Spatio-temporal pattern formation is fundamental to many phenomena in nature and the sciences, for examples water waves or the action potential in nerve axons. Mathematically, pattern formation occurs in models in the form of evolutionary equations, typically parabolic partial differential equations (PDE). In this course we discuss basic methods and tools for the study of simple patterns with focus on one physical space dimension and reaction-diffusion equations. Elementary building blocks of patterns are travelling waves which move with constant speed while maintaining shape. In one physical space dimension the profiles of travelling waves solve in ordinary differential equation (ODE). Hence, ODE methods are fundamental for the existence problem of travelling waves. Once existence has been established, the question of stability arises: does a perturbation from the travelling wave decay in time? This is mainly determined by the eigenvalue problem associated to the travelling wave, which can also be cast in an ODE framework.
Aantal college-uren
2
Tentaminering
Homework exercises and a final project
Verplichte literatuur
“PATTERNS AND WAVES: THE THEORY AND APPLICATIONS OF REACTION-DIFFUSION EQUATIONS” by Peter Grindrod. Oxford University Press 2nd Edition, see http://www.grindrodbook.com/
Voorkennis
Analyse 3