# Solving differential equations with Fourier extension

PGR-P-1374

## Key facts

Type of research degree
PhD
Project start date
Sunday 1 October 2023
Country eligibility
International (open to all nationalities, including the UK)
Funding
Competition funded
Supervisors
Dr Jitse Niesen
Schools
School of Mathematics
Research groups/institutes
Applied Mathematics, Applied Nonlinear Dynamics

Fourier series are a classical method for approximating functions which can be used in spectral methods to solve differential equations. However, Fourier series do not work well for non-periodic functions and they do not work at all on irregular domains in two or more dimensions.<br /> <br /> Fourier extension provide a way around this problem. The idea is to extend the irregular domain to a rectangle and approximate the unknown function by a Fourier series defined on the rectangle. Earlier research shows that this method can in principle be used to solve differential equations but there are some practical issues. Firstly, Fourier series form an orthogonal basis leading to very nice properties, but Fourier extension does not form a basis which means that the approximation problem is ill conditioned. Secondly, the method needs to be sped up if we want to compete with existing methods; the basic idea here is to use the Fast Fourier Transform but the details are not clear.<br /> <br /> The aim of this project is to design, analyze and implement a method based on Fourier extension for solving differential equations on irregular domains. We want a method with good performance for which we can prove spectral convergence. This project requires prior knowledge of numerical analysis (approximation theory, linear algebra, spectral methods) and the mathematical analysis that underpins this.

<p>Formal applications for research degree study should be made online through the&nbsp;<a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University&#39;s website</a>. Please state clearly in the Planned Course of Study section that you are applying for <em><strong>PHD Applied Mathematics FT</strong></em> and&nbsp;in the research information section&nbsp;that the research degree you wish to be considered for is <em><strong>Solving differential equations with Fourier extension</strong></em> as well as <a href="https://eps.leeds.ac.uk/maths/staff/4067/dr-jitse-niesen">Jitse Niesen</a> as your proposed supervisor.</p> <p>If English is not your first language, you must provide evidence that you meet the University&#39;s minimum English language requirements (below).</p> <p><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p> <p class="MsoNoSpacing">Applications will be considered on an ongoing basis. &nbsp;Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application. &nbsp;We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p>Please note that you must provide the following documents at the point you submit your application:</p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> <li>Funding information including any alternative sources of funding that you are applying for or if you are able to pay your own fees and maintenance</li> </ul>

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