## Mathematical Modelling

Researchers: John Butler; Alberto Caimo; Dana Mackey; Stephen O'Sullivan

Collaborators: Chris Hills; Rossen Ivanov; Brendan Redmond; Environmental Sustainability & Health Institute (ESHI); Industrial & Engineering Optics (IEO); Facility for Optical Characterisation & Spectroscopy (FOCAS)

### Description of area

Mathematical modelling covers a wide range of mathematical topics and disciplines and focuses on the application of advanced mathematical techniques to physical problems. For example, staff have engaged in mathematical modelling associated with: theoretical neuroscience & signal processing (John Butler); computational statistics (Alberto Caimo); problems from physics or biotechnology, nonlinear dynamical systems & differential equations (Dana Mackey); fluid mechanics & fluid processes, geophysical & environmental fluid flows (Brendan Redmond, Chris Hills, Rossen Ivanov); computational finance & numerical analysis (Stephen O'Sullivan). Details of individual research in these areas can also be found on the Fluid Mechanics, Mathematical Physics and Statistics research pages.

Examples of research projects include the following:

Mathematical modelling of holography (Dana Mackey)

A collaborative project with the Industrial & Engineering Optics (IEO) centre at DIT aims to develop a set of mathematical models based on partial differential equations that will characterise, and ultimately guide the design of, photosensitive materials capable of copying with high fidelity an illumination pattern into a holographic grating.

Optical patterning in considerably more complex systems such as hybrid materials containing zeolite nanoparticles are also considered, addressing issues such as shrinkage minimisation and optimisation of dopant redistribution for fabrication of environmental monitoring sensors.

Mathematical modelling of immunoassays (Dana Mackey)

An interdisciplinary project is using mathematical models to optimize the performance of immunodiagnostic devices (or immunoassays). It is a collaboration with the BDI (Biomedical Diagnostics Institute) and NCSR (National Centre for Sensor Research) at DCU. Such devices rely on the binding of antigens by antibodies and are used to detect biomarkers for a variety of diseases. Many immunoassay technologies involve immobilization of the antibody to solid surfaces: one of the modelling challenges in such cases is optimizing the concentration of capture antibody in order to achieve maximal antigen binding and, subsequently, improved sensitivity and limit of detection.

Computational finance (Stephen O'Sullivan)

Stiff systems of parabolic partial differential equations arise in financial models including Heston's stochastic volatility model and Bates' jump diffusion model. Stabilized explicit numerical integration techniques offer an effective approach to solving systems of this type.

For more information about any of these areas and to discuss opportunities for research please contact the individuals above.