Construction of explicit and generalized Runge-Kutta formulas of arbitrary order with rational parameters

summary:In the article containing the algorithm of explicit generalized Runge-Kutta formulas of arbitrary order with rational parameters two problems occuring in the solution of ordinary differential equaitions are investigated, namely the determination of rational coefficients and the derivation of the adaptive Runge-Kutta method. By introducing suitable substitutions into the nonlinear system of condition equations one obtains a system of linear equations, which has rational roots. The introduction of suitable symbols enables the authors to generalize the Runge-Kutta formulas. The starting point for the construction of adaptive R. K. method was the consistent $s$-stage R. K. formula. Finally, the S-stability of the ARK method is investigated

A contribution to Runge-Kutta formulas of the 7th order with rational coefficients for systems of differential equations of the first order

summary:The purpose of this article is to find the 7th order formulas with rational parameters. The formulas are of the 11th stage. If we compare the coefficients of the development $\sum^\infty_{i=1} \frac {h^i} {i!} \frac {d^{i-1}} {dx^{i-1}} \bold f\left[x,\bold y(x)\right]$ up to $h^7$ with the development given by successive insertion into the formula $h.f_i(k_0,k_1,\ldots, k_{i-1})$ for $i=1,2,\ldots, 10$ and $k=\sum^{10}_{i=0} p_i, k_i$ we obtain a system of 59 condition equations with 65 unknowns (except, the 1st one, all equations are nonlinear). As the solution of this system we get the parameters of the 7th order Runge-Kutta formulas as rational numbers