An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations

Duromola Monday Kolawole

doi:doi:10.11648/j.ajam.20251303.11

Research Article |

| Peer-Reviewed

An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations

Duromola Monday Kolawole^*

Published in American Journal of Applied Mathematics (Volume 13, Issue 3)

Received: 9 April 2025 Accepted: 23 April 2025 Published: 29 May 2025

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Furthering research on linear multistep hybrid block methods is essential to enhance accuracy, stability, and efficiency in solving ordinary differential equations, enabling advanced modelling of complex systems across science and engineering for better predictive analysis and real-world applications. In this study, an efficient linear multistep hybrid block method with a single-step and seven off-step points for the direct numerical integration of fifth-order initial value problems (IVPs) in ordinary differential equations (ODEs), eliminating the need for reduction to a system of first-order ODEs is proposed. The method is constructed using a collocation approach at both grid and off-grid points, alongside interpolation at five off-grid points, to approximate the solution via a power series polynomial. The resulting system of equations is solved to obtain the necessary discrete and additional formulae that constitute the block approach. A comprehensive theoretical analysis confirms that the method possesses desirable numerical properties, including a well-defined order, zero stability, consistency, convergence, and absolute stability. Comparative numerical experiments against existing methods demonstrate that the proposed approach achieves superior accuracy and efficiency, making it a promising tool for solving both linear and nonlinear fifth-order ODEs.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 3)
DOI	10.11648/j.ajam.20251303.11
Page(s)	174-193
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), 2025. Published by Science Publishing Group

Keywords

Power Series Polynomials, Grid Points, Off-grid Points, Convergence, Interpolation and Collocation

References

[1]	Lambert, J. D. (1973). Computational Methods in Ordinary Differential Equations. John Wiley, New York.
[2]	Fatunla, S. O., (1988). Numerical Methods for Initial value Problems in Ordinary Differential Equations. Academic Press Inc., New York.
[3]	Bun, R. A., and Vasil’Yer, Y. D., (1992). A Numerical Method for Solving Differential Equations of any orders. Comp. Math. Phys, 32, Pp 317-330.
[4]	Awoyemi, D. O (1999): A class of continuous Methods for general second order initial value problems in ordinary differential equations. International journal of computer mathematics 72; 29-39.
[5]	Kayode S. J. and Awoyemi D. O. (2010) A Multiderivative collocation method for 5th order ordinary differential equations. Journal of Mathematics and Statistics. 2010; 6(1): 60-63. https://doi.org/10.3844/jmssp.2010.60.63
[6]	Akbar M. A and Siddique T. A. (2010). Perturbation solutions of fifth order oscillatory non linear system. Nonlinear Anal. Mod. Control. 2011; 16(2): 123-134. https://doi.org/10.15388/na.16.2.14100
[7]	Olabode B. T. (2013) A block predictor-corrector method for the direct solution of general fifth order ordinary differential equations. Canadian Journal on Computing in Mathematics, Natural Sciences, Engineering and Medicine. 2013; 4(1): 133-139.
[8]	Kayode S. J. (2014) Symmetric implicit multiderivative numerical integrators for direct solution of fifth-order differential equations. Thammasat International Journal of Science and Technology. 2014; 19(2): 1-8.
[9]	Mechee M. S. and Kadhim M. A. (2016) Explicit Direct Integrators of RK Type for Solving Special Fifth-Order Ordinary Differential Equations. American Journal of Applied Sciences. 2016; 13(12): 1452-1460. https://doi.org/10.3844/ajassp.2016.1452.1460
[10]	Turki MY., Alku S. Y. and Mechee M. S. (2022) The general implicit-block method with two-points and extra derivatives for solving fifth-order ordinary differential equations. International J. Nonlinear Anal. Appl. 2021; 13(1): 1081-1097. https://doi.org/10.22075/ijnaa.2022.5650
[11]	M. K. Duromola, A. L. Momoh and J. M. Adeleke (2022). One-step Hybrid Block Method for Directly Solving Fifth-order Initial Value Problems of Ordinary Differential Equations. Asian Research Journal of Mathematics 18(1): Pp 53-64. https://doi.org/10.9734/arjom/2022/v18i130354
[12]	Duromola M. K. (2022) Single-Step Block Method of P-Stable for Solving Third Order Differential Equations (IVPs): Ninth Order of Accuracy. American Journal of Applied Mathematics and Statistics, 2022, Vol. 10, No. 1. Pp 4-13 http://pubs.sciepub.com/ajams/10/1/2
[13]	Duromola Monday Kolawole, Momoh Adelegan Lukuman and Akingbodi Oluwagbenga Joseph (2025). A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs. American Journal of Applied Mathematics. 13(2): 103-116. https://doi.org/10.11648/j.ajam.20251302.11
[14]	M. K. Duromola. A. L. Momoh and O. J. Akingbodi (2024). A Chebyshev Generated Block Method for Directly Solving Nonlinear and Ill-posed Fourth-Order ODEs. Earthline Journal of Mathematical Sciences, 14(6), 1267-1292. https://doi.org/10.34198/ejms.14624.12671292
[15]	D. O. Awoyemi, E. A. Adebile, A. O. Adesanya and T. A. Anake (2011). Modified block method for the direct solution of second order ordinary differential equations. International Journal of Applied Mathematics and Computation. Volume 3(3), pp 181 –188.
[16]	Y. A. Yahaya and A. M. Badmus (2009). A class of collocation for General Second Order Ordinary Differential Equations. African Journal of Mathematics and Computer Science Research, 2(4): 069-072.
[17]	Fatunla, S. O. (1991). Block method for Second Order IVPs. International Journal of Computer Mathematics, 41(9); 55-63.
[18]	Henrici P. (1962). Discrete Variable Method in Ordinary Differential Equations. John Wiley and Sons, New York.
[19]	Roseline Bosede Ogunrinde, Ololade Funmilayo Fayose and Taiwo Stephen Fayose (2021). Optimization of one-step hybrid method for direct solution of fifth order ordinary differential equations of initial value problems. World Journal of Advanced Research and Reviews, 2021, 09(01), 239–249. https://doi.org/10.30574/wjarr.2021.9.1.0012
[20]	Higinio Ramos and Adelegan Lukuman Momoh (2023). A Tenth - Order Sixth Derivative Block Method for Directly Solving Fifth-Order Initial Value Problems. International Journal of Computational Methods. World Scientific Publishing Company. https://doi.org/10.1142/S0219876223500111
[21]	Nazreen Waeleh and Zanariah Abdul Majid (2016) A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly. International Journal of Mathematics and Mathematical Sciences Volume 2016, Article ID 9823147, 8 pages. http://dx.doi.org/10.1155/2016/9823147
[22]	Animasaun I. L., Muhammad, T., & Yook, S.-J. (2025) Exploration of Half-Cycle Length of Converging Circular Wavy Duct with Diverging-Outlet: Turbulent Water Dynamics. Advanced Theory and Simulations 2025, 2500038. https://doi.org/10.1002/adts.202500038
[23]	Li, L., Animasaun, I. L., Koriko, O. K., Muhammad, T., & Elnaqeeb, T. (2024). Insight into turbulent Reynolds number at the regular, converging, and diverging outlets: Dynamics of air, water, and kerosene through y-shaped cylindrical copper ducts. International Communications in Heat and Mass Transfer, 159, 108044. https://doi.org/10.1016/j.icheatmasstransfer.2024.108044
[24]	Wang, F., Animasaun, I. L., Al-Mdallal, Q. M., Saranya, S., & Muhammad, T. (2023). Dynamics through three-inlets of t-shaped ducts: Significance of inlet velocity on transient air and water experiencing cold fronts subject to turbulence. International Communications in Heat and Mass Transfer, 148, 107034. https://doi.org/10.1016/j.icheatmasstransfer.2023.10703

