Three Stage MERK Methods for Delay Differential Equations

Olaniyan Adegoke Stephen; Aleshinloye Zainab Labi; Omolara Fatimah Bakre; Akanbi Moses Adebowale

doi:doi:10.11648/j.ijamtp.20261202.11

Research Article |

| Peer-Reviewed

Three Stage MERK Methods for Delay Differential Equations

Olaniyan Adegoke Stephen^*

, Aleshinloye Zainab Labi

, Omolara Fatimah Bakre

, Akanbi Moses Adebowale

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 12, Issue 2)

Received: 14 March 2026 Accepted: 26 March 2026 Published: 26 May 2026

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The application of Runge- Kutta (R-K) methods in obtaining numerical solutions to a wide class of ordinary and partial differential equations arising in various fields of applied sciences has been widely documented. Their efficiency, accuracy, and ease of implementation have made them one of the most popular techniques for solving initial value problems. This success has motivated their extension to more complex systems, particularly Delay Differential Equations (DDEs), which naturally arise in modeling real-life phenomena where time delays are inherent, such as in population dynamics, control systems, epidemiology, and engineering processes. In this paper, we present a numerical approach for solving DDEs by adapting a three-stage Multiderivative Explicit Runge-Kutta (MERK) method. The presence of delayed arguments in DDEs introduces additional computational challenges, especially in the evaluation of past states. To address this, Lagrange interpolation is employed to approximate the delayed terms. Furthermore, the stability properties of the proposed method are investigated through the derivation of the associated stability polynomials. These polynomials provide insight into the convergence behavior and robustness of the methods when applied to stiff and non-stiff delay systems. The performance of these methods are analyzed by solving DDEs, and comparisons are made with existing methods in the literature. The results demonstrate reliability of the three-stage MERK methods for solving DDEs.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 12, Issue 2)
DOI	10.11648/j.ijamtp.20261202.11
Page(s)	65-69
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Delay Differential Equations, Runge-Kutta Methods , MERK Methods, Lagrange Interpolation, Stability Region

References

[1]	Akanbi M. A. (2005). A Family of 2-Stage Multiderivative Explicit Runge-Kutta Methods for Nonstiff Ordinary Differential Equations. Proceedings of NMC on Recent Development in Numerical Methods for Solving Ordinary Differential Equations , 6(2005), 87-100.
[2]	Barewell V. K. (1975). Special Stability Problems for Functional Differential Equations. BIT, Vol. 15, pp: 130-135.
[3]	Butcher J. C.(1964). On Runge-Kutta processes of high Order. J. Austral. Math. Soc., Vol. 4, pp: 39–41.
[4]	Goeken D. and Johnson O. (1999). Fifth-order Runge-Kutta with Higher Order Derivative Approximations. Electronic Journal of Differential Equations, pp. 1-9.
[5]	Ismail F., Read A. A. (2002). Numerical Treatment of Delay Differential Equations by Runge-Kutta Method Using Hermite Interpolation. Matematika, No 2, 79-90.
[6]	Ismail F. and Suleiman M. B.(1988). Embedded Singly Diagonally Implicit Runge-Kutta Method. Int. J. Comput. Maths, Vol. 66, pp: 325-341.
[7]	Kumar D. C. and Pushpam E. K.(2020). Two Stage Multiderivative Explicit Runge-Kutta Methods for Delay Differential Equations. Advances in Mathematics: Scientific Journal, No. 3 1293-1299.
[8]	Lambert J. D. (1973). Computational Methods in Ordinary Differential Equations, Jojn Wiley and Sons Inc., New York.
[9]	Olaniyan A. S., Bakre O. F., Akanbi M. A. (2020). A 2- Stage Implicit Runge-Kutta method based on heronian mean for solving ordinary Differential Equations. Pure and Applied Mathematics Journal, Vol. 9(5).
[10]	Olaniyan A. S., Akanbi M. A., Wusu A. S., Shonibare K. A. (2024). A Four-Stage Multiderivative Explicit Runge-Kutta Method for the Solution of First Order Ordinary Differential EquationsAnnals of Mathematics and Computer Science, 20, 72-81.
[11]	Olaniyan A. S., Kazeem M. T., Aweda A. A. (2025). Numerical Solution of Delay Differential Equations with Heronian Implicit Runge-Kutta Method.International Journal of Research and Innovation in Applied Science (IJRIAS), Vol. 9, Pages: 326-330.
[12]	Wusu A. S., Akanbi M. A., Okunuga S. A.(2013). A Three-Stage Multiderivative Explicit Runge-Kutta Method. American Journal of Computational Mathematics, 3, 121-126.

