I finalize the discussion started in Part 1 with an example showing the behavior for a specific source, which is a simple Gaussian in space and in time. I comment on how to make sure the desired initial conditions are satisfied by adding a solution to the homogeneous equation. Video. Notes.
Category Archives: Fourier series and transforms
Green’s function for diffusion in 1D. Part 1.
I solve for the Green’s function of the diffusion operator on the whole real line. To that end, I use Fourier transforms in space and time and contour integration (more on that later!).Video. Notes.
Another Fourier way to diffusion and waves.
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.
The Fourier transform. Part 4: Waves!
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.
The Fourier transform. Part 3: Diffusion!
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.
The Fourier transform. Part 2.
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.
The Fourier transform. Part 1
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.
Green’s functions of the Laplacian: eigenfunction expansion.
Using the cartesian and spherical eigenfunctions of the Laplacian discussed in previous videos, we build the corresponding Green’s functions. What happens when we remove the boundaries? A step toward Fourier transforms and quantum field theory.Video. Notes.
Fourier series: generalizations & applications. Part 3.
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.”
Fourier series: generalizations & applications. Part 2.
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.