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 Zh. Vychisl. Mat. Mat. Fiz., 2014, Volume 54, Number 7, Pages 1096–1109 (Mi zvmmf10061)

Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations

V. M. Abdullaevab, K. R. Aida-zadeab

a Institute of Cybernetics, Academy of Sciences of Azerbaijan, ul. B. Vagabzade 9, Baku, AZ1141, Azerbaijan

Abstract: A numerical method is suggested for solving systems of nonautonomous loaded linear ordinary differential equations with nonseparated multipoint and integral conditions. The method is based on the convolution of integral conditions into local ones. As a result, the original problem is reduced to an initial value (Cauchy) problem for systems of ordinary differential equations and linear algebraic equations. The approach proposed is used in combination with the linearization method to solve systems of loaded nonlinear ordinary differential equations with nonlocal conditions. An example of a loaded parabolic equation with nonlocal initial and boundary conditions is used to show that the approach can be applied to partial differential equations. Numerous numerical experiments on test problems were performed with the use of the numerical formulas and schemes proposed.

Key words: loaded systems of ordinary differential equations, nonseparated conditions, integral conditions, nonlocal multipoint conditions, numerical method.

DOI: https://doi.org/10.7868/S0044466914070023

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English version:
Computational Mathematics and Mathematical Physics, 2014, 54:7, 1096–1109

Bibliographic databases:

UDC: 519.624
MSC: 34A30, 65L10

Citation: V. M. Abdullaev, K. R. Aida-zade, “Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations”, Zh. Vychisl. Mat. Mat. Fiz., 54:7 (2014), 1096–1109; Comput. Math. Math. Phys., 54:7 (2014), 1096–1109

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Aida-zade K.R., Abdullayev V.M., “Solution to a class of inverse problems for a system of loaded ordinary differential equations with integral conditions”, J. Inverse Ill-Posed Probl., 24:5 (2016), 543–558
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3. V. M. Abdullayev, K. R. Aida-zade, “Optimization of loading places and load response functions for stationary systems”, Comput. Math. Math. Phys., 57:4 (2017), 634–644
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5. D. S. Dzhumabaev, S. M. Temesheva, “Approximation of problem for finding the bounded solution to system of nonlinear loaded differential equations”, News Natl. Acad. Sci. Rep. Kazakhstan-Ser. Phys.-Math., 1:311 (2017), 13+
6. V. M. Abdullayev, “Identification of the functions of response to loading for stationary systems”, Cybern. Syst. Anal., 53:3 (2017), 417–425
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8. A. T. Asanova, Zh. M. Kadirbaeva, E. A. Bakirova, “On the unique solvability of a nonlocal boundary-value problem for systems of loaded hyperbolic equations with impulsive actions”, Ukr. Math. J., 69:8 (2018), 1175–1195
9. D. Dzhumabaev, “Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro-differential equations”, Math. Meth. Appl. Sci., 41:4 (2018), 1439–1462
10. D. S. Dzhuraabaev, “Well-posedness of nonlocal boundary value problem for a system of loaded hyperbolic equations and an algorithm for finding its solution”, J. Math. Anal. Appl., 461:1 (2018), 817–836
11. K. R. Aida-zade, V. A. Hashimov, “Optimization of measurement points positioning in a border control synthesis problem for the process of heating a rod”, Autom. Remote Control, 79:9 (2018), 1643–1660
12. A. T. Assanova, A. E. Imanchiyev, Zh. M. Kadirbaeva, “Numerical solution of systems of loaded ordinary differential equations with multipoint conditions”, Comput. Math. Math. Phys., 58:4 (2018), 508–516
13. K. R. Aida-Zade, V. M. Abdullayev, “Numerical solution to optimal control problems for loaded dynamic systems with integral conditions”, Proceedings of the 6th International Conference on Control and Optimization With Industrial Applications, v. I, eds. A. Fikret, B. Tamer, Baku State Univ., Inst. Applied Mathematics, 2018, 107–109
14. I. N. Parasidis, E. Providas, “An exact solution method for a class of nonlinear loaded difference equations with multipoint boundary conditions”, J. Differ. Equ. Appl., 24:10 (2018), 1649–1663
15. K. Aida-Zade, V. Hashimov, “On one problem of synthesis of lumped controls on heating the plate”, Proceedings of the 6th International Conference on Control and Optimization With Industrial Applications, v. II, eds. A. Fikret, B. Tamer, Baku State Univ., Inst. Applied Mathematics, 2018, 41–43
16. V. M. Abdullaev, “Chislennoe reshenie kraevoi zadachi dlya nagruzhennogo parabolicheskogo uravneniya s nelokalnymi granichnymi usloviyami”, Vestnik KRAUNTs. Fiz.-mat. nauki, 32:3 (2020), 15–28
17. K. R. Aida-zade, V. M. Abdullayev, “Optimization of right-hand sides of nonlocal boundary conditions in a controlled dynamical system”, Autom. Remote Control, 82:3 (2021), 375–397
18. A. T. Assanova, A. Zholamankyzy, “Problem with data on the characteristics for a loaded system of hyperbolic equations”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 31:3 (2021), 353–364
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