Solving ordinary differential equations on the Infinity Computer by working with infinitesimals numerically

摘要

There exists a huge number of numerical methods that iteratively constructapproximations to the solution $y(x)$ of an ordinary differential equation(ODE) $y'(x)=f(x,y)$ starting from an initial value $y_0=y(x_0)$ and using afinite approximation step $h$ that influences the accuracy of the obtainedapproximation. In this paper, a new framework for solving ODEs is presented fora new kind of a computer -- the Infinity Computer (it has been patented and itsworking prototype exists). The new computer is able to work numerically withfinite, infinite, and infinitesimal numbers giving so the possibility to usedifferent infinitesimals numerically and, in particular, to take advantage ofinfinitesimal values of $h$. To show the potential of the new framework anumber of results is established. It is proved that the Infinity Computer isable to calculate derivatives of the solution $y(x)$ and to reconstruct itsTaylor expansion of a desired order numerically without finding the respectivederivatives analytically (or symbolically) by the successive derivation of theODE as it is usually done when the Taylor method is applied. Methods usingapproximations of derivatives obtained thanks to infinitesimals are discussedand a technique for an automatic control of rounding errors is introduced.Numerical examples are given.