Scientific journal
European Journal of Natural History
ISSN 2073-4972

COMBINED METHODS OF NUMERICAL SOLUTION FOR ALGEBRA DIFFERENTIAL EQUATIONS

Vasсhenko G.V.

We consider algebra - differential equations of the normal form and Cauchy problem [1] as follows:

f                (1)

where t ∈[0, T], Y : GR nR m, F : GR nR n - m, x(t)R m, z(t)R n - m .

Combined numerical methods for solution (1): implicit Euiler´s scheme with simple iterations and implicit Runge -Kutta´s scheme with Newton´s iterations are presented. It is shown convergence and exact numerical solutions.

Asymptotic properties of the both combined methods are discussed. We also give examples in which the numerical and the exact solutions are compared.

References:

  1. Chistyakov V.F. Algebra - differential operators with finite dimension kernel. Novosibirsk, Hauka, 1996. - 279 p.
  2. Schropp J. Geometric properties of Runge-Kutta discretizations for index 2 differential-algebraic equations // SIAM J. Numer. Anal. - 2002. - vol. 40, N 3. - pp. 872 - 890.

The work is submitted to the IV Scientific International Conference "Basic research", Italy, October, 11-18, 2008, came to the editorial office on 19.08.2008.