Dr Tranter is responsible for teaching the modules Mathematical Methods and Topics in Applied Mathematics. His research involves the construction of weakly-nonlinear approximations to partial differential equations, particularly those governing nonlinear waves, as well as numerical simulations. He also has an interest in fluid dynamics.
Dr Tranter completed his PhD at Loughborough University in 2018, modelling the propagation of nonlinear waves in layered waveguides. He then continued to work at Loughborough University on the modelling of droplets rolling down an incline plane, with either smooth or rough topography, using the diffuse-interface Cahn-Hilliard-Navier-Stokes model. This model was used to simulate the effect of rain droplets on a solar panel.
In 2019 Dr Tranter started work as a Lecturer in Applied Mathematics at Nottingham Trent University, his current role. He continues to work on the propagation of nonlinear waves, such as solitons, in layered waveguides with a particular focus on the detection of finite delaminations within the structure, as well as continuing his research on droplet motion.
Dr Tranter's research interests are in nonlinear waves, fluid dynamics, mathematical modelling and numerical analysis, specifically:
- Localised and periodic nonlinear wave propagation in layered waveguides
- Pseudospectral and finite-difference techniques for partial differential equations
- Fluid-structure interaction problems
- Diffuse-interface models for droplet motion
- Multiscale methods
- Weakly-nonlinear solutions to nonlinear partial differential equations
If you are interested in applying for a MRes or PhD in any of the areas above, please e-mail Dr Tranter for further information. Some available PhD projects include:
- Initial-value Problem for a System of Boussinesq-type Equations
The propagation of nonlinear waves in a layered elastic waveguide can be described by Boussinesq-type equations. The type of bonding material between the layers determines the system of equations that arise. For a two-layered bar with a soft bonding layer, a system of coupled Boussinesq equations are derived. Solutions to these equations have been found, but the problem for three or more layers is more complicated.
The aim of this project is to study the initial-value problem describing wave propagation in a single section of a layered waveguide, for three or more layers. The first case will consider a three-layered waveguide and analysis of the dispersion curves will instruct the study of this system. A weakly-nonlinear solution will be constructed to this system of equations, with different regimes as instructed by the analysis of the dispersion curves. This will then be generalised to a given number of layers. Further analysis could be performed on the behaviour of the solutions to the system in a layered waveguide incorporating delamination.
- Construction of Weakly-Nonlinear Solutions Respecting the Apparent Zero-Mass Contradiction
Recent works have shown that construction of a weakly-nonlinear solution to a governing equation can lead to additional constraints on the solution. For a Boussinesq-Klein-Gordon equation, it was shown that the constructed solution must take account of the mass of the initial condition, as the derived equations in the constructed solution necessarily require zero mass.
This project will explore the issue of zero-mass for an equation or system of equations, ensuring that the constructed solution takes account of any additional restraints imposed by the derived equations. This will initially study a Boussinesq-type system of two or more equations, with different coupling conditions as derived for a layered waveguide with bonding between the layers. The project will then consider other equation systems where such a contradiction can arise. The Boussinesq system of equations can arise for surface water waves in shallow water channels or from layered elastic waveguides with bonding between the layers.
Opportunities arise to carry out postgraduate research towards an MPhil / PhD in the areas identified above. Further information may be obtained on the NTU Research Degrees website https://www.ntu.ac.uk/research/research-degrees-at-ntu.
Refereeing for scientific journals:
- Baghdad Science Journal
- Mathematical Reviews