## My research interests

My primary interests are partial differential equations, and spectral theory of ordinary differential operators. I am particularly interested in linear differential operators with complicated boundary conditions, and how the Fokas unified transform method can be used to study associated initial boundary value problems. I have recently become interested in the numerical study of differential operators and numerical evaluation of oscillatory integrals.

### Linear IBVP & generalisations

Much of my work is dedicated to solving initial boundary value problems, and their generalisations, for linear evolution equations. My first paper, [Smi2012a] gave a solution method for linear evolution equations of arbitrary spatial order subject to arbitrary two point boundary conditions, via the Fokas unified transform method. More recent works have generalised this method to more complicated classes of boundary conditions. In collaborations with Natalie Sheils, [SS2015a] & [DSS2016a], I studied various interface problems of second and third order. The more recent work [PS2018a] studies multipoint problems, in which the "boundary conditions" may feature couplings between the boundaries and points interior to the spatial interval. The further extension [MS2018a] generalised further to admit nonlocal problems, in which some weighted integral of the solution may be specified in place of a boundary condition. A Conservation of mass law is an example of such a nonlocal condition.

### Spectral theory of differential operators

The classical Sturm-Liouville theory of second order linear differential operators is inadequate to describe the spatial differential operators encountered in the above work. There are extensions to encompass multipoint boundary conditions and operators of higher order, but the success of the Fokas unified transform method highlighted a significant gap. There exists a significant class of differential operators associated with well posed initial boundary value problems but for which there was no known spectral theory. In works [FS2016a] & [PS2016a], my collaborators and I began to describe the missing spectral theory, for finite interval and half line differential operators, respectively. There is also an overview of these two papers, comparing and contrasting their results, in [Smi2015a]. In [PS2013a], we give another view on the more established spectral theory, ellucidated by the Fokas method.

### Dispersive revival phenomena

Certain dispersive equations, under periodic boundary conditions with steplike initial data, exhibit fractal solution profiles at irrational times and steplike solutions at rational times. The latter effect is also known as dispersive quantization. This behaviour was first observed in optical experiment by Talbot, and has been variously rediscovered by physicists and mathematicians ever since. The effect is known to occur for the linear Schrödinger equation and linearised Korteweg-de Vries equations, among many others. In [OSS2020a], we showed that, at least for the linear Schrödinger equation, the effect is not limited to the periodic regime, but occurs for a rather broad class of boundary conditions known as pseudoperiodic. In [Smi2020a], I give a brief overview of recent work in the field.

### Numerical evaluation of oscillatory integrals

Analytic solutions of initial boundary value problems are necessarily expressed as some kind of series. Often, these series take the form of oscillatory integrals, which are difficult to evaluate numerically, even with a modern computer. I have recently become interested in how analytic techniques may be combined with numerical algorithms to efficiently evaluate such integrals.

### Mathematics education

With efficiency of instructor time and quality of teaching two increasingly important constraints, I became interested in how peer instruction and peer assessment can be made into more effective instructional techniques. This paper is the result of a grant used to develop a new teaching technique. There are further details on my teaching page.

## 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.

## Research with undergraduate students

I have supervised seven summer research projects, and am supervising a further seven in summer 2020. I've also supervised semester projects and final year undergraduate capstone projects.

If you are a University of Newcastle student interested in doing research with me, you can find out more at the Unified transform lab website.

### Journal articles

On occasion, this research can lead to a paper published in a mathematics journal.