Cybernetics And Systems Analysis logo
Editorial Board Announcements Abstracts Authors Archive
Cybernetics And Systems Analysis
International Theoretical Science Journal
Volume 58 >>> № 2 MARCH — APRIL 2021
-->

UDC 519.8
Yu.I. Kharkevych1


1 Lesya Ukrainka Volyn National University, Lutsk, Ukraine

 kharkevich.juriy@gmail.com

ON SOME ASYMPTOTIC PROPERTIES OF SOLUTIONS TO BIHARMONIC EQUATIONS

Abstract. The author considers the application of the approximation theory methods to the principles of optimality in the decision-making theory. In finding optimal solutions, the risk function often has rather complex structure for studying its properties, which makes it necessary to approximate the risk function to another function with simple and clear characteristics. In this regard, the asymptotic properties of the solutions of biharmonic equations as approximate functions are investigated. Complete asymptotic expansions of the upper limits of deviations of the Sobolev class functions W (the set that the risk functions in decision-making optimization belong to) from operators that are solutions of biharmonic equations with certain boundary conditions are obtained. The expansions allow us to find the Kolmogorov-Nikol’skii constants of arbitrarily high degree of smallness that allows us to estimate the approximation error when solving optimization problems with arbitrary accuracy. It is mentioned that the biharmonic equations can be used to efficiently generate mathematical models of natural and social phenomena.

Keywords: approximation error, optimization properties of functions, biharmonic equations, complete asymptotic expansions, Sobolev classes.


FULL TEXT

REFERENCES

  1. Maksymuk O.V., Sobchuk V.V., Salanda I.P., Sachuk Yu.V. A system of indicators and criteria for evaluation of the level of functional stability of information heterogenic networks. Mathematical Modeling and Computing. 2020. Vol. 7, N 2. P. 285–292. https://doi.org/10.23939/mmc2020.02.285.

  2. Sobchuk V., Pichkur V., Barabash O., Laptiev O., Kovalchuk I., Zidan A. Algorithm of control of functionally stable manufacturing processes of enterprises. IEEE 2nd International Conference on Advanced Trends in Information Theory (ATIT). Kyiv, Ukraine, 2020. P. 206–210. https://doi.org/10.1109/ATIT50783.2020.9349332.

  3. Pichkur V.V., Sobchuk V.V. Mathematical models and control design of a functionally stable technological process. Journal of Optimization, Differential Equations and their Applications (JODEA). 2021. Vol. 29, N 1. P. 1–11. https://doi.org/10.15421/142102.

  4. Dzyubenko G.T., Pshenichnyi B.N. Discrete differential games with information lag. Cybernetics and Systems Analysis. 1972. N 6. P. 947–952. https://doi.org/10.1007/BF01068518.

  5. Vlasenko L.A., Rutkas A.G., Chikrii A.A. On a differential game in an abstract parabolic system. Proc. Steklov Inst. Math. 2016. Vol. 293. P. 254–269. https://doi.org/10.1134/S0081543816050229.

  6. Chikrii A.A., Chikrii V.K. Image structure of multi-valued mappings in game problems of motion control. Journal of Automation and Information Sciences. 2016. Vol. 48, N 3. P. 20–35. https://doi.org/10.1615/JAutomatInfScien.v48.i3.30.

  7. Chikrii A.A., Eidelman S.D. Control game problems for quasilinear systems with Riemann–Liouville fractional derivatives. Cybernetics and Systems Analysis. 2001. N 6. P. 836–864. https://doi.org/10.1023/A:1014529914874.

  8. Samoilenko A.M., Samoilenko V.G., Sobchuk V.V. On periodic solutions of the equation of a nonlinear oscillator with pulse influence. Ukrainian Math. J. 1999. Vol. 51, N 6. Р. 926–933. https://doi.org/10.1007/BF02591979.

  9. Vladimirov V.S. Equations of mathematical physics [in Russian]. 4th ed. Moscow: Nauka, 1981. 512&nbps;p.

  10. Kharkevych Yu.I., Pozharska K.V. Asymptotics of approximation of conjugate functions by Poisson integrals. Acta Comment. Univ. Tartu. Math. 2018. Vol. 22, N 2. P. 235–243. https://doi.org/10.12697/ACUTM.2018.22.19.

  11. Kharkevych Yu.I. On Approximation of the quasi-smooth functions by their Poisson type integrals. Journal of Automation and Information Sciences. 2017. Vol. 49, N 10. P. 74–81. https://doi.org/10.1615/JAutomatInfScien.v49.i10.80.

  12. Kal’chuk I.V., Kravets V.I., Hrabova U.Z. Approximation of the classes by three-harmonic Poisson integrals. J. Math. Sci. (N. Y.). 2020. Vol. 246, N 2. P. 39–50.
    https://doi.org/10.1615/JAutomatInfScien.v49.i10.80.

  13. Hrabova U.Z., Kal’chuk I.V. Approximation of the classes by three-harmonic Poisson integrals. Carpathian Math. Publ. 2019. Vol. 11, N 2, P. 10–23. https://doi.org/10.15330/cmp.11.2.321-334.

