Developing and applying cubic spline method for the solution of boundary value problems in complex physical and engineering systems

It has long been a concern of researchers to address the challenges of solving higher-order differential equations. In order to approximate 11th-order boundary value problems (BVPs), this work presents a novel numerical approach that combines decomposition techniques with polynomial and Non-Polynomial Splines of third order. The method starts with a decomposition process that breaks down 11th-order BVPs into a system of second-order BVPs, breaking the problem down into smaller, more manageable parts. First-order derivatives are approximated using finite central differences, and each second-order ordinary differential equation is solved using both spline methods. These methods improve accuracy and efficiency when handling complex BVPs by providing a thorough framework for solving high-order differential equations. Comparing numerical responses with the precise response on a variety of examples was part of the numerical evaluations.