Role
Dr Tranter is responsible for teaching the modules Calculus, Partial Differential Equations and Professional Development. His research involves the construction of weakly-nonlinear approximations to partial differential equations, particularly those governing nonlinear waves and solitons, as well as numerical solutions to PDEs. Dr Tranter also explores droplet motion on patterned surfaces. More recently, Dr Tranter has begun research in mathematical education, with a particular focus on Learning for Mastery.
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, progressing to Senior Lecturer in 2021. Since 2024 he has been Principal Lecturer and Postgraduate Courses Manager for Physics and Mathematics. 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 fluid-structure interaction and diffuse-interface models for droplet motion. He has also begun to work on Mathematical Education topics, including Learning for Mastery.
From 2020 to 2024, Dr Tranter was the Placement Manager in the Department of Mathematics and Physics , 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 nonlinear wave propagation in layered waveguides
- Pseudospectral and finite-difference numerical techniques for partial differential equations
- Fluid-structure interaction problems
- Diffuse-interface models for droplet motion
- Weakly-nonlinear solutions to nonlinear partial differential equations
Dr Tranter also has an interest in research on student learning and achievement. Areas include:
- Impact of Learning for Mastery assessments (formative/summative) on achievement and progression
- Links between attendance and attainment, in light of post-pandemic behaviours, and "Consistency Bias" across modules
- Impact on attainment from work placements, attendance/engagement and semi-open book assessments
As part of our MRes Mathematical Sciences course, you can undertake research in one of these areas, or propose your own. Please e-mail Dr Tranter for further information on the projects listed above. If you are interested in PhD study around these topics, you can visit the NTU Doctoral School website for further information and then . Some available MRes/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 layered waveguide for three or more layers, constructing a weakly-nonlinear solution with different regimes as instructed by the analysis of the dispersion curves. If possible, this could be generalised to a given number of layers.
- Delamination detection in layered waveguides
Layered structures are highly dependent on the material and bonding between each layer; a gap in the bonding (delamination) could lead to the collapse of an entire structure. We can use methods such as ultrasound and x-ray imaging to detect this defect, although their range is limited. Recent advancements in nonlinear wave modelling have shown that long waves, such as solitons (localised stable waves), can be used to detect and classify delaminations by determining their position and length.
This project will explore the models that can be used to describe waves in layered waveguides and how these can be used to identify delaminations in such structures. Waves are generated at one side of the structure and changes in the transmitted wave field are used to determine the length and position of the delaminated region.
- 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.
External activity
Refereeing for scientific journals:
- Wave Motion
- Science Advances
- Physica Scripta
- European Physical Journal Plus
- Heliyon
- Chaos, Solitons and Fractals
- SIAM Journal on Applied Mathematics
Dr Tranter has also served as a reviewer for the EPSRC.
Publications
J. S. Tamber, D. J. Chappell and M. R. Tranter, Delamination detection in layered waveguides using Ostrovsky wave packets. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 481 (2307) 20240574 (2025).
J. S. Tamber, D. J. Chappell, J. C. Poore and M. R. Tranter, Detecting delamination via nonlinear wave scattering in a bonded elastic bar. Nonlinear Dynamics 112 23-33 (2023)
K. R. Khusnutdinova and M. R, Tranter, Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction, Chaos 32 (11) 113132 (2022)
J. S. Tamber and M. R. Tranter, Scattering of an Ostrovsky wave packet in a delaminated waveguide, Wave Motion 114 103023 (2022)
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 and 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)