Initial Problems with Polynomials on Right-hand Sides

The application of Taylor series has become a standard concept in numerical analysis. Their ability of approximating functions arbitrarily close under certain conditions makes them an ideal tool for the integration of differential equations. Before the appearance of digital computers the analytical determination of Taylor series coefficients, i.e. the calculation of higher derivatives, was regarded as too complicated. In many modern text books on numerical or applied mathematics has even been stated that Taylor series can only be applied as implementation in very special cases. In this paper an outline is presented of establishing Taylor series coefficients of initial problems with polynomials on right-hand sides. The application of recurrent Taylor series to the integration of systems of ordinary differential equations is discussed. The software TKSL for the solution of initial value problems by means of recurrent Taylor series has been developed and many positive results have been obtained.