Date: Friday, 23 August, 2024 - 15:00 to 16:00
Title : General kernels and parametrised families: in search of semigroup properties
Speaker : Arran Fernandez, Eastern Mediterranean University, Northern Cyprus

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy
Organizers : Pavan Pranjivan Mehta* and Arran Fernandez**
SISSA, International School of Advanced Studies, Italy
** Eastern Mediterranean University, Northern Cyprus

Keywords: Operators with general kernels, Semigroup property, Fundamental theorem of fractional calculus, Operational calculus

Abstract

The field of fractional calculus has burgeoned recently with many newly defined operators roughly fitting the same mould: time-fractional operators in one dimension defined by convolution with some kernel function. In order to put some structure on the field, several researchers have suggested categorising these operators into classes defined by some general type of kernel function. However, so many different general kernel functions have been proposed that they too need to be categorised and understood in relation to each other.

Each type of general kernel has its own advantages and disadvantages; I will provide an overview of several proposals from the last fifteen years. Then I will pose a research question -- what is the most general kernel function that depends on a ``fractional order'' parameter α and gives fractional integral operators having a semigroup property in α? -- and answer it by using algebraic theory inspired by Mikusinski’s operational calculus. The general kernels proposed herein give rise to so-called one-parameter families and two-parameter families of fractional operators. These operators have a natural structure, in which fractional integrals and derivatives can be defined satisfying a fundamental theorem of fractional calculus, and differential equations can be posed and solved within the theory of such operators using operational calculus. Some potential extensions and open problems will also be mentioned.

Biography

Arran Fernandez is an associate professor at the Eastern Mediterranean University in Northern Cyprus. He was educated at the University of Cambridge, where he started as the youngest student in 2010 and graduated as the top student (senior wrangler) in 2013. Following his PhD, also at Cambridge in the Department of Applied Mathematics and Theoretical Physics, he has been working in Northern Cyprus. His research is on the mathematical analysis of fractional calculus, emphasising its connections with other fields of pure mathematics such as Clifford analysis, abstract algebra, analytic number theory, etc. He has published over 70 research papers and serves as an associate editor in 3 mathematical journals.

Bibliography

[1] Arran Fernandez. ”Abstract algebraic construction in fractional calculus: parametrised families with semigroup properties”. In: Complex Analysis and Operator Theory 18 (2024), p. 50