A STUDY OF ASYMPTOTIC EXPANSIONS IN APPLIED ANALYSIS

Abstract

Asymptotic expansions constitute a central analytical tool in applied analysis, enabling the systematic approximation of mathematical models characterized by the presence of small or large parameters. This paper presents a rigorous investigation of asymptotic expansion techniques, focusing on their formal derivation, structural properties, and analytical justification. Classical methodologies, including regular and singular perturbation theory, matched asymptotic expansions, and uniform asymptotic methods, are examined within a unified analytical framework. The study emphasizes the role of asymptotic scales, remainder estimates, and consistency conditions that govern the validity of expansions. Applications to differential equations and integral formulations are employed to illustrate the effectiveness and limitations of the methods, with particular attention to issues of accuracy, convergence behavior, and error control. By bridging formal asymptotics with rigorous analytical considerations, this work contributes to a deeper understanding of the applicability and reliability of asymptotic expansions in contemporary applied analysis.