Adomian’s decomposition method in the theory of nonlinear autonomous boundary-value problems

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

Abstract

UDC 517.9

For a nonlinear autonomous boundary-value problem for ordinary differential equation in the critical case, we establish constructive conditions for the solvability and propose a scheme for the construction of solutions based on the use of Adomian's 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).

A. Boichuk, S. Chuiko, Autonomous weakly nonlinear boundary value problems in critical cases, Different. Equat., № 10, 1353–1358 (1992).

И. Г. Малкин, Методы Ляпунова и Пуанкаре в теории нелинейных колебаний, Гостехиздат, Ленинград, Москва (1949).

А. Н. Тихонов, В. Я. Арсенин, Методы решения некорректных задач, Наука, Москва (1986).

S. M. Chuiko, On the regularization of a matrix differential-algebraic boundary-value problem, J. Math. Sci., 220, № 5, 591–602 (2017). DOI: https://doi.org/10.1007/s10958-016-3202-6

S. M. Chuiko, I. A. Boichuk, An autonomous Noetherian boundary-value problem in the critical case, Nonlinear Oscillations, 12, № 3, 405–416 (2009). DOI: https://doi.org/10.1007/s11072-010-0085-1

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

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

С. М. Чуйко, О. С. Чуйко, М. В. Попов, Метод декомпозиції Адомяна у теорії нелінійних періодичних крайових задач, Нелінійні коливання, 25, № 4, 413–425 (2022).

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

О. А. Бойчук, С. М. Чуйко, Конструктивні методи аналізу крайових задач теорії нелінійних коливань, Наук. думка, Київ (2023). DOI: https://doi.org/10.37863/6581477912-64

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

С. М. Чуйко, И. Ю. Курильченко, О положении равновесия автономной периодической задачи, Динамические системы, 23, 31–37 (2007).

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

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

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

Published
30.08.2023
How to Cite
BoichukO., ChuikoS., and DiachenkoD. “Adomian’s Decomposition Method in the Theory of Nonlinear Autonomous Boundary-Value Problems”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 8, Aug. 2023, pp. 1053 -67, doi:10.3842/umzh.v75i8.7624.
Section
Research articles