Partial answers - like when the derivation first occurred or who the major mathematicians working in this field at the appropriate time were - are also useful.
Who first derived the solution to the heat equation over the whole real line?
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0As far as I know this problem was one of the motivations for development of Fourier analysis. Another name which comes to mind when hearing about heat equation is Wiener. – 2017-02-22
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0Such a question might get more attention (i.e., not get lost among 600+ other questions arriving every day) at [hsm.se] – 2017-03-03
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It appears to me that Fourier did it. However, it appears that Laplace contributed a critical integral along the way, which Fourier discusses. The original manuscript of Fourier did not reduce everything to the form shown below, but this was found in the published version of the manuscript. Fourier's original work was denied publication for several years because of the controversial work on trigonometric expansions of general functions of the time.
The above is a snip from page 372 of the Google e-book scan that is in the public domain. Of course this is not Fourier's original manuscript, but this is a well-regarded translation of such.
The cover page is shown below:
