Some asymptotic properties of the solutions of Laplace equations in a unit disk. (English. Ukrainian original) Zbl 1530.49007
Cybern. Syst. Anal. 59, No. 3, 449-456 (2023); translation from Kibern. Sist. Anal. 59, No. 3, 106-114 (2023).
Summary: The authors consider the optimization problem related to the integral representation of the deviation of positive linear operators on the classes of \((\psi, \beta )\)-differentiable functions in the integral metric. The Poisson integral, which is the solution of the Laplace equation in polar coordinates with the corresponding initial conditions given on the boundary of the unit disk, is taken as a positive linear operator. The Poisson integral is an operator with a delta-like kernel; therefore, it is the best apparatus for solving many problems of applied mathematics, namely: methods of optimization and variational calculus, mathematical control theory, theory of dynamical systems and game problems of dynamics, applied nonlinear analysis and moving objects search. The classes of \((\psi, \beta )\)-differentiable functions on which the asymptotic properties of the solutions to Laplace equations in a unit disk are analyzed are generalizations of the Sobolev, Weyl-Nagy, etc. classes well-known in optimization problems. The problem solved in the article will make it possible to generate high-quality mathematical models of many natural and social processes.
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
References:
| [1] | Chikrii, AA; Ts, G., Chikrii, “Matrix resolving functions in game problems of dynamics”, Proc. Steklov Inst. Math., 291, 56-65 (2015) · Zbl 1344.49062 · doi:10.1134/S0081543815090047 |
| [2] | A. A. Chikrii and I. I. Matichin, “Game problems for fractional-order linear systems,” Proc. Steklov Inst. Math., Vol. 268, 54-70 (2010).doi:10.1134/S0081543810050056. · Zbl 1303.91042 |
| [3] | Kharkevych, YuI, On some asymptotic properties of solutions to biharmonic equations, Cybern. Syst. Analysis, 58, 2, 251-258 (2022) · Zbl 1527.35152 · doi:10.1007/s10559-022-00457-y |
| [4] | Chikrii, AA; Prokopovich, PV, Simple pursuit of one evader by a group, Cybern. Syst. Analysis, 28, 3, 438-444 (1992) · Zbl 0875.90350 · doi:10.1007/BF01125424 |
| [5] | Yu. I. Kharkevych, “Approximative properties of the generalized Poisson integrals on the classes of functions determined by a modulus of continuity,” J. Autom. Inform. Sci., Vol. 51, Iss. 4, 43-54 (2019).doi:10.1615/JAutomatInfScien.v51.i4.40. |
| [6] | T. V. Zhyhallo and Yu. I. Kharkevych, “Approximating properties of biharmonic Poisson operators in the classes \({\widehat{L}}_{\beta ,1}^{\psi },\) ” Ukr. Math. J., Vol. 69, No. 5, 757-765 (2017).https://doi.org/10.1007/s11253-017-1393-8. · Zbl 1499.41098 |
| [7] | I. V. Kal’chuk, “Approximation of (ψ, β) -differentiable functions defined on the real axis by Weierstrass operators,” Ukr. Math. J., Vol. 59, No. 9, 1342-1363 (2007).doi:10.1007/s11253-007-0091-3. |
| [8] | Stepanets, AI, Classification and Approximation of Periodic Functions (1987), Kyiv: Naukova Dumka, Kyiv · Zbl 0689.42002 |
| [9] | Zhyhallo, KM; Kharkevych, YuI, Approximation of conjugate differentiable functions by their Abel-Poisson integrals, Ukr. Math. J., 61, 1, 86-98 (2009) · Zbl 1224.41066 · doi:10.1007/s11253-009-0196-y |
