Helmholtz equation in unbounded domains with complicated boundary conditions

Helmholtz equation is an important partial differential equation (PDE) which has many applications in science and engenering. Despite being a well studied linear PDE simple methods such as separation of variables work only in rare case when the boundary conditions are of specific form and on specific domains. There are other analytical methods (i.e. methods that provide an exact solution) which have been used such as Wiener–Hopf method which extend the applicability of analytical methods. There are also many numerical methods that could be used to solve such problems.

The key advantage of analytical methods over numerical ones is that the analytic solution contain the parameter dependence of the problem. Unfortunately, for more complicated boundary conditions, typically analytical methods do not exist. The idea of this project is to develop semi-analytical solutions that preserve this important feature of analytical methods. Specifically, to build up the solution to the problem from simpler sub-problems that have analytical solutions. These sub-problems will be linked numerically but nevertheless when parameters of the problems are changed the analytical blocks could be re-used giving a huge saving on computational time.

Eligibility

Applicants should have, or expect to achieve, at least a 2.1 honours degree or a master’s (or international equivalent) in a relevant science or engineering related discipline.

Funding

At Manchester we offer a range of scholarships, studentships and awards at university, faculty and department level, to support both UK and overseas postgraduate researchers applying for competition and self-funded projects.

For more information, visit our funding page or search our funding database for specific scholarships, studentships and awards you may be eligible for.

Before you apply

We strongly recommend that you contact the supervisor(s) for this project before you apply. Please include details of your current level of study, academic background and any relevant experience and include a paragraph about your motivation to study this PhD project.

How to apply

Apply online through our website: https://uom.link/pgr-apply-2425

When applying, you’ll need to specify the full name of this project, the name of your supervisor, if you already having funding or if you wish to be considered for available funding through the university, details of your previous study, and names and contact details of two referees.

Your application will not be processed without all of the required documents submitted at the time of application, and we cannot accept responsibility for late or missed deadlines. Incomplete applications will not be considered.

After you have applied you will be asked to upload the following supporting documents:

• Final Transcript and certificates of all awarded university level qualifications
• Interim Transcript of any university level qualifications in progress
• CV
• Supporting statement: A one or two page statement outlining your motivation to pursue postgraduate research and why you want to undertake postgraduate research at Manchester, any relevant research or work experience, the key findings of your previous research experience, and techniques and skills you’ve developed. (This is mandatory for all applicants and the application will be put on hold without it).
• Contact details for two referees (please make sure that the contact email you provide is an official university/work email address as we may need to verify the reference)
• English Language certificate (if applicable)

If you have any questions about making an application, please contact our admissions team by emailing [email protected].

Equality, diversity and inclusion is fundamental to the success of The University of Manchester, and is at the heart of all of our activities. We know that diversity strengthens our research community, leading to enhanced research creativity, productivity and quality, and societal and economic impact.

We actively encourage applicants from diverse career paths and backgrounds and from all sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation and transgender status.

We also support applications from those returning from a career break or other roles. We consider offering flexible study arrangements (including part-time: 50%, 60% or 80%, depending on the project/funder).