EPSRC logo

Details of Grant 

EPSRC Reference: EP/L018802/2
Title: Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory
Principal Investigator: Cherednichenko, Dr K
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematical Sciences
Organisation: University of Bath
Scheme: EPSRC Fellowship
Starts: 23 July 2014 Ends: 22 June 2019 Value (£): 710,104
EPSRC Research Topic Classifications:
Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:  
Summary on Grant Application Form
In order to understand how various physical media behave under specific conditions, for example: a) how the earth surface

is deformed during an earthquake, or b) what image resolution one can achieve with a fibre-optic endoscope, mathematicians write

differential equations (DEs) and study analytical properties of their solutions, which are then interpreted to draw conclusions

about real-life objects. Part of this activity involves analysis of DEs that additionally depend on a parameter. In the above examples

this parameter could be: a) the ratio of the size of the rock-forming crystals to the thickness of layers of rock in the ground, or

b) the thickness-to-length ratio of the endoscope.

The project will develop a new approach to the analysis of solutions of parameter-dependent DEs, based on recent achievements

in the mathematical theory of ``operators'' and their spectra, which could roughly be thought of as the sets of the operator ``values''.

In general, the spectrum of an operator is found in the two-dimensional-plane of ``complex'' numbers. However, for many operators

representing DEs, the spectrum is a subset of a straight line in this plane. It turns out that considering such an operator as a member of a

wider operator family, whose spectra are not necessarily situated on the same line, brings about a lot of technical benefits, in a similar way

analysis in the complex plane helps understanding real numbers. In the last 50 years or so, many elegant mathematical results about

operators (and about DEs as their particular case) have been obtained by using this analogy. We will exploit these results in order to

improve our understanding of the behaviour of families of DEs. As a particular source of such families we will study equations representing

composites, i.e. media that have several simpler constituent parts. Many objects around us are composites, for example, wood, porous

rocks, foams, bubbly liquids, reinforced resins, polycrystal metals. Mathematical statements that we aim at will provide new information

about such real-life objects concerning, for example, their acoustic properties, or the way in which they interact with an electromagnetic

field. From the physics point of view, members of this wider operator family admit some dissipation (i.e. loss of energy) in comparison to

the original ``loss-free'' setup. The project will provide a general mathematical framework for such dissipative extensions in the case of DEs

describing composites, yielding a new analytic approach to the study of their effective properties.
Key Findings
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Potential use in non-academic contexts
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Description This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Date Materialised
Sectors submitted by the Researcher
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Project URL:  
Further Information:  
Organisation Website: http://www.bath.ac.uk