A Study on Fourth-Order Coupled Boundary Value Problems: Existence, Uniqueness and Approximations

This study examines the existence and approximation of solutions for a coupled system of fourth-order boundary value problems (4th-BVPs), which model the interactions between two distinct but interrelated physical systems. These coupled boundary value problems arise in various applications in engineering and physics, including the analysis of bending behaviors in beams and vibrations in interconnected structural components. By leveraging Green`s functions and building upon prior research in fourth-order differential equations, we derive sufficient conditions for the existence and uniqueness of solutions to the system. Additionally, we provide a numerical framework for approximating these solutions, offering practical insights for real-world applications.