| [10] | I. Kal’chuk and Yu. Kharkevych, “Approximation ðroperties of the generalized Abel-Poisson integrals on the Weyl-Nagy classes,” Axioms, Vol. 11, No. 4, 161 (2022).doi:10.3390/axioms11040161. |
| [11] | Yu. I. Kharkevych and I. V. Kal’chuk, “Approximation of (ψ, β) -differentiable functions by Weierstrass integrals,” Ukr. Math. J., Vol. 59, No. 7, 1059-1087 (2007). https://doi.org/10.1007/s11253-007-0069-1. |
| [12] | K. M. Zhyhallo and Yu. I. Kharkevych, “Approximation of functions from the classes \({C}_{\beta ,\infty }^{\psi }\) (EQ) by biharmonic Poisson integrals,” Ukr. Math. J., Vol. 63, No. 7, 1083-1107 (2011). https://doi.org/10.1007/s11253-011-0565-1. · Zbl 1231.41033 |
| [13] | Zhyhallo, KM; Kharkevych, YuI, Approximation of (ψ, β) -differentiable functions of low smoothness by biharmonic Poisson integrals, Ukr. Math. J., 63, 12, 1820-1844 (2012) · Zbl 1263.31001 · doi:10.1007/s11253-012-0616-2 |
| [14] | Vladimirov, VS, Equations of Mathematical Physics (1981), Moscow: Nauka, Moscow |
| [15] | Zhyhallo, KM; Kharkevych, YuI, Approximation of conjugate differentiable functions by biharmonic Poisson integrals, Ukr. Math. J., 61, 3, 399-413 (2009) · Zbl 1224.31004 · doi:10.1007/s11253-009-0217-x |
| [16] | Zhigallo, KM; Kharkevych, YuI, On the approximation of functions of the Hölder class by biharmonic Poisson integrals, Ukr. Math. J., 52, 7, 1113-1117 (2000) · Zbl 0978.31002 · doi:10.1023/A:1005285818550 |
| [17] | A. Chikrii and I. Matichin, “Riemann-Liouville, Caputo, and sequential fractional derivatives in differential games,” in: M. Breton and K. Szajowski (eds.), Advances in Dynamic Games, Annals of the Intern. Soc. of Dynamic Games, Vol. 11, Birkhäuser, Boston (2011), pp. 61-81. doi:10.1007/978-0-8176-8089-3_4. · Zbl 1221.49070 |
| [18] | Zajac, J.; Korenkov, ME; Kharkevych, YuI, On the asymptotics of some Weierstrass functions, Ukr. Math. J., 67, 1, 154-158 (2015) · Zbl 1350.30060 · doi:10.1007/s11253-015-1070-8 |
| [19] | Albus, J.; Meystel, A.; Chikrii, AA; Belousov, AA; Kozlov, AI, Analytical method for solution of the game problem of soft landing for moving objects, Cybern. Syst. Analysis, 37, 1, 75-91 (2001) · Zbl 1003.93027 · doi:10.1023/A:1016620201241 |
| [20] | Bushev, DN; Kharkevich, YuI, Finding solution subspaces of the Laplace and heat equations isometric to spaces of real functions, and some of their applications, Math. Notes, 103, 6, 869-880 (2018) · Zbl 1406.46018 · doi:10.1134/S0001434618050231 |
| [21] | Zhyhallo, KM; Kharkevych, YuI, Complete asymptotics of the deviation of a class of differentiable functions from the set of their harmonic Poisson integrals, Ukr. Math. J., 54, 1, 51-63 (2002) · Zbl 1019.41014 · doi:10.1023/A:1019789402502 |
| [22] | Zhyhallo, TV; Kharkevych, YuI, Fourier transform of the summatory Abel-Poisson function, Cybern. Syst. Analysis, 58, 6, 957-965 (2022) · Zbl 1508.42008 · doi:10.1007/s10559-023-00530-0 |
| [23] | I. V. Kal’chuk and Yu. I. Kharkevych, “Approximation of the classes \({W}_{\beta ,\infty }^r\) by generalized Abel-Poisson integrals,” Ukr. Math. J., Vol. 74, No. 9, 575-585 (2022).doi:10.1007/s11253-022-02084-4. · Zbl 1525.41016 |
