Global Opportunities

MAS3810 : Methods for Differential Equations & Partial Differential Equations & Non -Linear Waves

Semesters
Semester 1 Credit Value: 10
Semester 2 Credit Value: 10
ECTS Credits: 10.0

Aims

Introduce a range of advanced methods for solving ordinary and partial differential equations, with appropriate applications

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.


•       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;
•       Charpit's method for fully nonlinear 1st order PDEs ;
•       Nonlinear waves with applications to traffic flow;
•       Solitons and shockwaves.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture41:004:00Revision Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture101:0010:00Problem Classes – Synchronous On-Line
Guided Independent StudyAssessment preparation and completion301:0030:00Completion of in course assessments
Scheduled Learning And Teaching ActivitiesLecture401:0040:00Formal Lectures – Present in Person
Guided Independent StudyIndependent study1161:00116:00Preparation time for lectures, background reading, coursework review
Total200:00
Teaching Rationale And Relationship

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 Methods

The format of resits will be determined by the Board of Examiners

Exams
Description Length Semester When Set Percentage Comment
Written Examination1502A80N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M10Coursework assignment
Prob solv exercises2M10Coursework assignment
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. 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.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.

Reading Lists

Timetable