation and developing the methods for solving the equation, Fourier initiated many innovations. He visualized the problem in terms of three components: heat transport in space, heat storage within a small element of the solid, and boundary conditions. The differential equation itself pertained only to the interior of the flow domain. The interaction of the interior with the exterior across the boundary was handled in terms of "boundary conditions," conditions assumed to be known a priori."
In the book Fourier presented a solution in terms of trigonometric series. Trigonometric series were suggested for solving other equations earlier by Bernoulli, but accepting them as valid solutions was controversial because of the prevailing wisdom of treating functions as analytic expressions. A committee of luminaries, consisting of Laplace, Lagrange, Lacroix, Monge and Poisson, initially dismissed Fourier's solution as unsound. Fourier's approach led to a "crisis" and reconsideration of the foundations of calculus described by Bressoud in A Radical Approach to Real Analysis, which resulted in the more general modern concept of functions and rigorous analysis that goes with it.
"The crisis struck four days before Christmas 1807. The edifice of calculus was shaken to its foundations. In retrospect, the difficulties had been building for decades. Yet while most scientists realized that something had happened, it would take fifty years before the full impact of the event was understood... Here was the heart of the crisis. Infinite sums of trigonometric functions had appeared before. Daniel Bernoulli (1700-1782) proposed such sums in 1753 as solutions to the problem of modeling the vibrating string. They had been dismissed by the greatest mathematician of the time, Leonhard Euler (1707-1783). Perhaps Euler scented the danger they presented to his understanding of calculus... Well into the 1820s, Fourier series would remain suspect because they contradicted the established wisdom about the nature of functions. Fourier did more than suggest that the solution to the heat equation lay in his trigonometric series. He gave a simple and practical means of finding those coefficients, the ai, for any function. In so doing, he produced a vast array of verifiable solutions to specific problems."
The first mathematician who did this accurately and with all detail was Joseph Fourier. Though he had predecessors. (One can almost never answer the question "who did this first", so on my opinion, it is not even worth asking. "Nothing is new under the Moon", as Ecclesiastes wrote). Fourier wrote a very influential book titled "Analytic theory of heat" (there is an English translation), where he accurately derived the heat equation and proposed a method of its solution. For this he invented what is called "Fourier Analysis" today, though in this invention he also had predecessors.
