Numerical Study of Some Nonlinear Boundary Value Problems (BVPs) with an Efficient Numerical Method

Abstract

This study presents a comprehensive numerical investigation of fourth-order nonlinear boundary value problems (BVPs) using an efficient and accurate computational approach. The present work focuses on a class of nonlinear boundary value problems that commonly emerge in scientific and engineering applications, where the underlying models are often governed by complex nonlinear differential equations. Due to the difficulty of obtaining exact analytical solutions for such problems, numerical techniques become essential for reliable approximation. In this work, the Finite Difference Method (FDM) is adopted as the core numerical tool due to its robustness and effectiveness in solving such problems. A carefully designed finite difference scheme is developed to discretize the governing fourth-order nonlinear differential equations, converting them into a system of nonlinear algebraic equations. These systems are subsequently solved numerically using Maple software as the computational tool. The article includes two illustrative examples of nonlinear BVPs to demonstrate the applicability and performance of the proposed method. Numerical results, including graphical representations, are provided for various step sizes. Both absolute and relative errors are calculated to assess the accuracy of the solutions. The numerical findings are further validated by comparing them with known analytical or previously published approximate results. The outcomes confirm that the finite difference approach yields highly accurate and reliable solutions.