Differential Equations

Module Provider: Mathematics and Statistics
Number of credits: 20 [10 ECTS credits]
Level:5
Terms in which taught: Autumn / Spring / Summer module
Pre-requisites: MA1CA Calculus MA1LA Linear Algebra
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA2ODE Ordinary Differential Equations MA2PDE Partial Differential Equations
Module version for: 2016/7

Summary module description:
In this module, we continue the ODE work of Part 1 and consider more advanced topics such as ODEs with non-constant coefficients, integral and series solutions, Fourier series and the theory of boundary value problems. This is then extended into the study of partial differential equations, in particular the diffusion equation, the wave equation and Laplace?s equation, for which appropriate solution techniques are studied.

Aims:
To further develop the study of ordinary differential equations building on ODEs met in Part 1 and to introduce and develop the study of partial differential equations and their applications.

Assessable learning outcomes:
By the end of the module the student is expected to be able to:
- Solve non-constant coefficient ODEs
- Construct and use Green's function to solve appropriate ODEs and PDEs problems
- Use series solution techniques for ODEs
- Use integral transform techniques to solve IVPs for ODEs and PDEs
- Derive the Fourier series of a function
- Use eigenfunction expansions to solve appropriate BVPs for ODEs and PDEs
- Use Duhamel's principle and the heat kernel to solve homogeneous and inhomogeneous diffusion problems
- Solve the wave equation using D'Alembert's formula
- Appropriately use maximum principles
- Solve a variety of PDEs using the separation of variables technique
Additional outcomes:
The student will also achieve an improved understanding of the issues of existence and uniqueness of solutions and the ability to provide a physical interpretation of their mathematics.

Outline content:
Differential equations are at the heart of modern applied mathematics. For ODEs we continue the work of part 1 and consider more advanced topics such as ODEs with non-constant coefficients, Laplace transform and series solutions, Fourier series and the theory of boundary value problems including eigenfunction expansion techniques for simple Sturm Liouville problems. For PDEs the module uses the diffusion, wave and Laplace?s equations as exemplars. Their solution properties are explored, including the different type of problems (IVP, IBVP and BVP) for which they are well posed as well as such issues as maximum principles for elliptic and parabolic PDEs. Solution techniques such as the heat kernel, Duhamel?s principle, separation of variables and D?Alembert?s solution are introduced as well as extending the Laplace transform, Greens functions and eigenfunction expansions to PDE problems. The relationship of PDEs to mathematical modelling of the physical sciences is highlighted.

Brief description of teaching and learning methods:
Lectures, supported by problem sheets and weekly tutorials.

Contact hours:
lectures- 20
tutorials- 10
Guided independent study- 68
Total hours by term- 98
Total hours for module- 200

Summative Assessment Methods:
Written exam- 70%
Set exercise- 30%

Other information on summative assessment:
Six pieces of assessed work.

Formative assessment methods:
Problem sheets.

Length of examination:
3 hours.

Requirements for a pass:
A mark of 40% overall.

Reassessment arrangements:
One examination paper of 3 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous coursework marks (70% exam, 30% coursework).