DOI
10.34229/KCA2522-9664.25.5.12
УДК 519.8
А.М. ШУТОВСЬКИЙ
Волинський національний університет імені Лесі Українки, Луцьк, Україна,
sh93ar@gmail.com
ОПТИМІЗАЦІЙНІ ВЛАСТИВОСТІ ФОРМУЛИ ТЕЙЛОРА
ДЛЯ ТРИГАРМОНІЙНОГО ІНТЕГРАЛА ПУАССОНА
Анотація. Розглянуто оптимізаційну задачу про побудову поліноміальної структури для розв’язків полігармонійних рівнянь. За основу приймається тригармонійне рівняння у декартових координатах із трьома нетривіальними граничними умовами. Доведено, що розв’язок розглядуваної крайової задачі належить до класу саме додатних операторів. Показано, що поліноміальні розклади для граничних значень тригармонійної функції поблизу межі верхньої півплощини породжують формулу Тейлора тригармонійного інтеграла Пуассона. За отриманим результатом доведено наявність додаткових граничних умов для розв’язків тригармонійних рівнянь у декартових координатах.
Ключові слова: тригармонійне рівняння, оптимізаційна задача, верхня півплощина, тригармонійний інтеграл Пуассона, формула Тейлора.
повний текст
СПИСОК ЛІТЕРАТУРИ
- 1. Chikrii A.A., Belousov A.A. On linear differential games with integral constraints. Proceedings of the Steklov Institute of Mathematics. 2010. Vol. 269, Suppl 1. P. 69–80. https://doi.org/10.1134/S0081543810060076.
- 2. Chikrii A.A., Matychyn I.I., Chikrii K.A. Differential games with impulse control. Annals of the International Society of Dynamic Games. 2007. Vol. 9. P. 37–55. https://doi.org/10.1007/978-0-8176-4553-3_2.
- 3. Abdullayev F.G., Bushev D.M., Imashkyzy M., Kharkevych Yu.I. Isometry of the subspaces of solutions of systems of differential equations to the spaces of real functions. Ukrainian Mathematical Journal. 2020. Vol. 71, N 8. P. 1153–1172. https://doi.org/10.1007/s11253-019-01705-9.
- 4. Chikrij A.A., Bezmagorychnyj V.V. Method of resolving functions in linear differential games with integral restrictions. Soviet Automatic Control. 1993. Iss. 4. P. 26–36.
- 5. Vlasenko L.A., Chikrii A.A. On a differential game in a system with distributed parameters. Proceedings of the Steklov Institute of Mathematics. 2016. Vol. 292, Suppl 1. P. 276–285. https://doi.org/10.1134/S0081543816020243.
- 6. Pilipenko Yu.B., Chikrii A.A. Oscillatory conflict-control processes. Journal of Applied Mathematics and Mechanics. 1993. Vol. 57, N 3. P. 407–417. https://doi.org/10.1016/0021-8928(93)90119-7.
- 7. Kharkevych Yu.I., Khanin O.G. Asymptotic properties of the solutions of higher-order differential equations on generalized Hlder classes. Cybernetics and Systems Analysis. 2023. Vol. 59, N 4. P. 633–639. https://doi.org/10.1007/s10559-023-00598-8.
- 8. Kondratenko Yu., Kuntsevich V.M., Chikrii A.A., Gubarev V. Advanced control systems: Theory and applications. Advanced Control Systems: Theory and Applications. 2024. P. 1–441.
- 9. Kharkevych Yu. Approximation theory and related applications. Axioms. 2022. Vol. 11, Iss. 12. P. 736. https://doi.org/10.3390/axioms11120736.
- 10. Korenkov M.E., Kharkevych Yu.I. On the asymptotics of associated sigma-functions and Jacobi theta-functions. Ukrainian Mathematical Journal. 2019. Vol. 70, N 8. P. 1326–1330. https://doi.org/10.1007/s11253-018-1572-2.
- 11. Prokopovich P.V., Chikrii A.A. A linear evasion problem for interacting groups of objects. Journal of Applied Mathematics and Mechanics. 1994. Vol. 58, N 4. P. 583–591. https://doi.org/10.1016/0021-8928(94)90135-X.
- 12. Kal’chuk I.V., Kharkevych Yu.I., Pozharska K.V. Asymptotics of approximation of functions by conjugate Poisson integrals. Carpathian Mathematical Publications. 2020. Vol. 12, Iss. 1. P. 138–147. https://doi.org/10.15330/cmp.12.1.138-147.
- 13. Bushev D.N., Kharkevich Yu.I. Finding solution subspaces of the Laplace and heat equations isometric to spaces of real functions, and some of their applications. Mathematical Notes. 2018. Vol. 103, N 5–6. P. 869–880. https://doi.org/10.1134/S0001434618050231.
- 14. Shutovskyi A.M. Some applied aspects of the Dirac delta function. Journal of Mathematical Sciences (United States). 2023. Vol. 276, N 5. P. 685–694. https://doi.org/10.1007/s10958-023-06790-7.
- 15. Zhyhallo T.V., Kharkevych Yu.I. Some asymptotic properties of the solutions of Laplace equations in a unit disk. Cybernetics and Systems Analysis. 2023. Vol. 59, N 3. P. 449–456. https://doi.org/10.1007/s10559-023-00579-x.
- 16. Kharkevych Yu.I., Stepaniuk T.A. Approximate properties of Abel–Poisson integrals on classes of differentiable functions defined by moduli of continuity. Carpathian Mathematical Publications. 2023. Vol. 15, Iss. 1. P. 286–294. https://doi.org/10.15330/cmp.15.1.286-294.
