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Optimization of Parameters in the Generalized D’alembert Formula for a Function of Two Variables

  • SYSTEMS ANALYSIS
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Cybernetics and Systems Analysis Aims and scope

Abstract

The authors consider classes of functions that can be exactly reconstructed using the D’Alembert formula generalized by O. M. Lytvyn in 1989. This formula as a special case is known to yield the Taylor polynomial of the expansion of functions in one variable but, unlike the Taylor polynomial, it retains the same differentiability class to which the approximated function belongs, even if its partial derivatives of sth order (s = 1, 2, ⋯ N) do not belong to the same differentiability class. In such case, the system of parametersβ1, β0, ⋯ βN is used. The authors propose a method for the optimal choice of these parameters and provide and prove several theorems related to classes of functions that can be exactly reconstructed by the generalized D’Alembert operators.

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References

  1. O. M. Lytvyn, Interlineation of Functions and Some of Its Applications [in Ukrainian], Osnova, Kharkiv (2002).

    Google Scholar 

  2. O. M. Lytvyn, Calculation Methods. Additional Chapters [in Ukrainian], Naukova Dumka, Kyiv (2005).

  3. I. V. Sergienko, O. M. Lytvyn, O. O. Lytvyn, O. V. Tkachenko, and O. L. Grytsai, “Construction and analysis of the operator of approximation of functions of two variables with retaining the differentiability class with traces of their derivatives up to a fixed order on a given line,” Problemy Mashynobuduvannya, Vol. 19, No. 2, 50 –57 (2016).

    MATH  Google Scholar 

  4. I. V. Sergienko, V. K. Zadiraka, and O. M. Lytvyn, Elements of the General Theory of Optimal Algorithms and Related Issues [in Ukrainian], Naukova Dumka, Kyiv (2012).

    Google Scholar 

  5. I. V. Sergienko, O. M. Lytvyn, O. O. Lytvyn, O. V. Tkachenko, and O.L. Grytsai, “Reconstruction of functions of two variables with retaining the classCr (R2) by means of their traces and traces of their derivatives up to the fixed order on the given line,” Dopov. Nac. Akad. Nauk Ukr., No. 2, 50–55 (2014).

    MATH  Google Scholar 

  6. O. M. Lytvyn, “Interpolation of functions and their normal derivatives on smooth lines in Rn,” Dopovidi AN URSR, No. 7, 15–19 (1984).

    Google Scholar 

  7. O. M. Lytvyn, “Exact solution of the Cauchy problem for the equation \( \prod_{i=0}^n\left(\frac{\partial }{\partial t}-{a}_i^2\frac{\partial^2}{{\partial x}^2}\right)u\left(x,t\right)=g\left(x,t\right), \) Dopovidi AN URSR, No. 3, 12–17 (1991).

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Authors and Affiliations

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

    I. V. Sergienko

  2. Ukrainian Engineering Pedagogics Academy, Kharkiv, Ukraine

    O. M. Lytvyn & O. O. Lytvyn

  3. Zaporizhzhia Machine Building Design Bureau Progress, State Enterprise Named After Academician O. H. Ivchenko, Zaporizhzhia, Ukraine

    O. V. Tkachenko

  4. Kharkiv National University of Radio Electronics, Kharkiv, Ukraine

    A. A. Biloborodov

Authors
  1. I. V. Sergienko
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  2. O. M. Lytvyn
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  3. O. O. Lytvyn
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  4. O. V. Tkachenko
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  5. A. A. Biloborodov
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Corresponding author

Correspondence to I. V. Sergienko.

Additional information

Translated from Kibernetyka ta Systemnyi Analiz, No. 4, July–August, 2021, pp. 20–29.

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Sergienko, I.V., Lytvyn, O.M., Lytvyn, O.O. et al. Optimization of Parameters in the Generalized D’alembert Formula for a Function of Two Variables. Cybern Syst Anal 57, 521–529 (2021). https://doi.org/10.1007/s10559-021-00377-3

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  • DOI: https://doi.org/10.1007/s10559-021-00377-3

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