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.

According to [1] the doomsday is likely to happen around year 2026. No, we will not die of some epidemiological disaster, but instead our population count will explode towards infinity. How this could happen? Maths.

During the "Numerical Methods I" course together with students at Faculty of Physics we talk about experimental data fitting. Matlab's polyfit function will cover most common use cases, but in Physics we are often interested not only in the point estimates of measurements. We also care about associated errors and the documentation of polyfit function fails to deliver on that. Unless you are well versed and statistics and know that you need to calculate covariance matrix and take square root of values on its diagonal.

I was caught of guard by the students who were interested in actually estimating the measurement error using polyfit. As at the time I wasn't prepared, I have told them about Bootstrap method, which I like a lot. I like this method, because it has natural interpretation (it gives us a confidence interval) and is applicable even in highly complex situations, for which no analytical error estimation formulas exist.

More about this method in the following video by StatQuest with Josh Starmer.