Cite This Article

Plain Text BibTeX RIS

APA Style

Kolawole, D. M. (2025). An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations. American Journal of Applied Mathematics, 13(3), 174-193. https://doi.org/10.11648/j.ajam.20251303.11

Copy | Download

ACS Style

Kolawole, D. M. An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations. Am. J. Appl. Math. 2025, 13(3), 174-193. doi: 10.11648/j.ajam.20251303.11

Copy | Download

AMA Style

Kolawole DM. An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations. Am J Appl Math. 2025;13(3):174-193. doi: 10.11648/j.ajam.20251303.11

Copy | Download

@article{10.11648/j.ajam.20251303.11,
  author = {Duromola Monday Kolawole},
  title = {An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations
},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {3},
  pages = {174-193},
  doi = {10.11648/j.ajam.20251303.11},
  url = {https://doi.org/10.11648/j.ajam.20251303.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251303.11},
  abstract = {Furthering research on linear multistep hybrid block methods is essential to enhance accuracy, stability, and efficiency in solving ordinary differential equations, enabling advanced modelling of complex systems across science and engineering for better predictive analysis and real-world applications. In this study, an efficient linear multistep hybrid block method with a single-step and seven off-step points for the direct numerical integration of fifth-order initial value problems (IVPs) in ordinary differential equations (ODEs), eliminating the need for reduction to a system of first-order ODEs is proposed. The method is constructed using a collocation approach at both grid and off-grid points, alongside interpolation at five off-grid points, to approximate the solution via a power series polynomial. The resulting system of equations is solved to obtain the necessary discrete and additional formulae that constitute the block approach. A comprehensive theoretical analysis confirms that the method possesses desirable numerical properties, including a well-defined order, zero stability, consistency, convergence, and absolute stability. Comparative numerical experiments against existing methods demonstrate that the proposed approach achieves superior accuracy and efficiency, making it a promising tool for solving both linear and nonlinear fifth-order ODEs.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations

