Continuous approximate solution of ordinary differential equations and their systems (Q801641)

The fifth-order Taylor polynomial of the solution \(y(x_ 0+th)\), \(0<t\leq 1\), \(h=step\)-size of \(y'=f(x,y)\), \(y(x_ 0)=y_ 0\), is approximated by Runge-Kutta processes obtaining polynomials of n-th order \(y^*_ n(x_ 0+th)\), \(n=1,2,3,4,5\). The polynomial \(y^*_ 4(x_ 0+th)\) provides an approximation of the fourth order at the interior points and of the fifth order at each end point. Its derivative constitutes a third order approximation of \((y(x_ 0+th))'\) for any t. The formulae are valid for systems.