The Korteweg–De Vries (KdV) equation is a nonlinear PDE used to model shallow water waves. The linear counterpart omits the nonlinear term in the middle. This variant is useful in itself, but also for understanding the nonlinear KdV equation. Solitons Solutions to the linear KdV equation spread out over time. The nonlinear term in the […] The post Linear KdV dispersion first appeared on John D. Cook ..read more

Differential equations, especially nonlinear differential equations, rarely have a closed-form solution, but sometimes it happens. As I wrote about a year ago It is unusual for a nonlinear PDE to have a closed-form solution, but it is not unheard of. There are numerous examples of nonlinear PDEs, equations with important physical applications, that have closed-form […] The post Closed-form minimal surface solutions first appeared on John D. Cook ..read more

The “fundamental solution” to a PDE solves the equation with the right-hand side set to δ. Intuitively, you can think of the delta function as striking something with a hammer in order to see how it rings. An aside on rigor A novice might be OK with the explanation above. A sophomore might rightly object […] The post Fundamental solution first appeared on John D. Cook ..read more

The following quote stuck with me when I read it years ago. Looking back I appreciate it even more. Now, when solving differential equations, or indeed solving any problem, it is permissible to use any methods at all, no matter how dubious, provided that once the solution has been found it can be proved to […] The post No matter how dubious first appeared on John D. Cook ..read more

What to do first after scoping out and starting a data science project? I’ve started an unsupervised learning project based on textual data. The first thing I like to do is actually look at the data. Is it noisy? What are the features—complex feature engineering needed? How heterogeneous? What generalization and overfitting challenges? Analysis can […] The post Looking at Your Data first appeared on John D. Cook ..read more

An ellipse has equation and a hyperbola has equation Similarly the superellipse has equation and the superhyperbola When p = 2, the absolute value signs are unnecessary and the superellipse and superhyperbola reduce to the ellipse and hyperbola respectively. Increasing p makes the superellipse more like a rectangle. But unlike a rectangle with rounded corners, the […] The post Superhyperbola first appeared on John D. Cook ..read more

The glass disk game is played on a grid. You have translucent colored glass disks you can either place on a vertex or an edge. There are two kinds of disks that can be placed on a vertex: red or white. There are two kinds of disks that can be placed on an edge: blue […] The post The glass disk game first appeared on John D. Cook ..read more

This post started out as a thread on X. Here I’ve edited it into a blog post. The image below and the fact cited can be found in JPL Publication 89-24. In 1960 it didn’t seem that it would be possible to explore the solar system beyond Jupiter without greatly improved propulsion. Then the gravitational […] The post Voyager’s slingshot maneuvers first appeared on John D. Cook ..read more

When Voyager 1 and 2 left Earth, their internal computers were programmed to use Golay error correction codes. Images transmitted from Jupiter and Saturn were encoded using Golay codes. After leaving Saturn, the software was upgraded to use Reed-Solomon error correction codes. I didn’t realize how much difference the change of encoding made until I […] The post Advantages of Reed-Solomon codes over Golay codes first appeared on John D. Cook ..read more

The previous post mentioned that Martin Gardner announced that Ramanujan’s conjecture that exp(π√163) in an integer had been proven. This was an April Fool’s joke in 1975. Gardner said Working by hand, he [Ramanujan] found the value to be 262537412640768743.999999999999… The calculations were tedious, and he was unable to verify the next decimal digits. Calculating […] The post Hand calculating exp(x) first appeared on John D. Cook ..read more