Home Solving Fredholm integro-differential equations involving integral condition: A new numerical method
Article
Licensed
Unlicensed Requires Authentication

Solving Fredholm integro-differential equations involving integral condition: A new numerical method

  • Zhazira Kadirbayeva EMAIL logo , Elmira Bakirova and Agila Tleulessova
Published/Copyright: May 24, 2024
Become an author with De Gruyter Brill

Abstract

In this work we investigate a nonlocal problem for the Fredholm integro-differential equation involving integral condition. The main tool used in our considerations is Dzhumabaev parametrization method. We make use of the numerical implementation of the Dzhumabaev parametrization method to obtain the desired result, which is well-supported with numerical examples.

MSC 2010: 34K10; 45J99; 65L10

This research has been funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP19174331).


Acknowledgement

The authors thank the referees for his/her careful reading of the manuscript and useful suggestions.

  1. Communicated by Jozef Džurina

References

[1] Aida-zade, K. R.—Abdullayev, V. M.: To the solution of integro-differential equations with nonlocal conditions, Turkish J. Math. 46(1) (2022), 177–188.Search in Google Scholar

[2] Akyuz-Dascioglu, A.—Sezer, M.: Chebyshev polynomial solutions of systems of higher-order linear Fredholm–Volterra integro-differential equations, J. Franklin Inst. 342(6) (2005), 688–701.10.1016/j.jfranklin.2005.04.001Search in Google Scholar

[3] Ali, N.—Zaman, G.—Hyo Jung, I.: Stability analysis of delay integro-differential equations of HIV-1 infection mode, Georgian Math. J. 27(3) (2018), 1–10.10.1515/gmj-2018-0011Search in Google Scholar

[4] Amirali, I.—Durmaz, M. E.—Acar, H.—Amiraliyev, G. M.: First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition, J. Phys. Conf. Ser. 2514(1) (2023), Art. ID 012003.10.1088/1742-6596/2514/1/012003Search in Google Scholar

[5] Andrade, B.—Viana, A.: Integrodifferential equations with applications to a plate equation with memory, Math. Nachr. 289(17–18) (2016), 2159–2172.10.1002/mana.201500205Search in Google Scholar

[6] Arikoglu, A.—Ozkol, I.: Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. Math. Comput. 168(2) (2005), 1145–1158.10.1016/j.amc.2004.10.009Search in Google Scholar

[7] Assanova, A. T.—Bakirova, E. A.—Kadirbayeva, Zh. M.—Uteshova, R. E.: A computational method for solving a problem with parameter for linear systems of integro-differential equations, Comput. Appl. Math. 39(3) (2020), Art. No. 248.10.1007/s40314-020-01298-1Search in Google Scholar

[8] Assanova, A. T.—Kadirbayeva, Zh. M.: On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations, Comput. Appl. Math. 37(4) (2018), 4966–4976.10.1007/s40314-018-0611-9Search in Google Scholar

[9] Athavale, P.—Tadmor, E.: Novel integro-differential equations in image processing and its applications. In: Proc. SPIE 7533, Computational Imaging VIII, California, United States, 2010; https://doi.org/10.1117/12.850779.10.1117/12.850779Search in Google Scholar

[10] Baiburin, M.—Parasidis, I.: Exact solution to systems of linear first-order integro-differential equations with multipoint and integral conditions. In: Mathematical Analysis and Applications, (T. Rassias, P. Pardalos, eds.), Springer Optimization and Its Applications, Vol. 154, 2019.10.1007/978-3-030-31339-5_1Search in Google Scholar

[11] Bakirova, E. A.—Assanova, A. T.—Kadirbayeva, Zh. M.: A problem with parameter for the integro-differential equations, Math. Model. Anal. 26(1) (2021), 34–54.10.3846/mma.2021.11977Search in Google Scholar

[12] Bellomo, N.—Firmani, B.—Guerri, L.: Bifurcation analysis for a nonlinear system of integro-differential equations modelling tumor-immune cells competition, Appl. Math. Lett. 12(2) (1999), 39–44.10.1016/S0893-9659(98)00146-3Search in Google Scholar

[13] Dzhumabaev, D.: On one approach to solve the linear boundary value problems for Fredholm integro-differential equations, J. Comput. Appl. Math. 294 (2016), 342–357.10.1016/j.cam.2015.08.023Search in Google Scholar

[14] Dzhumabaev, D.: A method for solving the linear boundary value problem for an integro-differential equation, Comput. Math. Math. Phys. 50 (2010), 1150–1161.10.1134/S0965542510070043Search in Google Scholar

[15] Dzhumabaev, D.: Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation, USSR Comput. Math. Math. Phys. 29(1) (1989), 34–46.10.1016/0041-5553(89)90038-4Search in Google Scholar

[16] Gelmi, C. A.—Jorquera, H.: IDSOLVER: A general purpose solver for nth-order integro-differential equations, Computer Physics Communications 185 (2014), 392–397.10.1016/j.cpc.2013.09.008Search in Google Scholar

[17] Iskakova, N. B.—Temesheva, S. M.—Uteshova, R. E.: On a problem for a delay differential equation, Math. Methods Appl. Sci. 46(9) (2023), 11283–11297.10.1002/mma.9181Search in Google Scholar

[18] Jaradat, H.—Alsayyed, O.—Al-Shara, S.: Numerical solution of linear integro-differential equations, J. Math. Stat. 4(4) (2008), 250–254.10.3844/jmssp.2008.250.254Search in Google Scholar

