Modelling with ODEs and PDEs

‘Modelleren met ODE's en PDE's’ is one of the third year modelling courses. Student should at least choose one of this selection of courses. The course is recommended to students interested in partial differential equations and applied mathematics. Please find more information about the study advisory paths in the bachelor at the student website.

This course is an introduction to mathematical modelling using dynamical systems. This includes the process of translation real-world phenomena (arising for example in natural sciences, engineering or economics), into well-defined mathematical problems, analysing these models, and translating and interpreting the outcome of these mathematical analyses in the context of the real-world setting. A major challenge lies in finding a good compromise between accuracy of the model and accessibility of the resulting mathematical problem to analysis and computation.

In this course, we will cover basic concepts and modelling techniques, such as non-dimensionalization, scaling analysis and perturbation methods. Further, we first will focus on ordinary differential equations, covering both mathematical techniques to study them (e.g. using nullclines, fixed points, stability analysis, phase diagram analysis and perturbation methods) and real-world phenomena that can be modelled using them. The last part of the course is an introduction to partial differential equations and models that use them. Here, the focus is on the process of diffusion and of (traffic) flow, where we treat how to derive such models, and how to study them using techniques such as Fourier Transforms, Fourier Series and the method of characteristics. In the whole course, the focus is on modelling using mathematical techniques, and not on a rigorous treatment including proofs for all mathematical techniques and theorems that are used.

An integral part of the course is communicating about mathematical results and model output. Hence, also report writing in the context of modelling applications will be taught in this course.
 
Leerdoelen:
After this course, a student is able to

• create simple conceptual models for real world phenomena using ordinary differential equations and partial differential equations;
• study models consisting of ordinary differential equations, in particular their fixed points and their stability;
• use dimensional analysis, scaling and perturbation methods on ordinary and partial differential equations;
• use Fourier analysis to study the time and equilibrium behaviour of (reaction-) diffusion equations;
• use the method of characteristics to study the time behaviour of traffic flow equations;
• interpret and communicate about the results of mathematical analysis in the context of a real-world application.

Onderwijsvormen:
In general, there will be two 2 hour lectures and two 2 hour tutorials per week.
 
Toetsing:
Final exam (60% of course grade), a project report including peer feedback (20% of course grade), and hand-in assignments (20% of course grade). The exam grade must be at least 5.5. In case of a retake, only the grade for the final exam will be changed.

The grading rules can still change, due to AI. 
 
Herkansing en inspanningsverplichting:
Students with a final grade lower than 4 can participate the retake exam if the average of the report and the hand-in assignments is at least 5.5. Only the final exam can be retaken. The retake exam will only replace the final exam grade.

The retake rules can still change, due to AI. 
 
Taal van het vak:
The language of instruction is English.