Adams-Type Block Hybrid Method for Direct Solutions of Second-Order Differential Equations
DOI:
https://doi.org/10.62054/ijdm/0301.04Abstract
In this study, an eighth-order Adams-Type Block Hybrid Method (ATBHM) is derived for the solutions of second-order ordinary differential equations. The two-step method was developed using collocation and interpolation techniques, using five off-step points within the interval of integration to enhance accuracy. In contrast to most existing methods that reduce higher-order differential equations to first-order systems, often resulting in loss of inherent characteristics, the proposed ATBHM maintains essential properties of the equations by solving them directly. Basic stability and convergence properties of the ATBHM were also analysed to check its computational reliability in handling second-order ordinary differential equations. The results of the analysis showed that the method is consistent, zero-stable and convergent. Additionally, the method was tested on some problems and the results obtained show that it performed better than existing methods with which we compared our results.
References
Adeyeye, O. & Zurmi, O. (2016). Maximal order block method for the solution of second order ordinary differential equations. IAENG Int. J. Appl. Math., 46, 4.
Areo, E.A., Adeyanju, N.O. & Kayode, S.J. (2020). Direct solution of second order ordinary differential equations using a class of hybrid block methods. FUOYE Journal of Engineering and Technology (FUOYEJET), 5(2), 48-53. https://doi.org/10.46792/fuoyejet.v5i2.537.
Anemee, N.N.M.F. & Latif, N.A. (2022). The effectiveness of differential transform method on solving the Duffing equation. Enhanced Knowledge in Sciences and Technology, 2(1), 375-384.
James, A. A., Adesanya, A. O. & Sunday, J. (2012). Uniform Order Continuous Block Hybrid Method for the Solution of First Order Ordinary Differential Equations. IOSR Journal of Mathematics, 3(6): 8-14.
Jator, S.N., Ramos, H. & Modebei, M.I. (2023). A 14-order hybrid block method in variable step-size mode for solving second-order initial value problems. Mathematical Problems in Engineering, 2023, 5754475, 11 pages.
Lambert, J.D. (1973). Computational methods in ordinary differential equations. John Willey
and Sons, New York.
Lambert, J.D. (1991). Numerical methods for ordinary differential systems: The initial value
problem. John Wiley and Sons LTD, United Kingdom.
Fatunla, S.O. (1988). Numerical methods for initial value problems in ordinary differential
equations. Academic Press Inc., New York.
Nazreen, W. & A.M. Zanariah, A.M. (2017). Numerical algorithm of block method for general
second order ODEs using variable step size. Sains Malaysiana, 46(5), 817-824, 2017.
https://doi.org/10.17576/jsm-2017-4605-16
Obarhua, F.O. & Kayode, S.J. (2020). Continuous explicit hybrid method for solving second order ordinary differential equations. Pure Appl. Math. J., 9, 26.
Onyekonwu, A.C., Sunday, J. & Ndam, J.N. (2024). Formulation of Variable Step Block Backward Differentiation Formula for the Computation of Nonlinear Fuzzy Differential Equations. International Journal of Development Mathematics, 1(4), 38-60. https://doi.org/10.62054/ijdm/0104.04
Rasedee, A.F.N., Sathar, M.H.A., Ishak, N., Kamarudin, N.S., Nazri, M.A., Ramli, N.A., Ismail, I. & Sahrim, M. (2017). Solution for nonlinear Duffing oscillator using variable order variable stepsize block method. Matematika, 33(2), 165-175.
Skwame, Y., Sunday, J. & Sabo, J. (2018). On the Development of Two-step Implicit Second Derivative Block Methods for the Solution of Initial Value Problems of General Second Order ODEs. Journal of Scientific and Engineering Research, 5(3): 283-290.
Soraya, H. (2016). Pendulum with aerodynamic and viscous damping. J. Appl. Inf. Commun.
Technol. 2016, 3, 43–47.
Sunday, J., James, A. A., Odekunle, M. R. & Adesanya, A. O. (2015). Chebyshevian Basis Function-Type Block Method for the Solution of First-Order Initial Value Problems with Oscillating Solution. Journal of Mathematical and Computational Science, 5(4): 462-472.
Sunday, J., Kolawole, F. M., Ibijola, E. A. & Ogunrinde, R. B. (2015). Two-Step Laguerre Polynomial Hybrid Block Method for Stiff and Oscillatory First-Order ODEs. Journal of Mathematical and Computational Sciences, 5(5): 658-668.
Sunday, J., Kwanamu, J.A., Omole, E.O., Familua, A.B., Ajewole, K.P. & Amoyedo, F.E. (2025). An Eighth-Order Block Hybrid Integrator for the Simulations of Second-Order Differential Systems. 2025 IEEE International Conference of Science, Engineering and Business for Driving Sustainable Development Goals (SEB4SDG2025), Omu-Aran, Nigeria; NIPES-Journal of Science and Technology Research, 7(2), 2850-2857.
Sunday, J., Kwari, L.J., Sarjiyus, O., Omole, E.O., Oladimeji, S.O. & Amoyedo, F.E. (2025). Self-Starting Three-Step Algorithm for the Direct Computation of Second-Order Problems with Oscillating Solutions. 2025 IEEE International Conference of Science, Engineering and Business for Driving Sustainable Development Goals (SEB4SDG2025), Omu-Aran, Nigeria; NIPES-Journal of Science and Technology Research, 7(2), 2841-2849.
Yakusak, N.S. & Adeniyi, R.B. (2022). Hybrid Falkner-type block methods for the solution of second order boundary value problems. African Journal of Mathematics and Statistics Studies, 5(1), 67-81, 2022. https://doi.org/ 10.52589/AJMSS-I01PYJA7.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Joshua A. Kwanamu (Author)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors are solely responsible for obtaining permission to reproduce any copyrighted material contained in the manuscript as submitted. Any instance of possible prior publication in any form must be disclosed at the time the manuscript is submitted and a
copy or link to the publication must be provided.
The Journal articles are open access and are distributed under the terms of the Creative
Commons Attribution-NonCommercial-NoDerivs 4.0 IGO License, which permits use,
distribution, and reproduction in any medium, provided the original work is properly cited.
No modifications or commercial use of the articles are permitted.
