Analysis of a Block Method Developed with Six Generalised Grid Points for Solving Fourth Order Initial Value Problems

Abstract

This manuscript presents the development, analysis, and application of a novel six-step block method,
derived using a linear block algorithm (LBA), for the approximate solution of fourth-order initial value problems (IVPs). The proposed method is designed to overcome the shortcomings associated with traditional reduction methods, which involve converting the fourth-order IVP into a system of first-order ordinary differential equations (ODEs). Instead, the new method solves the problem directly, leveraging the advantages of hybrid block methods. A comprehensive theoretical analysis of the proposed method is provided, including proofs of its convergence and accuracy. The method’s performance is then compared to existing methods for solving fourth-order IVPs, using numerical examples and tables to illustrate the results. The comparative analysis demonstrates the accuracy, efficiency, and reliability of the proposed method, highlighting its potential as a viable alternative for solving fourth-order IVPs.