Cite This Article

Plain Text BibTeX RIS

APA Style

Stephen, O. A., Labi, A. Z., Bakre, O. F., Adebowale, A. M. (2026). Three Stage MERK Methods for Delay Differential Equations. International Journal of Applied Mathematics and Theoretical Physics, 12(2), 65-69. https://doi.org/10.11648/j.ijamtp.20261202.11

Copy | Download

ACS Style

Stephen, O. A.; Labi, A. Z.; Bakre, O. F.; Adebowale, A. M. Three Stage MERK Methods for Delay Differential Equations. Int. J. Appl. Math. Theor. Phys. 2026, 12(2), 65-69. doi: 10.11648/j.ijamtp.20261202.11

Copy | Download

AMA Style

Stephen OA, Labi AZ, Bakre OF, Adebowale AM. Three Stage MERK Methods for Delay Differential Equations. Int J Appl Math Theor Phys. 2026;12(2):65-69. doi: 10.11648/j.ijamtp.20261202.11

Copy | Download

@article{10.11648/j.ijamtp.20261202.11,
  author = {Olaniyan Adegoke Stephen and Aleshinloye Zainab Labi and Omolara Fatimah Bakre and Akanbi Moses Adebowale},
  title = {Three Stage MERK Methods for Delay Differential Equations},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {12},
  number = {2},
  pages = {65-69},
  doi = {10.11648/j.ijamtp.20261202.11},
  url = {https://doi.org/10.11648/j.ijamtp.20261202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20261202.11},
  abstract = {The application of Runge- Kutta (R-K) methods in obtaining numerical solutions to a wide class of ordinary and partial differential equations arising in various fields of applied sciences has been widely documented. Their efficiency, accuracy, and ease of implementation have made them one of the most popular techniques for solving initial value problems. This success has motivated their extension to more complex systems, particularly Delay Differential Equations (DDEs), which naturally arise in modeling real-life phenomena where time delays are inherent, such as in population dynamics, control systems, epidemiology, and engineering processes. In this paper, we present a numerical approach for solving DDEs by adapting a three-stage Multiderivative Explicit Runge-Kutta (MERK) method. The presence of delayed arguments in DDEs introduces additional computational challenges, especially in the evaluation of past states. To address this, Lagrange interpolation is employed to approximate the delayed terms. Furthermore, the stability properties of the proposed method are investigated through the derivation of the associated stability polynomials. These polynomials provide insight into the convergence behavior and robustness of the methods when applied to stiff and non-stiff delay systems. The performance of these methods are analyzed by solving DDEs, and comparisons are made with existing methods in the literature. The results demonstrate reliability of the three-stage MERK methods for solving DDEs.
},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Three Stage MERK Methods for Delay Differential Equations
AU  - Olaniyan Adegoke Stephen
AU  - Aleshinloye Zainab Labi
AU  - Omolara Fatimah Bakre
AU  - Akanbi Moses Adebowale
Y1  - 2026/05/26
PY  - 2026
N1  - https://doi.org/10.11648/j.ijamtp.20261202.11
DO  - 10.11648/j.ijamtp.20261202.11
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 65
EP  - 69
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20261202.11
AB  - The application of Runge- Kutta (R-K) methods in obtaining numerical solutions to a wide class of ordinary and partial differential equations arising in various fields of applied sciences has been widely documented. Their efficiency, accuracy, and ease of implementation have made them one of the most popular techniques for solving initial value problems. This success has motivated their extension to more complex systems, particularly Delay Differential Equations (DDEs), which naturally arise in modeling real-life phenomena where time delays are inherent, such as in population dynamics, control systems, epidemiology, and engineering processes. In this paper, we present a numerical approach for solving DDEs by adapting a three-stage Multiderivative Explicit Runge-Kutta (MERK) method. The presence of delayed arguments in DDEs introduces additional computational challenges, especially in the evaluation of past states. To address this, Lagrange interpolation is employed to approximate the delayed terms. Furthermore, the stability properties of the proposed method are investigated through the derivation of the associated stability polynomials. These polynomials provide insight into the convergence behavior and robustness of the methods when applied to stiff and non-stiff delay systems. The performance of these methods are analyzed by solving DDEs, and comparisons are made with existing methods in the literature. The results demonstrate reliability of the three-stage MERK methods for solving DDEs.

