I derive in detail the specific form of 2D Newton-Raphson for a simple root-finding problem on the complex plane. Try it out and create your own fractals! Video. Notes.
Generalization of the Newton-Raphson method to 2D, i.e. two equations with two unknowns. Basic algebraic setup, schematics of the algorithm, potential pitfalls, and generalizations.
A very brief introduction to the fixed-point method to calculate roots. I discuss the convergence criterion and its relation to the Newton-Raphson method.
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.
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.
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.
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.
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.
