The Gradient Methods for Solving the Cauchy Problem for a Nonlinear ODE System

The article considers the Cauchy problem for a nonlinear system of ODE. This problem is reduced to the variational problem of minimizing some functional on the whole space. For this functional necessary minimum conditions are presented. On the basis of these conditions the steepest descent method and the method of conjugate directions for the considered problem are described. Numerical examples of the implementation of these methods are presented. The Cauchy problem with the system which is not solved with respect to derivatives is additionally investigated.