Development and Implementation of a Top-order Multi-step Block Method for Boundary Value Problems: The advancement of numerical accuracy

This paper introduces the development and implementation of a Top-order Multi-step Block Method (TMBM) tailored for the numerical solution of third-order boundary value problems (BVPs) in ordinary differential equations (ODEs). Traditional numerical methods often face challenges in accurately solving such complex problems, especially in scenarios where higher-order accuracy and computational efficiency are paramount. The methodology adopts the principle of collocation and interpolation in its derivation. The TMBM combines the benefits of higher-order accuracy and block processing techniques to provide robust solutions while maintaining computational tractability. Through a comprehensive analysis and three numerical experiments, the efficacy and advantages of the TMBM are demonstrated, showcasing its potential for addressing a wide range of third-order BVPs encountered in various scientific and engineering applications.