Improved Finite Difference Methods for Solving Second-Order Boundary Value Problems of Ordinary Differential Equations Using Chebyshev Polynomials

Abstract

In this paper, two advance numerical techniques for solving second-order boundary value problems in ordinary differential equations (ODEs) are presented. The first method, the Chebyshev
Finite Difference Method (CFDM), which enhances the traditional Finite Difference Method
by utilizing Chebyshev Polynomials as basis functions, resulting in improved computational
performance. The second method developed is the Perturbed Chebyshev Finite Difference
Method (Perturbed CFDM), which incorporates perturbation techniques to further enhance
the accuracy and efficiency of the method. Both methods were applied to homogeneous and
non-homogeneous linear boundary value problems, with numerical results demonstrating that
the Perturbed CFDM significantly outperforms both standard CFDM and the traditional finite
difference method in terms of accuracy and computational efficiency. These findings establish
the Perturbed CFDM as a powerful and reliable tool for solving boundary value problems. All
computations were carried out using MATLAB, ensuring accurate approximation and numerical solutions of the tested problems.