A Comparative Study of Finite Difference, Shooting, and Collocation Methods for Linear Non-Stiff, Stiff, and Nonlinear Two-Point Boundary Value Problems with Dirichlet and Neumann Boundary Conditions

This study discusses the performance comparison of three numerical approaches, namely the Shooting method, the Finite Difference Method (FDM), and the Collocation method, in solving Boundary Value Problems (BVP) for three categories of problems: linear non-stiff, linear stiff, and non-linear. Each type of problem is tested with two types of boundary conditions, namely Dirichlet and Neumann. The evaluation is based on two main criteria: accuracy, measured using error norms with respect to the exact solution, and computational efficiency, quantified in terms of CPU execution time. The results show that in the non-stiff case with Dirichlet boundary conditions, the Shooting methods based on LSODA and RK5 provide very high accuracy with good efficiency, while the Finite Difference Method excels in efficiency but is slightly inferior in accuracy. Under Neumann boundary conditions, the Finite Difference Method tends to be less accurate, whereas the Collocation method delivers very good accuracy but with relatively lower efficiency. For stiff problems, the Shooting method maintains high accuracy, while the Finite Difference and Collocation methods show varying performance depending on the type of boundary condition. In the non-linear case, the Shooting method becomes the most accurate option, although with slightly lower efficiency compared to the Finite Difference Method. These findings provide practical guidance in selecting appropriate numerical methods for BVPs based on problem characteristics and boundary conditions.