Research Article
An Optimized hybrid one step method for numerical integration of second order initial value problems
- By Oluwatoyin Ayodele Edwin, Emmanuel Adegbemiro Areo, Sunday Jacob Kayode, Adebayo Oludare Adeniran - 22 May 2026
- Applied Mathematics on Science and Engineering, Volume: 3(2026), Issue: 1, Pages: 1 - 15
- https://doi.org/10.65036/amse311
- Received: 24.02.2026; Accepted: 08.05.2026; Published: 22.05.2026
Abstract: This article presents an optimized hybrid one-step block method for the direct numerical integration of second-order initial value problems (IVPs) of the form y′′ = f (x,y,y′). The method is derived using a power series basis function through a collocation and interpolation technique, allowing for direct integration without the need to reduce the second-order equation into a system of first-order ordinary differential equations. To maximize the accuracy of the scheme, the off-grid collocation points are optimized by analyzing the local truncation error (LTE). By forcing the leading terms of the error expansion to zero, we determined the optimal positions for the hybrid points at c = 1/2 − √21/14 and b = 12+√21/14 . The resulting hybrid block method achieves a high local truncation error of O(h8), ensuring superior precision and rapid convergence. The continuous formulation of the method facilitates the simultaneous generation of approximate solutions at both grid and off-grid nodes. The consistency and efficiency of the proposed scheme make it a robust computational tool for solving dynamical problems in fields such as physics, engineering, and biology.
Author's affiliation:
Oluwatoyin Ayodele Edwin (ORCID), Federal University of Technology, Department of Mathematical Sciences, Akure, Nigeria.
Emmanuel Adegbemiro Areo (ORCID), Federal University of Technology, Department of Mathematical Sciences, Akure, Nigeria.
Sunday Jacob Kayode (ORCID), Federal University of Technology, Department of Mathematical Sciences, Akure, Nigeria.
Adebayo Oludare Adeniran (ORCID), Federal Polytechnic, Department of General Studies, Ile Oluji, Nigeria.
How To Cite: O.A. Edwin, E.A. Areo, S.J. Kayode, A.O. Adeniran. An Optimized hybrid one step method for numerical integration of second order initial value problems. Applied Mathematics on Science and Engineering, 3(1):1–15, 2026. https://doi.org/10.65036/amse311