Numerical Analysis Assignment with Source Code: ODE Experiments and Extensions

This document presents a numerical analysis assignment with complete source code, organized into two primary sections. The first section contains three experimental problems focusing on ordinary differential equations (ODEs), while the second section extends the discussion to higher-order ODE solutions and boundary value problems (BVP). All algorithms and computational examples are implemented using MATLAB programming. From the perspective of stiffness characteristics, ODE problems are classified into non-stiff and stiff categories. For stiff problems such as Van der Pol equations with large coefficients, conventional methods like ODE45 demonstrate poor computational efficiency, whereas specialized solvers like ODE15S provide significantly better performance. Non-stiff problems typically achieve excellent results using ODE45 solver. Regarding equation order, ODEs are categorized into higher-order differential equations and first-order differential equations. Higher-order equations can generally be transformed into state-space representations for efficient numerical solution. From the numerical properties standpoint, when dealing with equations containing large first-derivative coefficients, smaller step sizes should be selected. If the problem characteristics are difficult to estimate precisely, conservative step-size selection is recommended. Comparative analysis reveals that with smaller step sizes, Adams multistep methods outperform Runge-Kutta methods in both efficiency and accuracy, though they possess smaller stability regions. Significant differences exist between initial value problems and boundary value problems in terms of solution approaches. For nonlinear ordinary differential equations, additional methods include shooting techniques and cell mapping approaches. Stability analysis tools such as phase plane diagrams remain essential for comprehensive ODE studies. Notably, beyond using MATLAB's ode function family, employing block diagram environments like Simulink provides an alternative and often advantageous approach for solving differential equations, particularly from application-oriented perspectives. The implementation includes proper error handling, step-size adaptation mechanisms, and visualization components for result analysis.