New Method for the Numerical Integration of Ordinary Differential Equations
The author introduces the concept of an /in-approximate solution of a problem with initial conditions for an ordinary differential equation of the first order and derives the integral equation of the error of this approximate solution. The segment of the existence of a real hn - approximate solution is established. An iterative process is proposed for the solution of the integral equation of the error, and the sufficient conditions for its convergence are established. The author shows that the proposed iterative process of finding the error of the A«-approximate solution of a problem with initial conditions for an ordinary differential equation of the first order makes it possible to find its numerical solution with any given precision and to determine the segment of existence of a solution of this problem with sufficient precision An estimate of the error of the numerical integration was obtained by the methods of. Euler, Runge and Adams.
Citation Example: Bondarenko P. S. New Method for the Numerical Integration of Ordinary Differential Equations // Ukr. Mat. Zh. - 1960. - 12, № 2. - pp. 118-131.