Global Opportunities

MAS3806 : Partial Differential Equations and Nonlinear Waves

Semesters
Semester 2 Credit Value: 10
ECTS Credits: 5.0

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 and touching on, and extending, the ideas of separation of variables for second-order PDEs. 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; the 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 ActivitiesLecture51:005:00Present in Person
Scheduled Learning And Teaching ActivitiesLecture41:004:00Synchronous on-line material
Structured Guided LearningLecture materials181:0018:00Non-Synchronous Activities
Guided Independent StudyAssessment preparation and completion151:0015:00Completion of in course assessment
Structured Guided LearningStructured non-synchronous discussion91:009:00Non Synchronous Discussion of Lecture Material
Scheduled Learning And Teaching ActivitiesDrop-in/surgery21:002:00Office Hour or Discussion Board Activity
Guided Independent StudyIndependent study471:0047:00Lecture preparation, background reading, coursework review, revision for exam
Total100:00
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 Methods

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

Exams
Description Length Semester When Set Percentage Comment
Written Examination602A80End of Semester exam
Other Assessment
Description Semester When Set Percentage Comment
Written exercise2M20Written Exercises
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.

Reading Lists

Timetable