An Efficient Approach for Solving Two-point Boundary Value Problems of Ordinary Differential Equations

Abstract

This study presents an efficient self-starting, single-step block hybrid technique for solving two-point boundary value problems (BVPs) in ordinary differential equations (ODEs). The proposed method integrates an optimized approach with higher derivative functions to address key challenges in two-point BVPs while ensuring essential numerical properties such as zero stability, consistency, and convergence. A polynomial function serves as the approximate solution, developed using interpolation and multistep collocation techniques. By interpolating at one point and collocating at all points, the scheme enhances robustness. The method was validated on six strongly nonlinear two-point BVPs, demonstrating reliability and suitability for solving such problems. Comparative analysis with existing techniques highlights its superior accuracy and efficiency. Ultimately, this study introduces a robust computational approach that not only enhances precision but also streamlines the solution process for two-point BVPs in ODEs, marking a significant advancement in numerical methods.