Adaptive Collocation Methods for Fractional Differential Equations (VC2656)

About the Project

The University of the West of Scotland (UWS) is seeking to attract a PhD candidate of outstanding ability and commitment to join its vibrant and growing programme of internationally excellent research.

Fractional differential equations provide a powerful framework for modelling complex systems with memory, nonlocal interactions, and multiscale behaviour. Such models are increasingly used across artificial intelligence, engineering, physics, and data science; however, their broader adoption is limited by the lack of robust and efficient numerical methods.

This project aims to develop adaptive collocation methods for fractional differential equations, combining ideas from approximation theory, numerical analysis, and scientific computing. A central aim is to design high-accuracy numerical schemes capable of resolving the nonlocal and potentially singular behaviour characteristic of fractional models.

The research will explore modern approximation techniques, including spline and rational representations of solutions, alongside adaptive strategies that concentrate computational effort where it is most beneficial. This includes investigating data-driven residual-based and neural-network based approaches to constructing efficient solvers.

The project will involve both theoretical and computational work, including the development and analysis of new algorithms, the implementation of software tools, and testing on representative applications in areas such as machine learning, energy systems, complex materials, biological modelling and physics informed neural networks.

The successful candidate will gain training in advanced numerical methods, machine learning, neural networks, data science, approximation theory, and scientific computing, with opportunities to contribute to interdisciplinary research and open-source software development.

The candidate/eligibility criteria

Applicants should hold, or expect to obtain, a first-class honours degree (or equivalent) in Mathematics, Electronic and Electrical Engineering, Physics, or a closely related discipline.

A strong background in calculus, differential equations, and linear algebra is essential. Familiarity with complex analysis and numerical analysis or scientific computing is highly desirable. Experience with programming (e.g. Python, Julia, MATLAB, or similar) will be advantageous, as the project involves the development and implementation of numerical algorithms.

The project will involve detailed use of fractional calculus; prior knowledge of this area would be beneficial but is not essential.

Applicants should have a clear interest in mathematical modelling, neural networks, approximation theory, machine learning, data science or computational science, and should demonstrate strong analytical ability, curiosity, and the capacity to work both independently and as part of a research team.

The successful candidate must meet the following criteria:

• be a UK National (meeting residency requirements),
• or have settled status,
• or have pre-settled status (meeting residency requirements),
• or have indefinite leave to remain

For more information, or to discuss the project informally, please contact Dr Ryan Meeten at ryan.meeten@uws.ac.uk.

Application Deadline: 15/06/2026

Start Date: 01/10/2026

Applications must be made via the UWS Online System.

Funding Notes

This is a fully funded PhD Studentship and includes payment of tuition fees for 36 months at the home/UK rate and an annual maintenance stipend equivalent to UKRI minimum stipend rate (£21,805pa from 01/10/2026).

The successful candidate must meet the following criteria:

• be a UK National (meeting residency requirements),
• or have settled status,
• or have pre-settled status (meeting residency requirements),
• or have indefinite leave to remain