The real numbers, it is taught at universities, correspond to our idea of a continuum, although the hyperreal numbers are located ``in between'' the real numbers. The number $x + dx$, where $dx$ should be an infinitesimal number and $x$ real, is infinitesimally close to $x$ but ``infinitely'' far away from all other real numbers. Analogously: If $f'(x_0)$ and $f(x_0)$ are given for a differentiable function $f:\mathbb{ R}\rightarrow\mathbb{ R}$ at $x_0\in\mathbb{ R}$, we can not determine $f(x)$ at { \em any} point $x\in \mathbb{ R}$ different from $x_0$. These points seem to be ``infinitely'' far away. That is one conceptual problem of solving differential equations in numerical mathematics. In this article, we will present a numerical algorithm to solve very simple initial value problems. However, the change of paradigm is, that we will not ``leave'' the point $x_0$. Solving ordinary differential equations is like searching for ``recipes'' $f$. Instead of trying to find these recipes for values $x\in\mathbb{ R}$, we will learn them from special relations in the ``monad'' of $x_0$.
Categories: math.CA