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# Mathematical Modelling Basics

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## Overview

How do populations grow? How do viruses spread? What is the trajectory of a glider? Many real-life problems can be described and solved by mathematical models.

This course will introduce you to the modelling cycle which includes: analyzing a problem, formulating it as a mathematical model, calculating solutions and validating your results.

All models are (systems of) ordinary differential equations, and you will learn more about those by watching videos and reading short texts, and more importantly, by completing well-crafted exercises.

You will learn how to implement Euler’s method in a (Python) program, and finally, you will learn how to write about your findings in a scientific way (with LaTeX).

In the verified track of this course you will additionally:

• Consolidate the new theoretical skills with graded problem sets about five real-life applications.
• Work on your own modelling project (individually or in a team). Because mathematical modelling is only learned by doing it yourself, you complete your own modelling project on a self-defined real-life problem. You will be guided through the project by completing a list of smaller tasks.

This course is aimed at Bachelor students from Mathematics, Engineering and Science disciplines.

The course is for anyone who would to use mathematical modelling for solving real world problems, including business owners, researchers and students.

The course materials of this course are Copyright Delft University of Technology and are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike (CC-BY-NC-SA) 4.0 International License.

## Syllabus

Module 1:

Introduction to the cycle of mathematical modelling. We will start describing a population of fish by a differential equation.

Verified Track: Two practice problems with other real-life applications to consolidate the theory learned. You start your personal modelling project. You can choose to work in a team of two.

Module 2:

Complete more modelling cycles by improving on the model and evaluating the consequences. Euler’s method is introduced for solving ordinary differential equations. You will run Python simulations.

Verified Track: A new application to practice the theory. For your project you specify a real-life problem. You implement a 1-dimensional model.

Module 3:

Predator fish are added to the model. How do the populations interact? Systems of differential equations. You also learn how to write about your project in a scientific report. You get an introduction to scientific and mathematical writing. You will learn how to write a preliminary report about mathematical modelling in LaTeX.

Verified Track: One more practice problem to consolidate the theory learned about systems. You do more simulations with your own mathematical model and complete the modelling cycle several times. You apply your writing skills by writing a scientific report about your modelling project. You submit both a preliminary version of the report and the final version. Both are peer reviewed.

#### Taught by

Marleen Keijzer, Dennis den Ouden-van der Horst, Iris Smit and Kees Vuik

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