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