  14. Tikhonov A.N., Samarsky A.A. Equations of mathematical physics [in Russian]. Moscow: Nauka, 1977. 735 p.

  15. Zhyhallo K.M., Kharkevych Yu.I. Approximation of functions from the classes by biharmonic Poisson integrals. Ukrainian Math. J. 2011. Vol. 63, N 7. P. 1083–1107. https://doi.org/10.1007/s11253-011-0565-1.

  16. Zhyhallo K.M., Kharkevych Yu.I. Approximation of -differentiable functions of low smoothness by biharmonic Poisson integrals. Ukrainian Math. J. 2012. Vol. 63, N 12. P. 1820–1844. https://doi.org/10.1007/s11253-012-0616-2.

  17. Chikrii A.A., Chikrii G.Ts. Matrix resolving functions in game problems of dynamics. Proc. Steklov Inst. Math. 2015. Vol. 291. P. 56–65. https://doi.org/10.1134/S0081543815090047.

  18. Chikrii A.A., Matichin I.I. Game problems for fractional-order linear systems. Proc. Steklov Inst. Math. 2010. Vol. 268. P. 54–70. https://doi.org/10.1134/S0081543810050056.

  19. Kal’chuk I.V., Kharkevych Yu.I., Pozharska K.V. Asymptotics of approximation of functions by conjugate Poisson integrals. Carpathian Math. Publ. 2020. Vol. 12, N 1. P. 138–147. https://doi.org/10.15330/cmp.12.1.138-147.

  20. Abdullayev F.G., Kharkevych Yu.I. Approximation of the classes by biharmonic Poisson integrals. Ukrainian Math. J. 2020. Vol. 72, N 1. P. 21–38. https://doi.org/10.1007/s11253-020-01761-6.

  21. Kharkevych Yu.I., Zhyhallo T.V. Approximation of functions from the class by Poisson biharmonic operators in the uniform metric. Ukrainian Math. J. 2008. Vol. 60, N 5. P. 769–798. https://doi.org/10.1007/s11253-008-0093-9.

  22. Baskakov V.A. Some properties of operators of Abel-Poisson type. Math. Notes. 1975. Vol. 17, N 2. P. 101–107. https://doi.org/10.1007/BF01161864.

  23. Stepanets A.I. Uniform approximations by trigonometric polynomials [in Russian]. Kiev: Nauk. dumka, 1981. 340 p.

  24. Zhyhallo K.M., Kharkevych Yu.I. On the approximation of functions of the Holder class by triharmonic Poisson integrals. Ukrainian Math. J. 2001. Vol. 53, N 6. P. 1012–1018. https://doi.org/10.1023/A:1013364321249.

  25. Zhyhallo K.M., Kharkevych Yu.I. Approximation of differentiable periodic functions by their biharmonic Poisson integrals. Ukrainian Math. J. 2002. Vol. 54, N 9. P. 1462–1470. https://doi.org/10.1023/A:1023463801914.

  26. Korovkin P.P. Linear operators and approximation theory [in Russian]. Moscow: Fizmatgiz, 1959. 213 p.

  27. Kharkevych Yu.I. Asymptotic expansions of upper bounds of deviations of functions of class from their generalized Poisson integrals. Journal of Automation and Information Sciences. 2018. Vol. 50, N 8. P. 38–49. https://doi.org/10.1615/jautomatinfscien.v50.i8.40.

  28. Kharkevych Yu.I. Approximative properties of the generalized Poisson integrals on the classes of functions determined by a modulus of continuity. Journal of Automation and Information Sciences. 2019. Vol. 51, N 4. P. 43–54. https://doi.org/10.1615/JAutomatInfScien.v51.i4.40.

  29. Gradshtein I.S., Ryzhik I.M. Tables of integrals, sums, series and products [in Russian]. Moscow: Fizmatgiz, 1963. 1100 p.

  30. Makarchuk A., Kal’chuk I., Kharkevych Y., Yakovleva A. The Usage of Interpolation Polynomials in the Studying of Data Transmission in Networks. IEEE 2nd International Conference on System Analysis & Intelligent Computing (SAIC). Kyiv, Ukraine. 2020. P. 1–4.
    https://doi.org/10.1109/SAIC51296.2020.9239180.

  31. Makarchuk A., Kal’chuk I., Kharkevych Y., Voloshyna T. Usage of Fourier transformation in theoretical studying of signals in data transmission. IEEE 2nd International Conference on Advanced Trends in Information Theory (IEEE ATIT 2020). Kyiv, Ukraine, 2020. P. 192–195.
    https://doi.org/10.1109/ATIT50783.2020.9349308.

  32. Kharkevych G., Kharkevych Y., Kal’chuk I. and Sobchuk V. Usage of Fourier transformation theory in machine translation. IEEE 2nd International Conference on Advanced Trends in Information Theory (IEEE ATIT 2020). Kyiv, Ukraine, 2020. P. 196–199.
    https://doi.org/10.1109/ATIT50783.2020.9349329.

  33. Tovkach R., Kharkevych Y., Kal’chuk I. Application of a Fourier Series for an Analysis of a network Signals. IEEE International Conference on Advanced Trends in Information Theory (IEEE ATIT 2019). Kyiv, Ukraine, 2019. P. 107–110. https://doi.org/10.1109/ATIT49449.2019.9030488.




Editorial Board Announcements Abstracts Authors Archive
about scope office editorial board instructions for authors subscriptions contacts
© 2022 Kibernetika.org. All rights reserved.