Archive of posts with tag “pde”

• Jul 17, 2025

  Multi-pass Gaussian blur filter

  According to the central limit theorem, the sum of some i.i.d. samples is normally distributed in the limit of large sample size. This fact can be used to implement a multi-pass Gaussian blur filter, where the total number of passes is equal to the number of samples used in the averaging. Through this, we can also see a nice relation to the heat equation, which is not surprising since the heat kernel is a Gaussian function.

  /programming #algorithm #shader #probability #pde
• Apr 11, 2025

  Eigenfunctions of the Laplacian on an annulus with homogeneous Neumann boundary condition

  Several features of the eigenfunctions of the Laplacian on an annulus with homogeneous Neumann boundary condition are discussed. The distribution of the eigenvalues is discussed in detail, making use of a phase angle function called $\tht$. The limiting cases of a disk and a circle are discussed.

  /math #mathematical physics #pde #ode #long paper
• Jan 18, 2023

  This is what will happen after you get a haircut

  Denote the length distribution of one’s hair to be $f(l,t)$, where $l$ is hair length, and $t$ is time. Considering that each hair may be lost naturally from time to time (there is a probability of $\lambda\,\mathrm dt$ for each hair to be lost within time range from $t$ to $t+\mathrm dt$) and then restart growing from zero length, how will the length distribution of hair evolve with time? It turns out that we may model it with a first-order PDE.

  /math #calculus #probability #pde

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