Unravelling Aspects Of Mathematical Modeling Via Differential Equations

Abstract

Mathematical modeling serves as a pivotal tool in representing the behavior of real-world systems, allowing scientists and engineers to gain insights, make predictions, and solve complex problems. This paper explores the process of generating mathematical representations or models, validating them, understanding their utility, and recognizing their limitations. Before delving into these crucial aspects, it is imperative to understand the motivations behind mathematical modeling. Both engineers and scientists utilize mathematical modeling for practical reasons, aiming to address challenges and advance knowledge in their respective fields. Moreover, the joy of formulating and solving mathematical problems contributes to the appeal of mathematical modeling. Over the past few decades, the importance of mathematical modeling has been widely recognized, leading to a surge in research and publications in this domain. Mathematical modeling often involves differential equations when modeling situations with continuous variables and reasonable hypotheses about their rates of change. Depending on the number of dependent and independent variables, mathematical models may manifest as ordinary differential equations or systems of partial differential equations. This paper aims to provide insights into the fundamentals of mathematical modeling via differential equations, highlighting its significance across various disciplines.