A Numerical Comparison of Finite Difference and Linear Shooting Methods for a Non-Homogeneous Cauchy-Euler Boundary Value Problem

Abstract

This study presents a comparative numerical analysis of the Finite Difference Method (FDM) and the Linear Shooting Method (LSM) for solving a second-order non-homogeneous Cauchy-Euler boundary value problem subject to Dirichlet, Neumann, and Robin boundary conditions. In contrast to previous studies that often employ different differential equations for different numerical experiments, this study uses an identical Cauchy-Euler equation while varying only the boundary conditions. This unified framework enables a more systematic investigation of the influence of boundary conditions on the performance of the numerical methods. Numerical solutions are computed using three mesh sizes, N = 10, N = 20, and N = 50. The accuracy of the methods is assessed by comparing the numerical solutions with the exact solution using Maximum Absolute Error (MaxAE) and Mean Absolute Error (MAE). In addition, graphical comparisons of the exact and numerical solutions, together with their corresponding error distributions, are presented to illustrate the solution behavior under different boundary conditions. The numerical results show that both methods produce accurate approximations with decreasing errors as the mesh is refined. The FDM exhibits approximately second-order convergence and requires less execution time, whereas the LSM achieves a higher convergence order and consistently produces smaller MaxAE and MAE values, indicating superior numerical accuracy.