Метод декомпозиції Адомяна у теорії нелінійних крайових задач

O. Boichuk; S. Chuiko; M. Popov

doi:10.3842/umzh.v76i5.7900

O. Boichuk Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv
S. Chuiko Donbas State Pedagogical University and the Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Slovyansk, Donetsk region; Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany
M. Popov Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Slovyansk, Donetsk region

DOI: https://doi.org/10.3842/umzh.v76i5.7900

Keywords: Тonlinear boundary-value problem, ordinary differential equation, Adomian decomposition method

Abstract

UDC 517.9

For the nonlinear boundary-value problem for an ordinary differential equation in the critical case, we obtain constructive conditions for the existence of solutions and propose a scheme for finding these solutions by using the Adomian decomposition method.

References

A. A. Boichuk, A. M. Samoilenko, Generalized inverse operators and Fredholm boundary-value problems, 2th ed., De Gruyter, Berlin, Boston (2016). DOI: https://doi.org/10.1515/9783110378443

A. A. Boichuk, Nonlinear boundary-value problems for systems of ordinary differential equations, Ukr. Math. J., 50, № 2, 186–195 (1998). DOI: https://doi.org/10.1007/BF02513444

А. А. Бойчук, Конструктивные методы анализа краевых задач, Наук. думка, Киев (1990).

P. Benner, A. Seidel-Morgenstern, A. Zuyev, Periodic switching strategies for an isoperimetric control problem with application to nonlinear chemical reactions, Appl. Math. Model., 69, 287–300 (2019). DOI: https://doi.org/10.1016/j.apm.2018.12.005

И. Г. Малкин, Некоторые задачи теории нелинейных колебаний, Гостехиздат, Москва (1956).

A. A. Boichuk, S. M. Chuiko, On approximate solutions of nonlinear boundary-value problems by the Newton–Kantorovich method, J. Math. Sci., 258, № 5, 594–617 (2021). DOI: https://doi.org/10.1007/s10958-021-05569-y

A. A. Boichuk, S. M. Chuiko, On approximate solutions of weakly nonlinear boundary-value problems by the Newton–Kantorovich method, J. Math. Sci., 261, № 2, 228–240 (2022). DOI: https://doi.org/10.1007/s10958-022-05748-5

Л. В. Канторович, Г. П. Акилов, Функциональный анализ, Наука, Москва (1977).

Ю. Д. Шлапак, О периодических решениях нелинейных уравнений второго порядка, не разрешенных относительно старшей производной, Укр. мат. журн., 26, № 6, 850–854 (1974).

A. M. Samoilenko, S. M. Chuiko, O. V. Starkova, Nonlinear boundary-value problem that is not solved with respect to the derivative, Ukr. Mat. Zh., 72, № 8, 1280–1293 (2020). DOI: https://doi.org/10.1007/s11253-020-01852-4

G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. and Appl., 135, 501–544 (1988). DOI: https://doi.org/10.1016/0022-247X(88)90170-9

G. Adomian, Polynomial nonlinearities in differential equations, J. Math. Anal. and Appl., 109, 90–95 (1985). DOI: https://doi.org/10.1016/0022-247X(85)90178-7

G. Adomian, Convergent series solution of nonlinear equations, J. Comput. and Appl. Math., 11, 225–230 (1984). DOI: https://doi.org/10.1016/0377-0427(84)90022-0

M. Mac, C. S. Leung, T. Harko, A brief introducion to the Adomian decomposition method, Roman. Astron. J., 1, № 1, 1–41 (2019).

S. M. Chuiko, O. S. Chuiko, M. V. Popov, Adomian decomposition method in the theory of nonlinear boundary-value problems, J. Math. Sci., 277, № 2, 338–351 (2023). DOI: https://doi.org/10.1007/s10958-023-06837-9

Е. А. Гребеников, Ю. А. Рябов, Конструктивные методы анализа нелинейных систем, Наука, Москва (1979).

S. M. Chuiko, Nonlinear matrix differential-algebraic boundary-value problem, Lobachevskii J. Math., 38, № 2, 236–244 (2017). DOI: https://doi.org/10.1134/S1995080217020056

O. Vejvoda, On perturbed nonlinear boundary-value problems, Czech. Math. J., № 11, 323–364 (1961). DOI: https://doi.org/10.21136/CMJ.1961.100464

S. M. Chuiko, O. V. Starkova, On the approximate solution of autonomous boundary-value рroblems by the least square method, Nonlinear Oscillations, 12, № 4, 556–573 (2009). DOI: https://doi.org/10.1007/s11072-010-0095-z

A. Boichuk, O. Strakh, Linear Fredholm boundary-value problems for dynamical systems on a time scale, J. Math. Sci., 208, № 5, 487–497 (2015). DOI: https://doi.org/10.1007/s10958-015-2463-9

A. Samoilenko, A. Boichuk, S. Chuiko, Hybrid difference differential boundary-value problem, Miskolc Math. Notes, 18, № 2, 1015–1031 (2017). DOI: https://doi.org/10.18514/MMN.2017.2280