Abstract
In this paper, we consider a class of delay differential systems with two-point boundary value conditions. First, we give a criterion to guarantee the existence and uniqueness of solutions for the system. Then, by making use of lower order Newton–Cotes rules, we present a computer-oriented iterative scheme to approximate the exact solution. The efficiency of algorithm is verified by two examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agarwal, R.P., Chow, Y.M.: Finite difference methods for boundary-value problems of differential equations with deviating arguments. Comput. Math. Appl. 12A(11), 1143–1153 (1986)
Aziz, I., Amin, R.: Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet. Appl. Math. Model. 40(23–24), 10286–10299 (2016)
Bica, A.M., Curila, M., Curila, S.: Two-point boundary value problems associated to functional differential equations of even order solved by iterated splines. Appl. Numer. Math. 110, 128–147 (2016)
Engelborghs, K., Luzyanina, T., IN ’T Hout, K.J., Roose, D.: Collocation methods for the computation of periodic solutions of delay differential equations. SIAM J. Sci. Comput. 22(5), 1593–1609 (2000)
Fridman, E., Fridman, L., Shustin, E.: Steady modes in relay control systems with time delay and periodic disturbances. J. Dyn. Sys. Meas. Control 122, 732–737 (2000)
Garey, L.E., Gladwin, C.J.: Numerical methods for second order Volterra integro-differential equations with two-point boundary conditions. Utilitas Math. 35, 103–109 (1989)
Khuri, S.A., Sayfy, A.: Numerical solution of functional differential equations: a Green’s function-based iterative approach. Int. J. Comput. Math. 95(10), 1937–1949 (2018)
Kress, R.: Numerical Analysis. Springer, New York (1998)
Lakshmanan, M., Senthilkumar, D.V.: Dynamics of Nonlinear Time-Delay Systems. Springer Series Synergetics, Springer, Berlin (2010)
Ma, W., Saito, Y., Takeuchi, Y.: \(M\)-matrix structure and harmless delays in a Hopfield-type neural network. Appl. Math. Lett. 22, 1066–1070 (2009)
Singh, M., Verma, A., Agarwal, R.P.: On an iterative method for a class of 2 point & 3 point nonlinear SBVPs. J. Appl. Anal. Comput. 9(4), 1242–1260 (2019)
Tewarson, R.P., Huslak, N.S.: An adaptive implementation of interpolation methods for boundary value ordinary differential equations. BIT 23(3), 382–387 (1983)
Walter, W.: Ordinary Differential Equations. Springer, New York (1998)
Acknowledgements
The authors would like to thank the editors and the referees for their very helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by the National Natural Science Foundation of China (No. 12071491)
Rights and permissions
About this article
Cite this article
Zhu, ZQ., Wang, QR. The Newton–Cotes quadratures for solving a delay differential system. J. Appl. Math. Comput. 68, 4589–4604 (2022). https://doi.org/10.1007/s12190-022-01718-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-022-01718-x
Keywords
- Delay differential systems
- Two-point boundary value problems
- Existence and uniqueness
- Numerical solutions
- The Newton–Cotes Quadratures
- Green functions