Fractional calculus and its application to movement ecology in a changing world

Project Summary:

The collaborative project seeks to construct an appropriate mathematical formalism to describe how spatial disorder affect the movement statistics of objects, being they physical particles or agents, that diffuse anomalously, that is when the variance of their occupation probability grows slower or faster than linear in time, that is does not follow the diffusive paradigm. The collaboration aims to exploit the recent development, championed by Prof. Luca Giuggioli work on lattice random walk, to quantify the movement dynamics of diffusive entities in the presence of the so-called inert spatial heterogeneities, that is those that do not affect a particle or agent lifetime, with the one of Prof. Mujeeb ur Rehman on fractional operators, that are naturally employed to describe anomalous diffusion. Applications of the formalism to empirical observations in animal experiments are also planned as an important outcome of the project.

Prof. ur Rehman will visit over the period 25 October - 24 November and subsequently over the period 25 May - 24 June. In both periods he will deliver two open seminars on fractional calculus for postgraduate students and academics (details to be announced). Additionally an international one-day workshop organised by Prof. Giuggioli on Applications of fractional calculus to movement across disciplines at UoB is planned for October 31st. The aim of the workshop is to bring together theoreticians working on application of fractional calculus to movement in ecology, cell biology and pedestrian dynamics.

Visitor Biography:

Dr. Mujeeb ur Rehman serves as a teacher, mentor, and researcher at the School of Natural Sciences, National University of Sciences and Technology (NUST), Islamabad, Pakistan. He earned his PhD in Mathematics from NUST in 2011 with the support of the Maga S&T Scholarship (2006). Since 2012, he has held progressive academic positions at NUST, including Assistant Professor (2012-2017), Associate Professor (2017-2023), and currently serves as Full Professor (2023-present) and Head of the Department of Mathematics (2023-present).

Dr. Rehman teaches undergraduate and graduate mathematics courses and has been honored with Best Teacher Award (2020). His mentorship extends to guiding graduate students, having supervised PhD and MS students, with several more currently under his supervision.

Dr. Rehman's research focuses on fractional calculus and fractional differential equations, spanning both theoretical developments and computational methods for fractional-order systems. In theoretical fractional calculus, he has worked on the development of generalized operators, including tempered fractional calculus and Hadamard-type operators, and generalized integral transforms applied to fractional differential equations. He has also worked on the theoretical analysis of fractional difference equations and the development of fractional difference operators. His research further extends to fractional Sturm-Liouville problems, impulsive fractional equations, delay differential equations, and boundary value problems, employing tools such as variational methods and fixed-point theorems.

In numerical methods, Dr. Rehman has worked on operational matrix techniques for solving fractional differential equations using orthogonal polynomials and wavelet methods. His work includes spectral and quadrature approaches, along with product integration methods for fractional ordinary and partial differential equations.

With over 75 publications, his scholarly impact is well-established. Beyond academia, he has contributed to institutional leadership, previously serving as Head of Research and Development (2020–2023), and has been actively engaged in organizing conferences at the Department of Mathematics.