In this course, students will learn the fundamentals of mathematical modeling. A model is a miniature representation of the real world. Mathematical modeling is a process of developing models in mathematical terms. It is an abstraction of the real world to help us understand its essence clearly by simplifying its complexity. It also helps to formulate various problems concerning our lives for finding their solutions. The concept of mathematical modeling is a common basis for virtually all empirical sciences—not only natural and engineering sciences but also human sciences such as economics, psychology, sociology, etc.
The lectures introduce three types of mathematical models: optimization models, dynamic models, and probability models. Part I begins with concepts of mathematical optimization, then moves on to the calculus for finding optimal solutions, focusing on computational techniques. Part II explores the behavior of both discrete and continuous dynamical systems. The fundamental concepts include state space and state variables. Analytic methods as well as simulation techniques for analysis of dynamical systems are presented. Part III deals with modeling techniques for probabilistic phenomena and stochastic processes. It overviews some of the key concepts of probability and statistics and also discusses some analytical techniques such as Markov chains and Monte Carlo simulation.
In each part, the students are given opportunities to work on computer programming and numerical algorithms by the use of software utilities in practice sessions.