Mathematical models II.
Description:
The aim of the course is to provide an introduction to the basics of mathematical modeling with a deterministic approach.
The first part of the course considers one-dimensional ordinary differential equations: after getting familiar with the basic concepts (initial value problem, autonomous equations, equilibrium, direction field), we solve simple types of equations (separable variables, linear equations). In the rest of this section, we investigate problems from biology, chemistry, and physics, writing down the modeling equations and answering simple questions about the considered phenomena. First, we solve the problems by hand, and later we implement interactive programs to analyze solutions, the stability of equilibria, and parameter dependencies.
In the second part, we consider deterministic models and provide a brief introduction to higher-dimensional and higher-order models used in physics and epidemiology.
Throughout the course, we mix the by-hand and programming approaches to analyze the underlying models, improving skills on both the quantitative (analytical solutions) and qualitative (parameter dependencies) sides. Additionally, we use state-of-the-art tools (Python, Google Colaboratory) in our work and aim to provide industry-ready knowledge to students.
Curriculum:
- Modeling with differential equations 1.
- theory of one-dimensional first-order ordinary differential equations (existence and uniqueness of the solution, equilibrium, and stability in autonomous equations)
- solution of special 1D equations: separable and linear differential equations
- application in biology, chemistry, physics
- implementing interactive analysis of the models in Python
- Modeling with differential equations 2.
- introduction to higher-dimensional differential equations, the SIR model as an example
- implementing the solution of higher-dimensional models
Evaluation:
- written test about solving exercises on differential equations (25 points)
- interactive project about a modeling problem (using differential equations) (25 points)
VZS 