Numerical assessment of fourth-order initial value problems using Butcher’s fifth-order Runge-Kutta method and a new iterative method

Authors

  • Abubakar Salihu Department of Mathematics, American University of Nigeria, Yola, Adamawa State, Nigeria https://orcid.org/0009-0008-3556-0354
  • Auwal Lawan Department of Mathematics, University of Verona, Italy
  • Dahiru Usman Department of Mathematics, Aliko Dangote University of Science and Technology, Wudil, Kano State, Nigeria
  • Ahmad Isma'il Department of Mathematics, Federal University Dutse, Jigawa State, Nigeria
  • Surajo Ahmad Department of Mathematics, Aliko Dangote University of Science and Technology, Wudil, Kano State, Nigeria
  • Musa Kasimu Abdallah Department of Mathematics, Aliko Dangote University of Science and Technology, Wudil, Kano State, Nigeria
  • Umar Abubakar Department of Mathematics, Aliko Dangote University of Science and Technology, Wudil, Kano State, Nigeria
  • Harisu Sulaiman Abubakar Department of Mathematics, Aliko Dangote University of Science and Technology, Wudil, Kano State, Nigeria

DOI:

https://doi.org/10.64497/jssci.108

Keywords:

Higher-order ordinary differential equations,, Runge–Kutta numerical schemes, Semi-analytical iterative methods, Error analysis, Numerical simulations, Initial value problems

Abstract

This paper presents a numerical assessment of linear homogeneous and non-homogeneous fourth-order initial value problems (IVPs) using Butcher’s fifth-order Runge–Kutta (BRK5) method and the New Iterative Method (NIM). The study highlights the importance of fourth- order ODEs in modeling real-world systems such as beam deflection, oscillations, and dynamic processes, where analytical solutions are often intractable. BRK5, known for its accuracy and stability, is compared with NIM, which avoids discretization and perturbation while providing rapidly convergent series solutions. Two numerical examples are investigated to evaluate the performance of both methods in terms of accuracy, stability, and error distribution, with results presented both graphically and in tabular form. The findings reveal that while BRK5 maintains higher accuracy and stability for small step sizes, NIM offers a flexible semi- analytical alternative with effective convergence properties, thereby providing complementary insights for solving higher-order IVPs in applied sciences and engineering.

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Author Biography

Abubakar Salihu, Department of Mathematics, American University of Nigeria, Yola, Adamawa State, Nigeria

Abubakar Salihu obtained his B.Sc. in Mathematics with First-Class Honours from Aliko Dangote University of Science and Technology, Wudil, Nigeria. His research focuses on numerical methods for differential equations, computational fluid dynamics, and applied mathematical modeling. He has published works in numerical analysis and is particularly interested in the development of efficient algorithms for solving oscillatory and nonlinear problems. He is currently serving as a Graduate Assistant in the Department of Mathematics at the American University of Nigeria, where he is engaged in teaching support and research activities. His future academic goal is to pursue graduate studies specializing in mathematical epidemiology and computational sciences.

References

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Published

2025-12-14

How to Cite

Salihu, A., Lawan, A., Usman, D., Isma'il, A., Ahmad, S., Kasimu Abdallah, M., … Sulaiman Abubakar, H. (2025). Numerical assessment of fourth-order initial value problems using Butcher’s fifth-order Runge-Kutta method and a new iterative method. Journal of Statistical Sciences and Computational Intelligence, 1(4), 383–395. https://doi.org/10.64497/jssci.108

Issue

Vol. 1 No. 4 (2025): Volume 1, Issue 4 (October – December 2025)

Section

Articles

License

Copyright (c) 2025 Abubakar Salihu, Auwal Lawan, Dahiru Usman, Ahmad Isma'il, Surajo Ahmad, Musa Kasimu Abdallah, Umar Abubakar, Harisu Sulaiman Abubakar

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

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