Analyzing Epsilon Effects on Exact and Composite Solutions via Matched Asymptotic Expansion

The method of matched asymptotic expansions (MAE) for solving singular perturbation problems is investigated in this study, emphasizing graphical data matching and simulation. By assessing the accuracy, computational efficiency, and impact of the perturbation parameter epsilon (ε), the research explores the agreement between exact and composite solutions across various problem types. Findings reveal that MAE-derived composite solutions closely align with exact solutions, exhibiting minimal deviations. The study observes that as ε decreases, both solutions converge to 1, demonstrating asymptotic stability, while larger ε values result in significant divergence. Consistent behavior is noted in the solutions, with convergence and divergence occurring at similar points with high accuracy. This thorough analysis not only highlights MAE's efficacy in addressing small parameter problems but also showcases its potential in graphical simulations, offering valuable contributions to future research in singular perturbation theory, especially in the domains of visual representation and numerical solution validation.