Abstract

This paper presents the development and analysis of a Toeplitz five-band monotone convergence boundary value method for the numerical solution of fourth-order boundary value problems (BVPs). The proposed approach is constructed by assembling linear multistep schemes into a block framework, derived systematically through the interpolation and collocation techniques. The resulting block method produces a Toeplitz coefficient structure with a five-band form, ensuring computational efficiency and numerical stability. The convergence properties of the method are rigorously examined, with proofs showing that the scheme is both consistent and stable, thereby guaranteeing monotone convergence. Special attention is given to the ability of the method to handle the complexity inherent in fourth-order BVPs, which frequently arise in engineering and applied sciences, particularly in beam theory, elasticity, and fluid mechanics. To demonstrate the efficiency and reliability of the method, several benchmark fourth-order boundary value problems are solved and compared against existing methods in the literature. Numerical results show that the proposed Toeplitz five-band block method achieves higher accuracy with fewer computational steps, while maintaining stability across a wide range of test problems. The performance improvements highlight its superiority in terms of accuracy, convergence rate, and computational cost. Overall, the Toeplitz five-band monotone convergence boundary value method provides a robust and efficient tool for solving fourth-order BVPs, contributing a significant advancement in numerical methods for higher-order differential equations.