# Research

My research interests lie in fractional calculus, fractional differential equations, operational calculus, Clifford analysis, zeta functions, asymptotic analysis, and Mittag-Leffler functions.

In fractional calculus, my philosophy as a pure mathematician is to work in the most general possible setting, and to enhance connections with other branches of mathematics. I also believe strongly in the importance of finding connections between different types of fractional calculus: each time a "new" fractional differintegral is invented, relating it to existing definitions can help its properties to be understood much more easily.

I have worked on fractional differential equations using a variety of different approaches, analytical and mostly constructive. I try to focus on problems which present some particular challenges mathematically, but also which arise naturally so that their solutions are interesting for scientists (there is no point in creating convoluted problems just for the sake of being able to solve them).

An interesting technique for differential equations is Mikusinski's operational calculus, which I have studied specifically in the setting of novel fractional operators and solving equations posed using such operators. This method uses ideas from abstract algebra to solve analytic problems, so it is interdisciplinary within pure mathematics. I also enjoy working on connections between fractional calculus and complex analysis, such as the fractional d-bar derivative, which I am currently extending into hypercomplex settings such as Clifford analysis.

My work on zeta functions has focused on, firstly, the asymptotic analysis of series and integral formulae for zeta functions, and, secondly, discovering fractional differintegral formulae for zeta functions. I have mostly studied the Hurwitz and Lerch zeta functions, of which the celebrated Riemann zeta function is a special case.

Although Mittag-Leffler functions are known classically for more than a century, there are still new types and generalisations emerging from fractional differential equations. Again, I try to focus on functions which arise naturally, rather than just adding new parameters willy-nilly, and on properties which will be useful for the scientists who need to use these functions.

My list of publications is here. Please feel free to contact me if you need the full text of any of my papers and cannot obtain it online, or if you have a serious research project where you feel that my knowledge could usefully be applied for a meaningful collaboration.