Title of project:
Qualitative Theory of Partial Differential Equations

Code:
ND2122B

Supervisor: Prof. N. Dirr

Project description:

Certain partial differential equations of the form

∂tu(x,t)=∆u(x,t)+f(x,u(x,t))

can be transformed in an integral equation and solved by a Peano-Iteration in a similar way as it is done for ODEs. The difference is however, that we have to work in infinite dimensional vector spaces instead of R. Combining ideas from ODEs and Functional Analysis, a lot can be said about the qualitative behaviour of such equations (stability, long-time behaviour etc.) similar to the ODE case.
These equations, called reaction-diffusion equations, have applications in chemistry, biology and physics.
A project could, depending on the interest of the student,  focus on numerics and/or on analysis.
Background Reading:
L.C. Evans, Partial Differential Equations, AMS Grad. Studies in Math. 19

Project offered as double module, single module, or both:
Double

Prerequisite 2nd year modules:
Real Analysis, Series and Transforms. Not prerequisite but recommended is Modelling with ODEs or a Year 2 Numerical Analysis Module.

Recommended 3rd year module for concurrent study:
Ordinary Differential Equations, Fourier and Functional Analysis

Number of students who can be supervised on this project:
1