Comparison of analytical and approximate approaches to solving differential equations
DOI:
https://doi.org/10.64226/sarj.v3i01.100Keywords:
Analytical method, Approximate method, Computational accuracy, Differential equations, Numerical solutionAbstract
Abstract
The solution of differential equations constitutes a fundamental problem in applied mathematics, playing a key role in the modeling of natural phenomena. Despite the high significance of analytical methods in providing exact solutions, many differential equations lack closed-form solutions. Therefore, the utilization of approximate methods as an effective and flexible alternative becomes essential. The research methodology is library-based, with information extracted from reputable scientific sources such as ScienceDirect, IEEE, Scopus, and Google Scholar, while the software Publish or Perish was employed for systematic collection and rigorous evaluation of scholarly references. This study adopts a mixed-methods approach (quantitative and qualitative), and the analysis is conducted through empirical comparisons among methods such as Taylor series, Euler, Runge–Kutta, and numerical modeling techniques. Findings indicate that analytical methods are effective and efficient in providing a precise theoretical framework and in explaining phenomena, yet they exhibit limited efficiency in solving complex problems. Conversely, approximate methods, despite their inherent errors, offer greater flexibility and computational speed, making them more suitable for practical applications. The most significant outcome of this investigation is that a judicious combination of analytical and approximate methods can enhance computational accuracy, improve stability, and reduce computational time in the numerical solution of differential equations. The present study is particularly important as it not only elucidates the capabilities and limitations of each approach but also provides a framework for optimal selection or integration of methods applicable in scientific and industrial contexts.
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