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 it applications to PDEs.
•       Green's functions for PDEs: application to Laplace and Poisson equations.

Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. Students who cannot attend a present-in-person session will be provided with an alternative activity allowing them to access the learning outcomes of that session. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.

Alternatives will be offered to students unable to be present-in-person due to the prevailing C-19 circumstances.
Student’s should consult their individual timetable for up-to-date delivery information.

Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will 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.