VL  - 12
IS  - 2
ER  -

Copy | Download

Author Information

Olaniyan Adegoke Stephen

Department of Mathematics, Lagos State University, Lagos, Nigeria

Contact Email

http://orcid.org/0000-0002-5784-5456
Aleshinloye Zainab Labi

Department of Mathematics, Lagos State University, Lagos, Nigeria

Contact Email

http://orcid.org/0009-0008-7245-1069
Omolara Fatimah Bakre

Department of Mathematics & Statistics, Federal College of Education(Technical), Akoka, Lagos, Nigeria

Contact Email

http://orcid.org/0000-0001-9122-6707
Akanbi Moses Adebowale

Department of Mathematics, Lagos State University, Lagos, Nigeria

Contact Email

http://orcid.org/0000-0002-5260-3060

Download PDF

Submit an Article

Sections

Plain Text BibTeX RIS

APA Style

Stephen, O. A., Labi, A. Z., Bakre, O. F., Adebowale, A. M. (2026). Three Stage MERK Methods for Delay Differential Equations. International Journal of Applied Mathematics and Theoretical Physics, 12(2), 65-69. https://doi.org/10.11648/j.ijamtp.20261202.11

Copy | Download

ACS Style

Stephen, O. A.; Labi, A. Z.; Bakre, O. F.; Adebowale, A. M. Three Stage MERK Methods for Delay Differential Equations. Int. J. Appl. Math. Theor. Phys. 2026, 12(2), 65-69. doi: 10.11648/j.ijamtp.20261202.11

Copy | Download

AMA Style

Stephen OA, Labi AZ, Bakre OF, Adebowale AM. Three Stage MERK Methods for Delay Differential Equations. Int J Appl Math Theor Phys. 2026;12(2):65-69. doi: 10.11648/j.ijamtp.20261202.11

Copy | Download

@article{10.11648/j.ijamtp.20261202.11,
  author = {Olaniyan Adegoke Stephen and Aleshinloye Zainab Labi and Omolara Fatimah Bakre and Akanbi Moses Adebowale},
  title = {Three Stage MERK Methods for Delay Differential Equations},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {12},
  number = {2},
  pages = {65-69},
  doi = {10.11648/j.ijamtp.20261202.11},
  url = {https://doi.org/10.11648/j.ijamtp.20261202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20261202.11},
  abstract = {The application of Runge- Kutta (R-K) methods in obtaining numerical solutions to a wide class of ordinary and partial differential equations arising in various fields of applied sciences has been widely documented. Their efficiency, accuracy, and ease of implementation have made them one of the most popular techniques for solving initial value problems. This success has motivated their extension to more complex systems, particularly Delay Differential Equations (DDEs), which naturally arise in modeling real-life phenomena where time delays are inherent, such as in population dynamics, control systems, epidemiology, and engineering processes. In this paper, we present a numerical approach for solving DDEs by adapting a three-stage Multiderivative Explicit Runge-Kutta (MERK) method. The presence of delayed arguments in DDEs introduces additional computational challenges, especially in the evaluation of past states. To address this, Lagrange interpolation is employed to approximate the delayed terms. Furthermore, the stability properties of the proposed method are investigated through the derivation of the associated stability polynomials. These polynomials provide insight into the convergence behavior and robustness of the methods when applied to stiff and non-stiff delay systems. The performance of these methods are analyzed by solving DDEs, and comparisons are made with existing methods in the literature. The results demonstrate reliability of the three-stage MERK methods for solving DDEs.
},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Three Stage MERK Methods for Delay Differential Equations
AU  - Olaniyan Adegoke Stephen
AU  - Aleshinloye Zainab Labi
AU  - Omolara Fatimah Bakre
AU  - Akanbi Moses Adebowale
Y1  - 2026/05/26
PY  - 2026
N1  - https://doi.org/10.11648/j.ijamtp.20261202.11
DO  - 10.11648/j.ijamtp.20261202.11
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 65
EP  - 69
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20261202.11
AB  - The application of Runge- Kutta (R-K) methods in obtaining numerical solutions to a wide class of ordinary and partial differential equations arising in various fields of applied sciences has been widely documented. Their efficiency, accuracy, and ease of implementation have made them one of the most popular techniques for solving initial value problems. This success has motivated their extension to more complex systems, particularly Delay Differential Equations (DDEs), which naturally arise in modeling real-life phenomena where time delays are inherent, such as in population dynamics, control systems, epidemiology, and engineering processes. In this paper, we present a numerical approach for solving DDEs by adapting a three-stage Multiderivative Explicit Runge-Kutta (MERK) method. The presence of delayed arguments in DDEs introduces additional computational challenges, especially in the evaluation of past states. To address this, Lagrange interpolation is employed to approximate the delayed terms. Furthermore, the stability properties of the proposed method are investigated through the derivation of the associated stability polynomials. These polynomials provide insight into the convergence behavior and robustness of the methods when applied to stiff and non-stiff delay systems. The performance of these methods are analyzed by solving DDEs, and comparisons are made with existing methods in the literature. The results demonstrate reliability of the three-stage MERK methods for solving DDEs.

VL  - 12
IS  - 2
ER  -

Copy | Download