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Cybernetics And Systems Analysis
International Theoretical Science Journal
Volume 57 >>> № 4 JULY — AUGUST 2021
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UDC 519.6
I.V. Sergienko1, O.M. Lytvyn2, O.O. Lytvyn3,
O.V. Tkachenko4 , A.A. Biloborodov5


1 V.M. Glushkov Institute of Cybernetics, National Academy
of Sciences of Ukraine, Kyiv, Ukraine

 incyb@incyb.kiev.ua

2 Ukrainian Engineering Pedagogics Academy,
Kharkiv, Ukraine

 academ_mail@ukr.net

3 Ukrainian Engineering Pedagogics Academy,
Kharkiv, Ukraine

 Olegolitvin55@gmail.com

4 Ivchenko-Progress ZMKB, Zaporizhzhia, Ukraine

5 Kharkiv National University of Radio Electronics, Kharkiv, Ukraine

 biloborodow23april@gmail.com

OPTIMIZATION OF PARAMETERS IN THE GENERALIZED D’ALEMBERT FORMULA
FOR A FUNCTION OF TWO VARIABLES

Abstract. The authors consider classes of functions that can be exactly restored using the d’Alembert formula generalized by O.M. Lytvyn in 1989. This formula as a special case is know to give the Taylor polynomial of the one variable function, but in opposite to the Taylor polynomial it preserves the same differentiability class to which the approximated function belongs, even if its partial derivatives of s  order (s =1, 2,..., N ) do not belong to the same differentiability class. In such case, the system of parameters β 0, β 1,..., β N  is used. The authors propose a method for the choice of optimal parameters and provide and prove several theorems related to classes of functions that can be exactly restored by the generalized d’Alembert operators.

Keywords: interpolation, operator, remainder, optimization.



FULL TEXT

REFERENCES

  1. Lytvyn O.M. Interlination of functions and some of its applications [in Ukrainian]. Kharkiv: Osnova, 2002. 544 p.

  2. Lytvyn O.M. Calculation methods. Additional sections [in Ukrainian]. Kyiv: Nauk. Dumka, 2005. 331 p.

  3. Sergienko I.V., Lytvyn O.M., Lytvyn O.O., Tkachenko O.V., Hrytsai O.L. Construction and research of the operator of approximation of functions of two variables with preservation of a class of differentiation on traces of their derivatives to the fixed order on the set line. Problemy mashynobuduvannya. 2016. Vol. 19, N 2. P. 50–57.

  4. Sergienko I.V., Zadiraka V.K., Lytvyn O.M. Elements of the general theory of optimal algorithms and related issues [in Ukrainian]. Kyiv: Nauk. Dumka, 2012. 404 p.

  5. Sergienko I.V., Lytvyn O.M., Lytvyn O.O., Tkachenko O.V., Hrytsai O.L. Restoring the functions of two variables while preserving the class C r ( R 2 ) using their traces and traces of their derivatives to a fixed order on a given line. Dop. NAN Ukrayiny. 2014. N 2. P. 50–55.

  6. Litvin O.M. Interpolation of functions and their normal derivatives on smooth lines in R n. Dop. AN URSR. 1984. N 7. P. 15–19.

  7. Litvin O.M. Exact solution of the Cauchy problem for the equation . Dop. AN URSR. 1991. N 3. P. 12–17.




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