## Role

Dr Tranter is responsible for teaching the modules Mathematical Methods and Partial Differential Equations. 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.

## Career overview

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 has been Placement Manager in the Department of Mathematics and Physics since August 2020, responsible for undergraduate placements and summer internships, including Mathematics Undergraduate Research Studentships (MURS).

## Research areas

Dr Tranter's research interests are in nonlinear waves, fluid dynamics, mathematical modelling and numerical analysis. This lies within the Imaging, Materials and Engineering Research Centre and the Computation and Simulation research area within the department. Specifically Dr Tranter research is in the following areas:

- 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, as well as the NTU Doctoral School website. 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.

**Solutions to the Periodic Korteweg-de Vries Equation with Varying Dispersion**

Recent work on the Korteweg-de Vries (KdV) equation has shown that a single-lobed periodic initial condition will generate several soliton-like excitations. The amplitudes of these soliton-like excitations can be found by studying the spectrum of the Schrödinger equation for a periodic potential (known as Hill's equation) using the Wentzel-Kramers-Brillouin (WKB) method. This leads to a series of band widths and band gaps, and solitons emerge when the band width shrinks to zero. Experimental results have confirmed these estimates.

Of interest is the behaviour of the solution to the periodic KdV equation with a varying dispersion coefficient. This project will explore whether the behaviour of the solution and the number of solitary waves present can be predicted from the change in the dispersion coefficient. The predictions will be checked against numerical simulations for various changes to the dispersion coefficient. For the non-periodic KdV equation, this situation of a changing dispersion coefficient can arise in the scattering problem for a layered elastic waveguide with a perfect bond between the layers, as the dispersion coefficient will change as a longitudinal strain wave passes from a bonded section to a delaminated section (no bonding between the layers). Successful results in this part of the project will lead to the consideration of the scattering problem in the context of periodic waves, with the aim of creating a non-destructive testing method for layered waveguides with delamination.

**Optimising Ambulance Dispatching using Networks**

The Ambulance Service receives, on average, over 20,000 calls per day across England, although this varies across each region. This poses a significant challenge in ensuring that the response to each call is as quick as possible. One challenge is determining the best way to use the limited resources to answer calls in the most efficient manner, taking account of the severity of the calls. Ambulance Quality Indicators (AQI) are used to determine the efficiency of the system, with the highest priority calls (C1) requiring, for example, an average response time of 7 minutes.

This project will consider the optimisation problem on a network with variable edge weights, simulating the changing traffic conditions. Ambulances on the network will be located at nodes and will need to be allocated to incoming calls, also present at nodes on the network. A statistical model will be used to simulate the random nature of incoming calls to the ambulance service, as well as randomising their category and therefore the response time required. The type of call will also determine the time required to address the call, and therefore the time during which the ambulance cannot respond to other calls. The project will start from a simple model of this network, building more complexity such as "peak" times as the project evolves, with the aim of reducing the mean response time in all categories.

## External activity

Refereeing for scientific journals:

- Baghdad Science Journal
- Mathematical Reviews

## Publications

K. R. Khusnutdinova, M. R. Tranter, "D'Alembert-type solution of the Cauchy problem for the Boussinesq-Klein-Gordon equation", *Studies in Applied Mathematics* **142** 551-585 (2019)

M. R. Tranter, "Solitary wave propagation in elastic bars with multiple sections and layers", *Wave Motion* **86** 21-31 (2019)

K. R. Khusnutdinova, Y. A. Stepanyants, M. R. Tranter, "Soliton solutions to the fifth-order Korteweg–de Vries equation and their applications to surface and internal water waves", *Physics of Fluids* **30** 022104 (2018)

K. R. Khusnutdinova, M. R. Tranter, "On radiating solitary waves in bi-layers with delamination and coupled Ostrovsky equations", *Chaos* **27** 013112 (2017)