Aims

Introduce a range of advanced methods for solving ordinary and partial differential equations.

Module Summary
Most mathematical models are formulated in terms of differential equations. This module will
introduce a range of topics from the theory of differential equations that have proved to be useful in
solving practical problems. Equal emphasis will be placed on the theorems that underly the methods,
the technical skills required to apply them and the meaning of the results. Illustrative problems will be drawn from a wide range of practical applications.

Outline Of Syllabus

• Eigenfunction methods: Hermitian operators, Sturm-Liouville equations.
• Special functions: Legendre functions, Bessel functions.
• Well-posed problems: uniqueness and existence of solutions.
• Separation of variables for 2nd order PDEs in cylindrical and spherical coordinates: Laplace equation and spherical harmonics.
• The Fourier transform and its applications to PDEs.
• Green's functions for PDEs: application to Laplace and Poisson equations.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-level thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.