- 17. Zhyhallo T.V., Kharkevych Yu.I. Fourier transform of the summatory Abel–Poisson function. Cybernetics and Systems Analysis. 2022. Vol. 58, N 6. P. 957–965. https://doi.org/10.1007/s10559-023-00530-0.
- 18. Kharkevych Yu.I., Zhyhallo T.V. Approximation of –differentiable functions defined on the real axis by Abel–Poisson operators. Ukrainian Mathematical Journal. 2005. Vol. 57, N 8. P. 1297–1315. https://doi.org/10.1007/s11253-005-0262-z.
- 19. Bushev D.M., Kharkevych Yu.I. Approximation of classes of periodic multivariable functions by linear positive operators. Ukrainian Mathematical Journal. 2006. Vol. 58, N 1. P. 12–21. https://doi.org/10.1007/s11253-006-0048-y.
- 20. Zhyhallo T.V., Kharkevych Yu.I. Approximation of –differentiable functions by Poisson integrals in the uniform metric. Ukrainian Mathematical Journal. 2009. Vol. 61, N 11. P. 1757–1779. https://doi.org/10.1007/s11253-010-0311-0.
- 21. Kharkevich Yu.I., Stepanyuk T.A. Approximation properties of Poisson integrals for the classes Mathematical Notes. 2014. Vol. 96, N 5–6. P. 1008–1019. https://doi.org/10.1134/S0001434614110406.
- 22. Zhyhallo K.M., Kharkevych Yu.I. Approximation of conjugate differentiable functions by their Abel–Poisson integrals. Ukrainian Mathematical Journal. 2009. Vol. 61, N 1. P. 86–98. https://doi.org/10.1007/s11253-009-0196-y.
- 23. Kharkevych Yu.I. Exact values of the approximations of differentiable functions by Poisson-type integrals. Cybernetics and Systems Analysis. 2023. Vol. 59, N 2. P. 274–282. https://doi.org/10.1007/s10559-023-00561-7.
- 24. Zhyhallo K.M., Kharkevych Yu.I. On some approximate properties of biharmonic Poisson integrals in the integral metric. Carpathian Mathematical Publications. 2024. Vol. 16, Iss. 1. P. 303–308. https://doi.org/10.15330/cmp.16.1.303-308.
- 25. Shutovskyi A.M., Pryt V.V. On the estimate of the biharmonic Poisson integral deviation from its boundary values in terms of the modulus of continuity. Journal of Mathematical Sciences (United States). 2024. Vol. 285, N 5. P. 732–742. https://doi.org/10.1007/s10958-024-07471-9.
- 26. Kharkevych Yu.I., Shutovskyi A.M. Approximation of functions from Hlder class by biharmonic Poisson integrals. Carpathian Mathematical Publications. 2024. Vol. 16, Iss. 2. P. 631–637. https://doi.org/10.15330/cmp.16.2.631-637.
- 27. Zjac J., Korenkov M.E., Kharkevych Yu.I. On the asymptotics of some Weierstrass functions. Ukrainian Mathematical Journal. 2015. Vol. 67, N 1. P. 154–158. https://doi.org/10.1007/s11253-015-1070-8.
- 28. Korenkov M., Kharkevych Yu. On the asymptotics and distribution of values of the Jacobi theta functions and the estimate of the type of the Weierstrass sigma functions. Axioms. 2022. Vol. 11, Iss. 1. P. 12. https://doi.org/10.3390/axioms11010012.
- 29. Shutovskyi A.M. On approximation of functions from Hlder classes by biharmonic Poisson integrals defined in the upper half-plane. Journal of Mathematical Sciences (United States). 2024. Vol. 282, N 1. P. 74–82. https://doi.org/10.1007/s10958-024-07169-y.
- 30. Zhyhallo T.V., Kharkevych Yu.I. Approximation of functions from the class by Poisson integrals in the uniform metric. Ukrainian Mathematical Journal. 2009. Vol. 61, N 12. P. 1893–1914. https://doi.org/10.1007/s11253-010-0321-y.
- 31. Vlasenko L.A., Rutkas A.G., Semenets V.V., Chikrii A.A. On the optimal impulse control in descriptor systems. Journal of Automation and Information Sciences. 2019. Vol. 51, N 5. P. 1–15. https://doi.org/10.1615/JAutomatInfScien.v51.i5.10.
- 32. Korenkov M.E., Zajac J., Kharkevych Yu.I. Nevanlinna characteristics and defective values of the Weierstrass zeta function. Ukrainian Mathematical Journal. 2011. Vol. 63, N 5. P. 838–841. https://doi.org/10.1007/s11253-011-0547-3.
- 33. Nakonechnyi A.G., Kapustian E.A., Chikrii A.A. Control of impulse systems in conflict situation. Journal of Automation and Information Sciences. 2019. Vol. 51, N 9. P. 1–11. https://doi.org/10.1615/JAutomatInfScien.v51.i9.10.
- 34. Baranovskaya L.V., Chikrii A.A., Chikrii Al.A. Inverse Minkowski functional in a nonstationary problem of group pursuit. Journal of Computer and Systems Sciences International. 1997. Vol. 36, Iss. 1. P. 101–106.
- 35. Chikrii A.A., Chikrii G.Ts. Game problems of approach for quasilinear systems of general form. Proceedings of the Steklov Institute of Mathematics. 2019. Vol. 304, Suppl 1. P. S44–S58. https://doi.org/10.1134/S0081543819020068.