Abstract
The numerical solution of systems of linear ordinary differential equations with nonlocal nonlinear conditions depending on the values of the desired function at intermediate points is investigated. Conditions for the existence of a solution to the problem under consideration are given. For the numerical solution, an approach is proposed that reduces the problem to two auxiliary linear systems of differential equations with linear conditions and to a single nonlinear algebraic system with a dimension depending only on the number of given intermediate points. The proposed approach is illustrated by solving two problems. The auxiliary problems are solved analytically in one of them and numerically in the other.
Similar content being viewed by others
REFERENCES
O. Nicoletti, “Sulle condizioni inizali che determinano gli integrali delle equazioni differenziali ordinarie,” Atti R., Torino 33, 746–748 (1897).
Ya. D. Tamarkin, On Certain General Problems in the Theory of Linear Ordinary Differential Equations and on Series Expansions of Arbitrary Functions (Petrograd, 1917) [in Russian].
Ch. J. de la Vallée Poussin, “Sur l'équation différentielle linéaire du second ordre: Détermination d’une intégrale par deux valeurs assignées: Extension aux équations d’ordre n,” J. Math. Pures Appl. 8 (9), 125–144 (1929).
A. A. Abramov, N. G. Burago, et al., “Software package for solving linear two-point boundary value problems,” Reports on Computer Software (Akad. Nauk SSSR, Moscow, 1982) [in Russian].
K. Moszyński, “A method of solving the boundary value problem for a system of linear ordinary differential equation,” Algorytmy 2 (3), 25–43 (1964).
I. T. Kiguradze, “Boundary value problems for system of ordinary differential equations,” Itogi Nauki Tekh. Sovrem. Probl. Mat. Nov. Dostizheniya 30, 3–103 (1987).
A. M. Samoilenko, V. M. Laptinskii, and K. K. Kenzhebaev, Constructive Methods for Studying Periodic Multipoint Boundary Value Problems (Kiev, 1999) [in Russian].
D. S. Dzhumabaev, “Well-posedness of linear multipoint boundary value problem,” Mat. Zh. Almaty 5 (1) (15), 30–38 (2005).
A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations (Nauka, Moscow, 1978; Birkhäuser, Basel, 1989).
A. S. Antipin, “Terminal control of boundary models, Comput. Math. Math. Phys. 54 (2), 275–302 (2014).
R. E. Bellman and R. E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems (Elsevier, New York, 1965).
J. M. Ortega and W. C. Reinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 2000).
P.-F. Dai, Q-B. Wu, and M.-H. Chen, “Modified Newton–NSS method for solving systems of nonlinear equations,” Numer. Algorithms 77 (1), 1–21 (2018). https://doi.org/10.1007/s11075-017-0301-5
J. M. Martinez, “Practical quasi-Newton methods for solving nonlinear systems,” J. Comput. Appl. Math. 124 (1–2), 97–121 (2000).
W. W. Young and Q. Ni, “A new cubic convergent method for solving a system of nonlinear equations,” Int. J. Comput. Math. 94 (10), 1960–1980 (2017). doi 10.1080, 00207160.2016, 1274740.2017
Yu. G. Evtushenko and A. A. Tret’yakov, “Pth-order numerical methods for solving systems of nonlinear equations,” Dokl. Math. 89 (2), 214–217 (2014).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Aida-zade, K.R. Numerical Solution of Linear Differential Equations with Nonlocal Nonlinear Conditions. Comput. Math. and Math. Phys. 60, 808–816 (2020). https://doi.org/10.1134/S0965542520030033
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542520030033