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

  • 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
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

Published
03.07.2024
How to Cite
BoichukO., ChuikoS., and PopovM. “Adomian Decomposition Method in the Theory of Nonlinear Boundary-Value Problems”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, no. 6, July 2024, pp. 820–831, doi:10.3842/umzh.v76i5.7900.
Section
Research articles