Monthly Archives: December 2022

In this short video I show how to generalize our Fourier approach from space to spacetime, to solve the diffusion and wave equations and obtain the same expressions as in previous videos in a more unified fashion. Video. Notes.

I solve the wave equation on the whole real line using the same Fourier approach I used for the diffusion equation in part 3. I give two examples to show how wave profiles propagate from given initial conditions. Video. Notes.

I solve the diffusion equation in 1D on the whole real line using Fourier transforms. A specific example is solved in full detail and an exercise is left for practice.Video. Notes.

As promised, I derive the Fourier transform of a Gaussian, which is… just another Gaussian! Their widths are inversely related, reflecting the uncertainty principle of quantum mechanics, which is actually valid for any signal you may want to study with Fourier transforms.Video. Notes.

I introduce the Fourier transform as a generalization of the Fourier series from a finite interval to the whole real line. I give two basic examples that start to show the mathematics underlying Heisenberg’s uncertainty principle in quantum mechanics.Video. Notes.

Last part! I give a few more details on how to find the radial part of the eigenfunctions of the Laplacian with vanishing boundary conditions on a sphere. This radial part is given by the spherical Bessel functions and their roots, and the latter quantize the eigenvalues of the full problem. A brief example is …

Continue reading “Fourier series: generalizations & applications. Part 3.”

More details on solving the diffusion equation in finite interval in 1D. Generalization to 3D in cartesian and spherical coordinates. Further details on the latter in part 3!Video. Notes.

Generalized Fourier series. A few comments and two examples: spherical harmonics and Chebyshev polynomials. An application to PDEs: diffusion in a finite interval in one spatial dimension. More in part 2! Video. Notes.