Supervisor

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Abstract

Fractional differential equations (where standard derivatives are generalised to derivatives of non-integer, real order) have become a fundamental modelling approach for understanding and simulating the many aspects of spatial heterogeneity of complex materials and systems [1,2]. Very recently [3], we have been able to further extend such ideas to derivatives of complex-order. This project aims to further explore the capabilities of these novel complex-order fractional operators in modulating the spatiotemporal dynamics of different systems of interest.

To investigate how complex-order fractional operators can advance the description of multiscale transport phenomena in physical and biological processes highly influenced by the heterogeneity of complex media.

Students will investigate, by means of scientific computing and modelling and simulation, how complex-order fractional operators modulate the response of different systems of physical and biological interest. Different research options will be available, including the analysis of: (i) novel Turing patterns in physical and biological processes of morphogenesis; (ii) periodic precipitation patterns in the manufacturing of micro- and nano-structures; or (iii) models of population dynamics and epidemiology.

1. Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization role of structural heterogeneity in dispersion of repolarization https://doi.org/10.1098/rsif.2014.0352 2. Fourier spectral methods for fractional-in-space reastion- diffusion equations. https://doi.org/10.1007/s10543-014-0484-2 3. The Complex-Order Fractional Laplacian: Stable Operators, Spectral Methods, and Applications to Heterogeneous Media. https://doi.org/10.20944/preprints202107.0569.v1