Geodesic
The Gradient Methods for Solving the Cauchy Problem for a Nonlinear ODE System
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 3, pp. 311-316 
A. V. Fominyh. The Gradient Methods for Solving the Cauchy Problem for a Nonlinear ODE System. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 3, pp. 311-316. http://geodesic.mathdoc.fr/item/ISU_2014_14_3_a9/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Tamasyan G. Sh., “The gradient methods for solving the Cauchy problem”, Vestnik St. Petersburg University. Ser. 10, 2009, no. 4, 224–230 (in Russian)

[2] Vasilyev L. V., Demyanov V. F., Nondifferentiable optimization, Nauka, M., 1981, 384 pp. (in Russian)

[3] Kantorovich L. V., Akilov G. P., Functional analysis, Nauka, M., 1977, 741 pp. (in Russian)

[4] Demyanov V. F., Extremum conditions and variation calculus, Vysshaya shkola, M., 2005, 335 pp. (in Russian)

[5] Vasilyev F. P., Optimization methods, Faktorial Press, M., 2002, 824 pp. (in Russian)

[6] Arrowsmith D. K., Place C. M., Ordinary differential equations. A qualitative approach with applications, Chapman and Hall, London, 1982, 243 pp. | MR | Zbl