Averaging in boundary-value problems for systems of differential and integrodifferential equations

  • S. T. Mynbayeva K.Zhubanov Aktobe Regional State University; Institute of Mathematics and Mathematical Modeling
  • A. N. Stanzhitskii Kiyev. nats. un-t im. T. Shevchenko
  • N. A. Marchuk Podil. agrarian-technical University of Kamyanets-Podilsky
Keywords: Averaging, boundary value problem, Volterra type, convergence, variation

Abstract

UDC 517.9

The averaging method is applied to the investigation of the problem of existence of solutions of boundary-value problems for systems of differential and integrodifferential equations.  It is shown that if the averaged boundary-value problem has a solution, then the original problem also has a solution.  It is worth noting that, in this case, the system obtained as a result of averaging of a system of integrodifferential equations has the form of a simpler  system of ordinary differential equations.

References

Alawneh, Ameen; Al-Khaled, Kamel; Al-Towaiq, Mohammed. Reliable algorithms for solving integro-differential equations with applications. Int. J. Comput. Math. 87 (2010), no. 7, 1538--1554. doi: 10.1080/00207160802385818

Kythe, Prem K.; Puri, Pratap. Computational methods for linear integral equations. Birkhouser Boston, Inc., Boston, MA, 2002. xviii+508 pp. ISBN: 0-8176-4192-0 doi: 10.1007/978-1-4612-0101-4

Wazwaz, Abdul-Majid. A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equations. Appl. Math. Comput. 181 (2006), no. 2, 1703--1712. doi: 10.1016/j.amc.2006.03.023

J. M. Kean, Nidel D. Barlow, A spatial model for the succesful biological control of sitona discoidens by Microctonus aethiopoides, J. Appl. Ecology, 38, 162 – 169 (2001). doi: 10.1046/j.1365-2664.2001.00579.x

Thieme, H. R. A model for the spatial spread of an epidemic. J. Math. Biol. 4 (1977), no. 4, 337--351. doi: 10.1007/BF00275082

O. A. Boichuk, I. A. Holovatska, Слабконелiнiйнi системи iнтегро-диференцiальних рiвнянь. (Russian) [[Slabkoneliniini systemy intehro-dyferentsialnykh rivnian]], Neliniini kolyvannia, 16, № 3, 314 – 321 (2013). ,

O. A. Boichuk, I. A. Holovatska, Крайовi задачi для систем iнтегро-диференцiальних рiвнянь. (Russian) [[Kraiovi zadachi dlia system intehro-dyferentsialnykh rivnian]], Neliniini kolyvannia, 16, № 4, 460 – 474 (2013).

Dzhumabaev, D. S. On a method for solving a linear boundary value problem for an integrodifferential equation. (Russian) ; translated from Zh. Vychisl. Mat. Mat. Fiz. 50 (2010), no. 7, 1209--1221 Comput. Math. Math. Phys. 50 (2010), no. 7, 1150--1161 doi: 10.1134/S0965542510070043

Dzhumabaev, D. S. Necessary and sufficient conditions for the solvability of linear boundary-value problems for the Fredholm integrodifferential equations. Translation of Ukraïn. Mat. Zh. 66 (2014), no. 8, 1074–1091. Ukrainian Math. J. 66 (2015), no. 8, 1200--1219. doi: 10.1007/s11253-015-1003-6

Dzhumabaev, D. S. On one approach to solve the linear boundary value problems for Fredholm integro-differential equations. J. Comput. Appl. Math. 294 (2016), 342--357. doi: 10.1016/j.cam.2015.08.023

Dzhumabaev, Dulat S. New general solutions to linear Fredholm integro-differential equations and their applications on solving the boundary value problems. J. Comput. Appl. Math. 327 (2018), 79--108. doi: 10.1016/j.cam.2017.06.010

A. M. Samoilenko, O. A. Boichuk, S. A. Kryvosheia, Крайовi задачi для систем лiнiйних iнтегро-диференцiальних рiвнянь з виродженим ядром. (Russian) [[Kraiovi zadachi dlia system liniinykh intehro-dyferentsialnykh rivnian z vyrodzhenym yadrom]], Ukr. mat. zhurn., 48, № 11, 1576 – 1579 (1996).

A. M. Samoilenko, R. Y. Petryshyn, Метод усреднения в некоторых краевых задачах. (Russian) [[Metod usrednenyia v nekotorыkh kraevыkh zadachakh]], Dyfferents. uravnenyia, 25, № 6, 956 – 964 (1989).

Yu. A. Mitropol`skij, D. D. Bajnov, S. D. Milusheva, Применение метода усреднения для решения краевых задач для обыкновенных дифференциальных уравнений и интегро-дифференциальных уравнений. (Russian) [[Primenenie metoda usredneniya dlya resheniya kraevy`kh zadach dlya oby`knovenny`kh differenczial`ny`kh uravnenij i integro-differenczial`ny`kh uravnenij]], Mat. fyzyka, vыp.25, 3 – 22 (1979).

Filatov, A. N.; Sharova, L. V. Интегральные неравенства и теория нелиней ных колебаний. (Russian) [[Integral inequalities and the theory of nonlinear oscillations]] Izdat. ``Nauka'', Moscow, 1976. 152 pp. MR0492576

Z. Shmarda, Существование и единственность решения задачи Коши для сингулярных систем интегро-

дифференциальных уравнений. (Russian) [[Sushhestvovanie i edinstvennost` resheniya zadachi Koshi dlya singulyarny`kh sistem integro-differenczial`ny`kh uravnenij]], Ukr. mat. zhurn, 45, # 12, 1716 – 1720 (1993).

Published
15.02.2020
How to Cite
MynbayevaS. T., StanzhitskiiA. N., and MarchukN. A. “Averaging in Boundary-Value Problems for Systems of Differential and Integrodifferential Equations”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 2, Feb. 2020, pp. 245-66, https://umj.imath.kiev.ua/index.php/umj/article/view/1071.
Section
Research articles