The first example of this you must see is solving partial differential equations with the Fourier series.
- Fourier series for the heat equation as shown at Fourier basis is complete for and solving partial differential equations with the Fourier series
- Hermite functions for the quantum harmonic oscillator
- Legendre polynomials for Laplace's equation in spherical coordinates
- bessel function for the 2D wave equation on a circular domain in polar coordinates