Construction of polynomial approximations for numerical solution of ordinary differential equations

The Cauchy problem for systems of first and second order ordinary differential equations is solved on the basis of local polynomial approximations. The method is based on the approximation of the right-hand sides of differential equations in a segment (whose length is equal to the integration step) by an algebraic interpolation polynomial followed by its integration. This interpolation polynomial is constructed without the use of divided differences as follows: an equation for unknowns that define the polynomial is introduced and, then, an iteration process for solving this equation is applied; the convergence of this process is proved. A peculiarity of our approach consists in the fact that the divided differences of the right-hand sides of differential equations are not calculated; this allows us to decrease computational errors of the sought-for solution and its derivative.