| [24] | T. V. Zhyhallo and Yu. I. Kharkevych, “Approximation of functions from the class \({C}_{\beta ,\infty }^r\) by Poisson integrals in the uniform metric,” Ukr. Math. J., Vol. 61, No. 12, 1893-1914 (2009). https://doi.org/10.1007/s11253-010-0321-y. · Zbl 1224.41048 |
| [25] | Zhyhallo, TV; Kharkevych, YuI, Approximation of (ψ, β) -differentiable functions by Poisson integrals in the uniform metric, Ukr. Math. J., 61, 11, 1757-1779 (2009) · Zbl 1224.41076 · doi:10.1007/s11253-010-0311-0 |
| [26] | T. V. Zhyhallo and Yu. I. Kharkevych, “On approximation of functions from the class \({L}_{\beta ,1}^{\psi }\) by the Abel-Poisson integrals in the integral metric,” Carpathian Math. Publ., Vol. 14, No. 1, 223-229 (2022).doi:10.15330/cmp.14.1.223-229. · Zbl 1505.42001 |
| [27] | F. G. Abdullayev and Yu. I. Kharkevych, “Approximation of the classes \({C}_{\beta }^{\psi }{H}^{\alpha }\) by biharmonic Poisson integrals,” Ukr. Math. J., Vol. 72, No. 1, 21-38 (2020). https://doi.org/10.1007/s11253-020-01761-6. · Zbl 1448.42003 |
| [28] | Bushev, DM; Kharkevych, YuI, Conditions of convergence almost everywhere for the convolution of a function with delta-shaped kernel to this function, Ukr. Math. J., 67, 11, 1643-1661 (2016) · Zbl 1384.42005 · doi:10.1007/s11253-016-1180-y |
| [29] | U. Z. Hrabova, I. V. Kal’chuk, and L. I. Filozof, “Approximative properties of the three-harmonic Poisson integrals on the classes \({W}_{\beta }^r{H}^{\alpha }\) ,” J. Math. Sci. (N.Y.), Vol. 254, No. 3, 397-405 (2021).doi:10.1007/s10958-021-05311-8. · Zbl 1461.42002 |
| [30] | Zhyhallo, KM; Kharkevych, YuI, Approximation of differentiable periodic functions by their biharmonic Poisson integrals, Ukr. Math. J., 54, 9, 1462-1470 (2002) · Zbl 1019.41012 · doi:10.1023/A:1023463801914 |
| [31] | I. V. Kal’chuk and Yu. I. Kharkevych, “Approximating properties of biharmonic Poisson integrals in the classes \({W}_{\beta }^r{H}^{\alpha }\) ,” Ukr. Math. J., Vol. 68, No. 11, 1727-1740 (2017).doi:10.1007/s11253-017-1323-9. · Zbl 1490.42004 |
| [32] | Yu. I. Kharkevych and I. V. Kal’chuk, “Asymptotics of the values of approximations in the mean for classes of differentiable functions by using biharmonic Poisson integrals,” Ukr. Math. J., Vol. 59, No. 8, 1224-1237 (2007).doi:10.1007/s11253-007-0082-4. · Zbl 1150.41333 |
| [33] | Kharkevych, Yu, Approximation theory and related applications, Axioms, 11, 12, 736 (2022) · doi:10.3390/axioms11120736 |
| [34] | U. Z. Hrabova, I. V. Kal’chuk, and T. A. Stepanyuk, “Approximation of functions from the classes \({W}_{\beta }^r{H}^{\alpha }\) by Weierstrass integrals,” Ukr. Math. J., Vol. 69, No. 4, 598-608 (2017).doi:10.1007/s11253-017-1383-x. · Zbl 1498.42006 |
| [35] | Vlasenko, LA; Rutkas, AG; Chikrii, AA, On a differential game in an abstract parabolic system, Proc. Steklov Inst. Math., 293, Suppl 1, 254-269 (2016) · Zbl 1351.49046 · doi:10.1134/S0081543816050229 |
| [36] | Pilipenko, YuV; Chikrij, AA, The oscillation processes of conflict control, Prikl. Matem. Mekh., 57, 3, 3-14 (1993) · Zbl 0805.90139 |
| [37] | Chikrii, AA; Rappoport, IS, Method of resolving functions in the theory of conflict-controlled processes, Cybern. Syst. Analysis, 48, 4, 512-531 (2012) · Zbl 1311.91038 · doi:10.1007/s10559-012-9430-y |
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