# MA3J4 Mathematical modelling with PDE

**Lecturer: **Marie-Therese Wolfram

**Term(s): **Term 1

**Status for Mathematics students:**

**Commitment: **30 Lectures

**Assessment: **3 hour exam 100%.

**Prerequisites: **MA250 PDE

**Leads To: **The students will be given a general overview on the derivation and use of partial differential equations modeling real world applications. By the end of the course they should have aquired knowledge about the physical interpretation of PDE models and how the learned techniques can be applied to similar problems.

**Content: **

- Mathematical modelling

- Math. modelling in physics, chemistry, biology, medicine, economy, finance, art, transport, architecture, sports
- Qualitative/quantitative models, discrete/continuum models
- Scaling, dimensionless variables, sensitivity analysis
- Examples: projectile motion, chemical reactions

- Diffusion and drift

- Microscopic derivation
- Continuity equation and Fick's law
- Heat equation: scaling, properties of solutions
- Reaction diffusion systems: Turing instabilities
- Fokker-Planck equation

- Transport and flows

- Conservation of mass, momentum and energy
- Euler and Navier-Stokes equations

- From Newton to Boltzmann

- Newton's laws of motion
- Vlasov and Boltzmann equation
- Traffic flow models

**Aims: **The module focuses on mathematical modelling with the help of PDEs and the general concepts and techniques behind it. It gives an introduction to PDE modelling in general and provides the neccesary basics.

**Objectives: **By the end of the module students should be able to:

- Understand the nature of micro- and macroscopic models.
- Formulate models in dimensionless quantities
- Have an overview of well known PDE models in physics and continuum mechanics
- Calculate solutions for simple PDE models
- Use and adapt Matlab programs provided during the module

**Books: **

- J. David Logan, Applied Mathematics: A Contemporary Approach
- C.C.Lin, A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, 1988
- A. Aw, A. Klar, Rascle and T Materne, Derivation of continuum traffic flow models from microscopic follow the leader models, SIAM Appl Math., 2002
- B. Perthame, Transport equations in biology, Birkhäuser, 2007
- R. Illner, Mathematical Modelling: A Case Study Approach, SIAM, 2005
- C. Eck, H. Garcke, P. Knaber, Mathematische Modellierung, Springer, 2008
- L. Pareschi and G. Toscani, Interacting Multiagent Systems, Oxford University Press, 2013