Aims

To develop further the theory of partial differential equations, including methods of solution and more general results, with appropriate applications.

Module Summary

Almost all studies of physical phenomena lead to partial differential equations (PDEs), which have been studied for over 250 years; they are at the heart of modern applied mathematics, physics and engineering. It was soon noticed that many very similar – often identical – equations arise in many and varied applications, all with correspondingly similar solutions and methods of solution. This module continues the study of differential equations undertaken at Stage 2, bringing all these ideas together, developing more general methods for first-order PDEs with a particular focus on nonlinear wave equations. In addition, some of the standard results and theorems relating to classical PDEs will also be discussed. Examples of these equations, and methods of solution, will be taken from various practical, relevant and important applications.

Outline Of Syllabus

•       Classification and methods of solution for some classes of first-order partial differential equations, including the Cauchy problem, and Lagrange’s and the parametric methods of solution;

•       Classification of second-order semi-linear PDEs;

•       Nonlinear waves with applications to traffic flow;

•       Solitons and shockwaves;

•       Introduction to numerical modelling of PDEs.

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-order 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 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.