Epsilon variations and their impact on solution accuracy: A matched asymptotic approach

Abstract

This study investigates the method of matched asymptotic expansions (MAE) for solving singular perturbation problems, focusing on graphical data matching and simulation. We assess the accuracy, computational efficiency, and impact of the perturbation parameter epsilon (ε) on the agreement between exact and composite solutions across various problem types. Results demonstrate that MAE-derived composite solutions closely match exact solutions, with minimal deviations observed. As ε decreases, both solutions converge to 1, indicating asymptotic stability, but larger ε values lead to significant deviations, signaling divergence. The solutions exhibit consistent behavior, converging and diverging at similar points with high accuracy. This comprehensive analysis highlights MAE's effectiveness in solving small parameter problems and its utility in graphical simulations, offering valuable insights for future research in singular perturbation theory, particularly in visual representation and numerical validation of solutions.