The Properties of Green's Function and Variation of Constants in Damped Oscillations Using Numerical Approach

Abstract

Contemporarily, Green function is considered as a basic mathematical formular in various applications in advanced physics. On this basis, this study analyzes the efficacy and precision of Green's Function as well as the Variation of Constants (VoC) techniques in addressing damped oscillation issues, particularly in systems affected by an impulsive external force, exemplified by a Dirac delta function. To be specific, the analysis examines the duration necessary for a damped oscillator to be deemed at rest, contrasting the efficacy of Green's Function and VoC in forecasting critical moments and addressing numerical difficulties spanning underdamped, overdamped, and severely damped conditions. According to the analysis, numerical simulations were performed based on Python to evaluate the convergence, stability, and computational efficiency of each method. The results highlight the comparative advantages as well as disadvantages of each method, providing important insights for applications in oscillatory systems where accurate and prompt response predictions are critical.