Exploring the Theory of Mathematical Modelling: Unlocking the Power of Abstraction

Zhenyao Ortiz^*
*Correspondence: Zhenyao Ortiz, Department of Mathematics and Statistics, York University, Toronto, Canada, Email:

Description

Mathematical modelling is a powerful tool that enables us to describe and understand complex phenomena in various fields, including physics, engineering, biology, economics, and more. It provides a means to represent real-world systems using mathematical equations, allowing us to simulate, analyze, and predict their behavior. At the core of mathematical modelling lies a rich theory that encompasses a wide range of techniques and approaches. In this article, we will delve into the theory of mathematical modelling, exploring its key concepts, methodologies, and applications [1].

To comprehend the theory of mathematical modelling, we must first understand the concept of abstraction. Abstraction involves simplifying complex real-world systems into mathematical models that capture the essential features and relationships. By focusing on key variables and ignoring less significant details, we can create models that retain the fundamental aspects of the system under investigation. Mathematical models consist of variables, parameters, and functions. Variables represent quantities that can change, while parameters are fixed values that define the characteristics of the system. Functions establish relationships between variables and parameters, providing a mathematical description of how they interact. Mathematical models can be classified into various types, depending on their structure and purpose. Deterministic models, which use precise equations, aim to predict the exact behavior of a system. Stochastic models, on the other hand, incorporate randomness to account for inherent uncertainties in the system. Other classifications include discrete versus continuous models, linear versus nonlinear models, and static versus dynamic models [2].

Analytical modelling involves solving mathematical equations analytically to derive exact solutions. This approach often relies on well-established mathematical techniques such as calculus, differential equations, and linear algebra. Analytical models provide explicit formulas that express the relationships between variables, enabling direct interpretations and insights into the system's behavior. Numerical modelling encompasses techniques that employ computational methods to solve mathematical models when analytical solutions are infeasible or unavailable. These methods include finite difference, finite element, and Monte Carlo simulations, among others. Numerical models provide approximate solutions by discretizing the system into manageable elements and employing algorithms to compute numerical approximations.

Empirical modelling relies on experimental data to construct mathematical models. It involves collecting observations, identifying patterns, and formulating mathematical relationships that best fit the data. Statistical techniques, such as regression analysis, are often used to determine the optimal mathematical representation of the empirical data. Empirical models are particularly useful when the underlying system is complex or not well understood. Hybrid modelling combines elements of different modelling approaches to leverage their strengths. It integrates analytical, numerical, and empirical techniques to create comprehensive models that capture both the known theoretical aspects and the observed empirical behavior of a system. Hybrid models are often used when dealing with complex systems that require a multidimensional understanding. Mathematical modelling plays a fundamental role in physics and engineering, enabling the study of physical phenomena and the design of efficient and safe systems. From classical mechanics to quantum field theory, mathematical models provide insights into the behavior of particles, waves, and energy transfer. In engineering, models are used to optimize designs, simulate fluid flow, predict structural integrity, and more [3].

In biology and medicine, mathematical modelling aids in understanding complex biological processes and disease dynamics. Models are employed to study population dynamics, disease spread, genetic interactions, and drug efficacy. They assist in predicting the outcome of medical interventions, optimizing treatment protocols, and analyzing biological networks. Mathematical models are extensively employed in economics and finance to analyze market behavior, predict economic indicators, and develop financial strategies. Models capture supply and demand dynamics, macroeconomic trends, investment portfolio performance, risk analysis, and other factors that influence economic decision-making. The study of environmental systems relies heavily on mathematical modelling to assess the impact of human activities, predict climate change patterns, and develop sustainable strategies. Models simulate atmospheric circulation, ocean currents, ecological interactions, and pollutant dispersion, aiding in policy-making and environmental management [4].

The theory of mathematical modelling provides a powerful framework for understanding, simulating, and predicting complex systems across various disciplines. By abstracting real-world phenomena into mathematical representations, we gain insights and develop solutions that would otherwise be challenging or impossible to obtain. Whether in physics, biology, economics, or other fields, mathematical modelling continues to advance our understanding and shape the world we live in. As technology and computational power continue to evolve, the theory of mathematical modelling will undoubtedly remain at the forefront of scientific and engineering endeavors, unlocking new avenues of exploration and innovation [5].

Acknowledgement

None.

Conflict of Interest

None.

References

1. Casey, Beth M, Caitlin M. Lombardi, Dana Thomson and Hoa Nha Nguyen, et al. "Maternal support of children's early numerical concept learning predicts preschool and first‐grade math achievement." Child Dev 89 (2018): 156-173.

Google Scholar, Crossref, Indexed at

2. Clements, Douglas H, Julie Sarama, Carolyn Layzer and Fatih Unlu, et al. "Effects on mathematics and executive function of a mathematics and play intervention versus mathematics alone." J Res Math Educ 51 (2020): 301-333.

Google Scholar, Crossref, Indexed at

3. Duncan, Greg J, Chantelle J. Dowsett, Amy Claessens and Katherine Magnuson, et al. "School readiness and later achievement." Develop Psychol 43 (2007): 1428.

Google Scholar, Crossref, Indexed at

4. Hassinger-Das, Brenna, Tamara S. Toub, Jennifer M. Zosh and Jessica Michnick, et al. "More than just fun: A place for games in playful learning/Más que diversión: El lugar de los juegos reglados en el aprendizaje lúdico." Infanc Aprendiz 40 (2017): 191-218.

Google Scholar, Crossref, Indexed at

5. Huntsinger, Carol S, Paul E. Jose and Zupei Luo. "Parental facilitation of early mathematics and reading skills and knowledge through encouragement of home-based activities." Early Child Res Q 37 (2016): 1-15.

Google Scholar, Crossref, Indexed at