# Bondarenko P. S.

### N. P. Erugin. Book reviews

Bondarenko P. S., Martynyuk D. I., Pavlyuk I. A., ShkiI N. I., Shtokalo I. Z.

Ukr. Mat. Zh. - 1970. - 22, № 6. - pp. 848—851

### On a class of computation algorithms for approximate integration of ordinary differential equations with initial conditions

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 3-21

The author first introduces the concepts of computation and real computation algorithms for approximate integration, by the method of finite differences, of systems of ordinary differential equations with initial conditions and investigates the general properties of these algorithms, in particular the property of their stability.

Then a study is made of the class of computation algorithms based on methods of the Euler, Runge and Adams type for numerical integration of these equations, and it is shown that this class of computation algorithms makes it possible to determine with sufficient precision the guaranteed interval of existence of the solution sought for, to estimate the position of the graph of this solution and to find a guaranteed real estimate of the error of the numerical solution obtained.

### New Method for the Numerical Integration of Ordinary Differential Equations

Ukr. Mat. Zh. - 1960. - 12, № 2. - pp. 118-131

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.