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

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.