Надіслати цю статтю Поліноміальні розв’язки рівнянь Гаммерштейна в задачах з вільною фазою
Анотація
Розглянуто інтегральні рівняння Гаммерштейна, що виникають у задачах з вільною фазою. Запропоновано метод аналітично-числового розв’язування таких рівнянь. В основу методу покладено властивість логарифмічної похідної полінома скінченного степеня, що дає змогу аналітично обчислювати аргумент функції та звести вихідне нелінійне інтегральне рівняння до розв’язання систем трансцендентних рівнянь малої розмірності. Наведено числові результати, які свідчать про можливість застосування запропонованого підходу до інтегральних рівнянь, що досліджуються.
Зразок для цитування: Ю. П. Тополюк, “Поліноміальні розв’язки рівнянь Гаммерштейна в задачах з вільною фазою”, Мат. методи та фіз.-мех. поля, 67, №3-4, 84-96 (2024), https://doi.org/10.15407/mmpmf2024.67.3-4.84-96
Ключові слова
Посилання
M. I. Andriychuk, N. N. Voitovich, P. A. Savenko, V. P. Tkachuk, Antenna Synthesis According to the Amplitude Directivity Pattern. Numerical Methods and Algorithms [in Russian], Nauk. Dumka, Kyiv (1993).
O. O. Bulatsyk, “Investigation of branching of solutions to nonlinear integral equation of the modified phase problem in the case of discrete Fourier transform,” Visn. Lviv. Univ. Ser. Prykl. Mat. Inf., Issue 7, 20–32 (2003) (in Ukrainian).
O. O. Bulatsyk, M. M. Voitovich, “Analytical representation of solutions of a nonlinear integral equation of the problem of approximation of finite functions with a free phase by functions with a finite discrete spectrum,” Mat. Met. Fiz.-Mekh. Polya, 44, No. 2, 55–63 (2001) (in Ukrainian).
O. O. Bulatsyk, M. M. Voitovich, O. M. Gis, “Branching of solutions of nonlinear equations of modified phase problem,” Mat. Met. Fiz.-Mekh. Polya, 45, No. 2, 64–74 (2002) (in Ukrainian).
M. M. Voitovich, O. M. Gis, Yu. P. Topolyuk, “Mean-square approximation of finite functions with free phase by functions with finite spectrum,” Dop. Nats. Akad. Nauk Ukr., No. 3, 7–10 (1999) (in Ukrainian).
N. N. Voitovich, P. A. Savenko, “On one integral equation of the theory of antenna synthesis,” Mat. Met. Fiz.-Mekh. Polya, No. 2, 161–163 (1975) (in Russian).
O. M. Gis, Yu. P. Topolyuk, “Mean square approximation of nonnegative finite functions by the modulus of functions with a finite spectrum (the case of vanishing solutions of approximation on the finite function supports),” Mat. Met. Fiz.-Mekh. Polya, 44, No. 2, 48–54 (2001) (in Ukrainian).
O. D. Melkumyan, Analysis of Systems by the Logarithmic Derivative Method [in Russian], Energoizdat, Moscow (1981).
P. O. Savenko, Nonlinear Problems of Synthesis of Radiating Systems (Theory and Methods of Solution) [in Ukrainian], Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv (2014).
