# Applied Mathematics projects

April 21, 2014

Below is a list of projects available under each supervisor.

a) Asymptotic analysis of boundary value problems in singularly perturbed domains
The project involves analysis of the Dirichlet and Neumann boundary value problems for the Laplace operator posed in domains with singularly pertubed boundaries. Examples include bodies with small holes or inclusions. The objective is to study the asymptotic approximations of solutions to these boundary value problems and to obtain the estimates for the error of the approximations. We shall also discuss applications in electrostatics, elasticity and theory of steady flow of inviscid fluid.

b) Integro-differential equations of the Peierls type in models of dislocations
Classical examples of equations of the Peierls type include the Peierls-Nabarro equation describing a 1D model for an edge (or screw) dislocation in a crystal and the Benjamin-Ono equation that occurs in models of fluid mechanics for stratified fluids. The objective of this project is to study the mathematical models of dislocations that can be formulated in the form of equations of the Peierls type. We shall also work with the discrete lattice models obtained as a result of regularization of the continuum Peierls-Nabarro equation. Finally, stability of dislocations in a discrete lattice will be considered.

a) Spectral problems for elliptic operators in domains with small defects
The objective is to study spectral problems for the Laplace or the Lamé operators in domains with small inclusions or cavities. Assuming that the solution of the unperturbed problem (for a domain without defect) is given we shall obtain the asymptotic approximation for the first positive eigenvalue and the corresponding eigenfunction for a problem posed in a domain with a defect. This is a singularly perturbed problem and the asymptotic algorithm involves analysis of the boundary layer near the inclusion. Applications will be considered for models of electro-magnetism and elasticity.

b) Mathematical models of conical cracks
The Neumann boundary value problem will be considered for the Laplace (or the Lame) operator in an infinite domain containing a crack with a non-smooth front. The objective is to study the asymptotic behaviour of solutions near the edge of the crack and near the irregular point of the crack front. This project involves analysis of boundary value problems posed in domains with non-smooth boundaries and the spectral asymptotics in the vicinity of conical points. Applications will be considered for problems of elasticity and fracture mechanics.

a) Variational Image Segmentation and Automatic Feature Extraction by Differential Equations

There is a rich source of mathematics associated with digitisation. The traditional tasks include transmission, compression and filtering. Modern tasks tend to centre more on intelligent use of image techniques e.g. digital matching, high resolution restoration and medical pattern recognition and reconstruction. This project will study a particular technique called segmentation for image feature extraction. On the modelling side, we need to tackle the challenges of low intensity contrast and the presence of noise. Training in Matlab will be given, applications to real images from the Local Hospitals will be considered and on-site visits will be arranged.

b) Modeling and Enhancing Fluorescein Angiography

Fluorescein angiography (FA) is an indispensible tool in the diagnosis and management of disorders of the fundus in the eye. The image quality of FA is a decisive factor for ophthalmologists in their assessment and treatment planning for retinal disorders

This project aims to address a practically important problem of enhancing quality of retinal image by improving as well as developing a class of novel variational techniques. Both deblurring and blind deconvolution methods including effective algorithms will be investigated. The project will involve interactions with Prof S Harding’s group in St Paul’s Eye Unit (SPEU), Royal Liverpool University Hospital.

4. Dr R. Bearon (rm 404)

Dr R. Bearon

a) Flashing & Jumping: hydromechanical signalling in the plankton
Many plankton (small plants and animals living in aquatic environments) escape being eaten by flashing or jumping. This project will investigate what hydromechanical signals are created by the predator and what signals are detected by the prey. The module mathematical biology (math426) should be taken in conjunction with doing this project.

b) Random searching for food: bacterial chemotaxis
Bacteria are typically too small to sense chemical gradients directly and so they undergo a biased random walk consisting of smooth swimming segments (runs) interspersed by changes of direction (tumbles). This project will investigate mechanisms for biasing the random walk in order for bacteria to reach regions of high concentration of chemical attractant.

c) Dynamic modelling of cell signalling pathways
Many cell signalling pathways display periodic oscillations and transient pulses, and ordinary differential equations can be useful to describe such dynamic behaviour. In close collaboration with experimentalists, this project will explore how mathematics can be used to understand the dynamics of biological systems.

Source: www.liverpool.ac.uk
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