[19] Lakshmikantham, V.: Theory of Integro-Differential Equations, CRC Press, Amsterdam, The Netherlands, 1995.Search in Google Scholar

[20] Kadirbayeva, Zh. M.—Kabdrakhova, S. S.: A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition, Open Math. 20(1) (2022), 1173–1183.10.1515/math-2022-0496Search in Google Scholar

[21] Kheybari, S.—Darvishi, M. T.—Wazwaz, A. M.: A semi-analytical approach to solve integro-differential equations, J. Comput. Appl. Math. 317 (2017), 17–30.10.1016/j.cam.2016.11.011Search in Google Scholar

[22] Kumar, K. S.: Electric Circuits & Networks, Pearson, Delhi, India, 2009.Search in Google Scholar

[23] MacCamy, R. C.: An integro-differential equation with application in heat flow, Quart. Appl. Math. 35(1) (1977), 1–19,10.1090/qam/452184Search in Google Scholar

[24] Maleknejad, K.—Mirzaee, F.: Numerical solution of integro-differential equations by using rationalized Haar functions method, Kybernetes 35(10) (2006), 1735–1744.10.1108/03684920610688694Search in Google Scholar

[25] Parasidis, I. N.—Providas, E.—Dafopoulos, V.: Loaded differential and Fredholm integro-differential equations with nonlocal integral boundary conditions, Appl. Math. Control Sci. 3(1) (2018), 50–68.10.15593/2499-9873/2018.3.04Search in Google Scholar

[26] Pour-Mahmoud, J.—Rahimi-Ardabili, M. Y.—Shahmorad, S.: Numerical solution of the system of Fredholm integro-differential equations by the Tau method, Appl. Math. Comput. 168(1) (2005), 465–478.10.1016/j.amc.2004.09.026Search in Google Scholar

[27] Ramdani, N. E.—Pinelas, S.: Solving nonlinear integro-differential equations using numerical method, Turkish J. Math. 46(2) (2022), 675–687.Search in Google Scholar

[28] Reutskiy, S. Yu.: The backward substitution method for multipoint problems with linear Volterra-Fredholm integro-differential equations of the neutral type, J. Comput. Appl. Math. 296 (2016), 724–738.10.1016/j.cam.2015.10.013Search in Google Scholar

[29] Rohaninasab, N.—Maleknejad, K.—Ezzati, R.: Numerical solution of high-order VolterraFredholm integro-differential equations by using Legendre collocation method, Appl. Math. Comput. 328 (2018), 171–188.10.1016/j.amc.2018.01.032Search in Google Scholar

[30] Saadatmandi, A.—Dehghan, M.: Numerical solution of the higher-order linear Fredholm integro-differential–difference equation with variable coefficients, Comput. Math. Appl. 59(8) (2010), 2996–3004.10.1016/j.camwa.2010.02.018Search in Google Scholar

[31] Samei, M. E.—Zanganeh, H.—Aydogan, S. M.: Investigation of a Class of the Singular Fractional Integro-differential Quantum Equations with Multi-Step Methods, J. Math. Ext. 15 (2021), 1–54.Search in Google Scholar

[32] Seba, D.—Rebai, H.—Henderson, J.: Existence result for nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions in Banach spaces, Georgian Math. J. 28 (2019), 1–7.10.1515/gmj-2019-2009Search in Google Scholar

[33] Shavlakadze, N.—Jokhadze, O.: Solutions of a singular integro-differential equation related to the adhesive contact problems of elasticity theory, Georgian Math. J. 29(2) (2021), 285–293.10.1515/gmj-2021-2126Search in Google Scholar

[34] Tavassoli Kajani, M.—Ghasemi, M.—Babolian, E.: Numerical solution of linear integro-differential equation by using sine–cosine wavelets, Appl. Math. Comput. 180(2) (2006), 569–574.10.1016/j.amc.2005.12.044Search in Google Scholar

[35] Wang, W.: An algorithm for solving the higher-order nonlinear Volterra-Fredholm integro-differential equation with mechanization, Appl. Math. Comput. 172(1) (2006), 1–23.10.1016/j.amc.2005.01.116Search in Google Scholar

[36] Xing, Y.—Jiao, F.—Liu, F.: On the generalization of a solution for a class of integro-differential equations with nonseparated integral boundary conditions, Math. Probl. Eng. 2020 (2020), Art. ID 8679465.10.1155/2020/8679465Search in Google Scholar

[37] Yalcinbas, S.—Sezer, M.: The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112(2–3) (2000), 291–308.10.1016/S0096-3003(99)00059-4Search in Google Scholar

[38] Yuldashev, T.: Spectral features of the solving of a Fredholm homogeneous integro-differential equation with integral conditions and reflecting deviation, Lobachevskii J. Math. 40 (2019), 2116–2123.10.1134/S1995080219120138Search in Google Scholar

[39] Yuldashev, T.: On the solvability of a boundary value problem for the ordinary Fredholm integrodifferential equation with a degenerate kernel, Comput. Math. Math. Phys. 59 (2019), 241–252.10.1134/S0965542519020167Search in Google Scholar

[40] Yusufoglu, E.: Improved homotopy perturbation method for solving Fredholm type integro-differential equations, Chaos Solitons Fractals 41(1) (2009), 28–37.10.1016/j.chaos.2007.11.005Search in Google Scholar

Received: 2023-06-12
Accepted: 2023-08-21
Published Online: 2024-05-24
Published in Print: 2024-04-25

© 2024 Mathematical Institute Slovak Academy of Sciences

Downloaded on 23.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2024-0031/html
Scroll to top button