# MA3J4 Mathematical Modelling with PDE

Lecturer: Tobias Grafke

Term(s): Term 1

Status for Mathematics students:

Commitment: 30 Lectures

Assessment: 3 hour exam 100%

Formal registration prerequisites: None

Assumed knowledge:

Useful background:

Synergies: The following modules go well together with Mathematical Modelling:

Leads to: The following modules have this module listed as assumed knowledge or useful background:

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 necessary 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