Undergraduate

Waves, diffusion and variational principles

On this page

• What you will study
• Entry
• If you have a disability or additional requirement
• Study materials
• Teaching and assessment
• Professional recognition
• Students also studied
• Future availability

This course focuses on three areas of applied mathematics. Waves builds your understanding of wave motion using vibrating strings and sound waves as examples. Techniques for solving linear partial differential equations are also developed. The diffusion section describes heat flow, and the flow of particles which follow random walks. Connections between random processes and deterministic diffusion processes are explained. The third section introduces variational principles and calculus through simple problems, such as determining the shortest line between two points on a curved surface. The Euler-Lagrange equation and the Lagrangian re-formulation of Newtonian mechanics are then explored.

Course facts

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What you will study

The three topics in this course are chosen to cover a broad range of applied mathematics techniques complementary to those developed in MST322. The course will start with some preparatory material discussing complex numbers, ordinary differential equations, and multivariable and vector calculus.

Waves are most familiar in the context of waves on the surface of water, but many other phenomena (sound, light, vibrations of mechanical structures, and even matter itself) can be understood as wave phenomena. The course will touch on these aspects, but the emphasis will be on gaining a clear understanding of the most easily understood examples of wave motion. This section of the course will introduce Fourier transforms and the method of separation of variables as techniques for solving linear partial differential equations. This block concludes with a discussion of how some of the characteristic phenomena of wave propagation (such as reflection, refraction, and Doppler shifts) can be understood from the solutions of the wave equation.

The diffusion equation is most commonly encountered in discussions of the flow of heat and of molecules moving in liquids, but diffusion equations arise from many different areas of applied mathematics. As well as considering the solutions of diffusion equations in detail, we also discuss the microscopic mechanism underlying the diffusion equation, namely that particles of matter or heat move erratically. This involves a discussion of elementary probability and statistics, which are used to develop a description of random walk processes and of the central limit theorem. These concepts are used to show that if particles follow random walk trajectories, their density obeys the diffusion equation.

Variational principles are most easily explained by giving an example: what is the shape of a chain which is held at the same height at each end? The principle of the solution is to find the shape which minimises the gravitational potential energy. The calculus of variations develops general methods for finding functions which minimise the value of a quantity which depends upon the function. The course develops the Euler-Lagrange equations, which give a general method for solving problems of this type. It also introduces Lagrangian mechanics, a re-formulation of Newtonian mechanics based upon a variational principle, which leads to powerful methods for simplifying complex mechanical problems.

You are assumed to have a sound knowledge of applied mathematics at the level of Mathematical methods and models (MST209).

You will learn

Successful study of this course should enhance your skills in communicating mathematical ideas clearly and succinctly, expressing problems in mathematical language and interpreting mathematical results in real-world terms.

Entry

This is a Level 3 course and you need a good knowledge of the subject area, obtained either from Level 2 study with The Open University or from equivalent work at another university. 

The course is designed to follow Using mathematics (MST121) and Exploring mathematics (MS221) and Mathematical methods and models (MST209). Your regional or national centre will be able to tell you where you can see reference copies of the courses, or you can buy selected texts from Open University Worldwide Ltd.

Applied mathematics at the level of MST209 would be suitable preparation, and it is assumed that you have sufficient knowledge of algebra, calculus and geometry (at the level of MS221) to have tackled such a Level 2 applied mathematics course. Some exposure to ideas of probability and statistics (at the level of MST121) would be an advantage. Students are more likely to complete this course successfully if they have acquired their prerequisite knowledge through passing MST209 (preferably with a grade 1 or 2).

If you have any doubt about the suitability of the course, please contact our Student Registration & Enquiry Service.

Preparatory work

Although the course revises the necessary topics in the first block, you need to have a good pass in MST209 (preferably grade 1 or 2), or an equivalent background before you start the course.

If you have a disability or additional requirement

If you are a new student, or new to courses using a computer or the internet, you will need to inform us of your particular needs as soon as possible, as some of our support services may take several weeks to arrange. Details of how to do this and our range of support services are described in our publication Meeting Your Needs.

You can also find information about accessible study materials, financial support and the Disabled Students' Allowance, equipment and other services, on our Services for disabled students website. It also includes our contact details for advice and support both before you register and while you are studying.

Study materials

What's included

Course books, supplementary printed materials and website.

You will need

Scientific calculator with basic mathematical functions (exp(x), ln(x), sin(x), arcsin(x) and so on) and memory.

Computing requirements

This course includes online computer activities – you can access these using a web browser that can play Flash and Shockwave.

You will need internet access and a computer. If you have purchased a new computer since 2005 it should meet your course computing requirements. Check our Technical Requirements section if your computer is older than this or is otherwise unusual.

Teaching and assessment

Support from your tutor

You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials or day schools that you are encouraged, but not obliged, to attend. Where your tutorials are held will depend on the distribution of students taking the course. 

Contact our Student Registration & Enquiry Service if you want to know more about study with The Open University before you register.

Assessment

The assessment details for this course can be found in the facts box above.

Please note that TMAs for all undergraduate mathematics and statistics courses must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.

Professional recognition

This course may help you to gain recognition from a professional body. You can view or download our Recognition leaflets 3.6 Institute of Mathematics and its Applications, 3.7 Computing, and 3.8 Scientific Institutions for further information.

Students also studied

Students who studied this course also studied at some time:

• Mathematical methods and models (MST209)
• Pure mathematics (M208)
• Electromagnetism (SMT359)
• Mathematical modelling (MSXR209)
• Exploring mathematics (MS221)

Future availability

The details given here are for the course that starts in February 2012. It will be available again in October 2013. We then expect it to be available once a year, in October.

Fees 2012

See fees and funding options for study from September 2012.

Study explained

Financial support

- find out if you qualify for support with your fees with our eligibility checker.

Study explained

- all you need to know about distance learning with the OU.

Student Reviews

I found this course both very interesting and very useful. Differential equations, and partial differential equations, are essential tools for ...
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I found this course significantly more difficult than other Level 3 maths courses, despite being very well prepared for it. ...
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Your questions

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Or contact an adviser in our Student Registration & Enquiry Service Email or call +44(0) 845 300 60 90+44(0) 845 366 60 35

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