Mathematical Modeling ( Powerpoint)

This Mathematical Modeling course is designed to introduce students to the art and science of constructing mathematical models to describe, analyze, and solve real-world problems. Over the span of a semester, students will learn how mathematical theories and tools can be applied across various disciplines, from science and engineering to economics and social sciences.

The course begins with an introduction to the concept of modeling, outlining different types of models (descriptive and predictive) and the fundamental steps involved in the modeling process, such as identifying the real-world problem, formulating a model, and interpreting results.

Students will then explore the mathematical tools necessary for modeling, such as linear algebra, calculus, and probability, and understand how these can be applied to solve specific problems. From here, the course delves into the modeling of dynamic systems using differential equations and discrete models using difference equations, equipping students to tackle various practical applications like population growth, financial modeling, and resource management.

A significant portion of the course focuses on optimization models, where students will learn linear programming and optimization techniques to solve real-world problems, such as resource allocation and logistics. The course also introduces probabilistic models, covering topics like Markov chains and Monte Carlo simulations, which are crucial for understanding randomness and uncertainty in processes like queueing systems and risk analysis.

The growing importance of data in today’s world is covered through data-driven models. Students will learn to build and apply regression models, time series forecasting, and even machine learning techniques, with applications ranging from predictive analytics to weather forecasting.

In addition to theoretical models, the course covers dynamic systems and their analysis, exploring concepts like phase planes and chaos theory, with practical applications in fields like epidemiology and control systems. Students will also engage with real-world case studies from biology, physics, engineering, and environmental science, learning how models are applied in these fields to solve complex problems.

The course emphasizes numerical methods to solve models that cannot be addressed analytically. Students will learn techniques like finite difference methods and numerical solutions of differential equations, using simulation tools to model processes such as heat transfer and financial systems.

As they progress, students will also focus on the verification and validation of models, learning how to test and refine their models to ensure accuracy and reliability. Case studies will help students debug and improve their models through sensitivity analysis.

The course culminates with a project where students, individually or in groups, will develop a mathematical model for a real-world problem of their choice. They will present their models, demonstrating their ability to apply the techniques learned throughout the course to practical, meaningful problems.

This course not only provides students with a strong foundation in mathematical theory but also equips them with practical skills in modeling, optimization, and data analysis, preparing them for careers in various fields, from engineering to data science.

Throughout the course, assignments, mid-term exams, and final projects will evaluate students’ understanding and application of the material. By the end, students will have mastered the ability to construct and analyze mathematical models, using numerical methods and computational tools to tackle real-world challenges across a wide array of disciplines.