Traffic problems are ones we all face every day and we can explore using Physics of Risk tools such as agent-based models. Here we will take a look at a couple simple parking strategies from [1, 2].

In the last post we have provided you with interactive app, which allows you to compare Euler method and Runge-Kutta 4th order method. Now we share a video by Good Vibrations, which discusses the exact derivation of the Runge-Kutta methods.

Apps in the last two posts (see here and here) differ in another important regard: the first app plots analytical solution of one ODE, while the second plots numerical solution of another ODE. Numerical solution is chosen, because it is impossible to obtain analytical solution.

Here, in this post, we will numerically solve ODE:

\begin{equation} \frac{d}{d t} x = - \left( x + a \right)^2 . \label{eq:rk4-ode} \end{equation}

It is simple enough to have analytical solution and thus we can compare two different numerical solution methods: Euler method and Runge-Kutta 4th order method.

In a previous post we have taken a look at a model, which predicts doomsday to fall on November 13, 2026 [1]. The prediction was made based on the best fit to the data available in 1960. Recent available data deviates from the prediction made in [1]. Instead it seems to be more consistent with Verhulst model.

Here in this post we modify the doomsday model by introducing correction, inspired by Verhulst's improvement upon Malthus model, to the driving ordinary differential equation.

Last week we were lucky to avoid doomsday, but how lucky can one get? In this Stand-up Maths video Matt Parker talks about few quite improbable lucky streaks experienced by a particular Minecraft speedrunner and few other people.