Solving partial differential equations with the Fourier series

See: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from heat equation solution with Fourier series.

Separation of variables of certain equations like the heat equation and wave equation are solved immediately by calculating the Fourier series of initial conditions!

Other basis besides the Fourier series show up for other equations, e.g.:

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