AU  - Duromola Monday Kolawole
Y1  - 2025/05/29
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251303.11
DO  - 10.11648/j.ajam.20251303.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 174
EP  - 193
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251303.11
AB  - Furthering research on linear multistep hybrid block methods is essential to enhance accuracy, stability, and efficiency in solving ordinary differential equations, enabling advanced modelling of complex systems across science and engineering for better predictive analysis and real-world applications. In this study, an efficient linear multistep hybrid block method with a single-step and seven off-step points for the direct numerical integration of fifth-order initial value problems (IVPs) in ordinary differential equations (ODEs), eliminating the need for reduction to a system of first-order ODEs is proposed. The method is constructed using a collocation approach at both grid and off-grid points, alongside interpolation at five off-grid points, to approximate the solution via a power series polynomial. The resulting system of equations is solved to obtain the necessary discrete and additional formulae that constitute the block approach. A comprehensive theoretical analysis confirms that the method possesses desirable numerical properties, including a well-defined order, zero stability, consistency, convergence, and absolute stability. Comparative numerical experiments against existing methods demonstrate that the proposed approach achieves superior accuracy and efficiency, making it a promising tool for solving both linear and nonlinear fifth-order ODEs.

VL  - 13
IS  - 3
ER  -

Copy | Download

Author Information

Duromola Monday Kolawole

Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria

Contact Email

http://orcid.org/0000-0002-7117-8418

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Kolawole, D. M. (2025). An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations. American Journal of Applied Mathematics, 13(3), 174-193. https://doi.org/10.11648/j.ajam.20251303.11

Copy | Download

ACS Style

Kolawole, D. M. An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations. Am. J. Appl. Math. 2025, 13(3), 174-193. doi: 10.11648/j.ajam.20251303.11

Copy | Download

AMA Style

Kolawole DM. An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations. Am J Appl Math. 2025;13(3):174-193. doi: 10.11648/j.ajam.20251303.11

Copy | Download

@article{10.11648/j.ajam.20251303.11,
  author = {Duromola Monday Kolawole},
  title = {An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations
},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {3},
  pages = {174-193},
  doi = {10.11648/j.ajam.20251303.11},
  url = {https://doi.org/10.11648/j.ajam.20251303.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251303.11},
  abstract = {Furthering research on linear multistep hybrid block methods is essential to enhance accuracy, stability, and efficiency in solving ordinary differential equations, enabling advanced modelling of complex systems across science and engineering for better predictive analysis and real-world applications. In this study, an efficient linear multistep hybrid block method with a single-step and seven off-step points for the direct numerical integration of fifth-order initial value problems (IVPs) in ordinary differential equations (ODEs), eliminating the need for reduction to a system of first-order ODEs is proposed. The method is constructed using a collocation approach at both grid and off-grid points, alongside interpolation at five off-grid points, to approximate the solution via a power series polynomial. The resulting system of equations is solved to obtain the necessary discrete and additional formulae that constitute the block approach. A comprehensive theoretical analysis confirms that the method possesses desirable numerical properties, including a well-defined order, zero stability, consistency, convergence, and absolute stability. Comparative numerical experiments against existing methods demonstrate that the proposed approach achieves superior accuracy and efficiency, making it a promising tool for solving both linear and nonlinear fifth-order ODEs.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations

AU  - Duromola Monday Kolawole
Y1  - 2025/05/29
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251303.11
DO  - 10.11648/j.ajam.20251303.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 174
EP  - 193
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251303.11
AB  - Furthering research on linear multistep hybrid block methods is essential to enhance accuracy, stability, and efficiency in solving ordinary differential equations, enabling advanced modelling of complex systems across science and engineering for better predictive analysis and real-world applications. In this study, an efficient linear multistep hybrid block method with a single-step and seven off-step points for the direct numerical integration of fifth-order initial value problems (IVPs) in ordinary differential equations (ODEs), eliminating the need for reduction to a system of first-order ODEs is proposed. The method is constructed using a collocation approach at both grid and off-grid points, alongside interpolation at five off-grid points, to approximate the solution via a power series polynomial. The resulting system of equations is solved to obtain the necessary discrete and additional formulae that constitute the block approach. A comprehensive theoretical analysis confirms that the method possesses desirable numerical properties, including a well-defined order, zero stability, consistency, convergence, and absolute stability. Comparative numerical experiments against existing methods demonstrate that the proposed approach achieves superior accuracy and efficiency, making it a promising tool for solving both linear and nonlinear fifth-order ODEs.

VL  - 13
IS  - 3
ER  -

Copy | Download