K. E. Atkinson, “A survey of numerical methods for solving nonlinear integral equations,” J. Integr. Equat. Appl., 4, No. 1, 15–46 (1992), https://doi.org/10.1216/jiea/1181075664
O. O. Bulatsyk, O. M. Gis, N. N. Voitovich, “Some peculiar properties of solutions of the generalized phase problem,” in: Proc. of VI Int. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory DIPED-2001 (September 18–20, 2001, Lviv, Ukraine), pp. 38–41 (2001), https://doi.org/10.1109/DIPED.2001.965029
O. O. Bulatsyk, B. Z. Katsenelenbaum, Yu. P. Topolyuk, N. N. Voitovich, Phase Optimization Problems: Applications in Wave Field Theory, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim (2010, https://doi.org/10.1002/9783527629824
O. O. Bulatsyk, N. N. Voitovich, “Modified phase problem with nonsymmetrical data,” in: Proc. of VII Int. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory DIPED-2002 (October 10–13, 2002, Lviv–Tbilisi), pp. 137–140 (2002), https://doi.org/10.1109/DIPED.2002.1049191
J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Optics., 21, No. 15, 2758–2769 (1982), https://doi.org/10.1364/AO.21.002758
O. M. Gis, “Some new vanishing solutions of linear antenna synthesis problem according to prescribed amplitude radiation pattern,” in: Proc. of III Int. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory DIPED–1998 (November 2–5, 1998, Tbilisi, Georgia), pp. 72–75 (1998), https://doi.org/10.1109/DIPED.1998.730947
O. M. Gis, Yu. P. Topolyuk, “Closed vanishing solution of the nonlinear problems of the antenna synthesis,” in: Proc. of IV Int. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory DIPED–1999 (September 20–23, 1999, Lviv, Ukraine), pp. 75–79 (1999), https://doi.org/10.1109/DIPED.1999.822133
A. Hammerstein, “Nichtlineare Integralgleichungen nebst Anwendungen,” Acta Math., 54, 117–176 (1930), https://doi.org/10.1007/BF02547519
R. Iglisch, “Existenz- und Eindeutigkeitssätze bei nichtlineare Integralgleichungen,” Math. Ann., 108, 161–189 (1933), https://doi.org/10.1007/BF01452830
S. Kumar, “Superconvergence of a collocation-type method for Hammerstein equations,” IMA J. Numer. Anal., 7, No. 3, 313–325 (1987), https://doi.org/10.1093/imanum/7.3.313
S. Kumar, I. H. Sloan, “A new collocation-type method for Hammerstein integral equations,” Math. Comput., 48, No. 178, 585–593 (1987), https://doi.org/10.2307/2007829
L. J. Lardy, “A variation of Nyström’s method for Hammerstein equations,” J. Integr. Equat., 3, No. 1, 43–60 (1981), https://www.jstor.org/stable/26164052
N. N. Voitovich, O. M. Gis, O. O. Reshnyak, Yu. P. Topolyuk, “New rigorous solution of some antenna synthesis problems according to prescribed amplitude radiation pattern,” in: Proc. of Int. Conf. on Mathematical Methods in Electromagnetic Theory MMET–1998 (June 2–5, 1998, Kharkiv, Ukraine), Vol. 1, pp. 133–135 (1998), https://doi.org/10.1109/MMET.1998.709699
N. N. Voitovich, O. O. Reshnyak, “Closed solutions of a nonlinear integral equation arisen in the antenna arrays synthesis theory,” in: Proc. of III Int. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory DIPED–1998 (November 2–5, 1998, Tbilisi, Georgia), pp. 67–71 (1998), https://doi.org/10.1109/DIPED.1998.730946
N. N. Voitovich, O. O. Reshnyak, “Solutions of nonlinear integral equation of synthesis of the linear antenna arrays,” J. Appl. Electromagn. (BSUAE), 2, No. 1, 43–52 (1999).
N. N. Voitovich, Yu. P. Topolyuk, “Antenna synthesis according to prescribed amplitude radiation pattern and the phase problem,” in: Proc. of II Int. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, Proc. of DIPED–1997 (September 15–17, 1997, Lviv, Ukraine), pp. 90–92 (1997), https://doi.org/10.1109/DIPED.1997.710923
N. N. Voitovich,, Yu. P. Topolyuk, O. O. Reshnyak, “Approximation of compactly supported functions with free phase by functions with bounded spectrum,” in A. G. Ramm, P. N. Shivakumar, A. V. Strauss (eds.), Operator Theory and its Applications, Vol. 25, Fields Institute Communications, AMS (2000), pp. 531–541, https://doi.org/10.1090/fic/025
N. N. Voitovich, Yu. P. Topolyuk, O. O. Reshnyak, J. Jarkowski, “New closed solution of the linear antenna synthesis problem according to amplitude radiation pattern,” in: Proc. of 12th Int. Conference on Microwave and Radar MIKON–98 (May 20–22, 1998, Cracow, Poland), Vol. 2, pp. 304–308 (1998), https://doi.org/10.1109/MIKON.1998.740794
Посилання
- Поки немає зовнішніх посилань.
Ця робота ліцензована Creative Commons Attribution 3.0 License.