See: https://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.:
* <Bessel function>
* <Hermite polynomials>


 Solving partial differential equations with the Fourier series