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

There exists a huge number of numerical methods that iteratively construct\napproximations to the solution $y(x)$ of an ordinary differential equation\n(ODE) $y'(x)=f(x,y)$ starting from an initial value $y_0=y(x_0)$ and using a\nfinite approximation step $h$ that influences the accuracy of the obtained\napproximation. In this paper, a new framework for solving ODEs is presented for\na new kind of a computer -- the Infinity Computer (it has been patented and its\nworking prototype exists). The new computer is able to work numerically with\nfinite, infinite, and infinitesimal numbers giving so the possibility to use\ndifferent infinitesimals numerically and, in particular, to take advantage of\ninfinitesimal values of $h$. To show the potential of the new framework a\nnumber of results is established. It is proved that the Infinity Computer is\nable to calculate derivatives of the solution $y(x)$ and to reconstruct its\nTaylor expansion of a desired order numerically without finding the respective\nderivatives analytically (or symbolically) by the successive derivation of the\nODE as it is usually done when the Taylor method is applied. Methods using\napproximations of derivatives obtained thanks to infinitesimals are discussed\nand a technique for an automatic control of rounding errors is introduced.\nNumerical examples are given.\n