# Research

My primary interests are boundary value problems for high order linear partial differential equations, and spectral theory of ordinary differential operators.

## Overview for non-mathematical audience

I am interested in initial-boundary value problems (IBVP), and studying the ways in which we can understand their solutions. An IBVP means a collection of equations that together describe how a certain quantity varies in position and time. For example, IBVPs describe the temperature of a metal object at different points within the metal and at different times, the way light propogates through optical fibres, and the heights of water waves.

One of the equations of an IBVP is a partial differential equation (PDE), which relates the way a quantity q changes in time, ∂tq, to the way it varies with position, ∂xq. In the simplest PDEs, these two rates of change might be proportional, but more interesting examples would instead relate the rate of change of q in time to the rate of change of the rate of change of q in space, that is ∂tq = k ∂x2q, where k is a constant of proportionality. Clearly, one could imagine ∂xnq for arbitrary integer n≥1 called the spatial order, and natural systems often have n>2.

Another important equation in the IBVP is the boundary condition (BC). The BC describes what happens to the quantity q at the physical edge of the system under study. For example, if the quantity is heat, and the boundary is insulated, then there can be no flow of heat over the boundary, so ∂xq=0 at the boundary. Alternatively, it might be that the two ends of a metal rod are held at constant, but different temperatures, say q=20°C at the left boundary, and q=100°C at the right boundary.

Provided 1≤n≤2, or the boundary conditions are very simple in a certain sense, these problems are classical. Moreover, it is understood by any mathematics graduate that the solution q can be expressed as an infinite sum of eigenfunctions of the spatial differential operator, where eigenfunction means a function φ which satisfies ∂xnφ=λφ, for some constant λ.

If n≥3 and the BC are more complicated, it was not known how to solve these problems until the recent introduction of the Unified Transform Method of Fokas. Further, the solution representation obtained through Fokas' method is not simply a sum of eigenfunctions and, in many cases, it is impossible to express the solution in such a way.

My contributions to the theory include the most general implementation of Fokas' method to date, and the derivation of criteria for the problem to be well-posed (have a unique solution). More recently, I have discovered a new generalization of eigenfunctions, called augmented eigenfunctions, which describe the solution representation obtained through Fokas' method.

## Unified transform method portal

I also run a website that acts as a central portal for researchers working on the unified transform method. An ever growing list of research papers and abstracts is organised thematically. There is also an introduction to the method.

Coming soon