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

• Offered for Year: 2020/21
• Module Leader(s): Dr Andrew Baggaley
• Lecturer: Dr Clive Emary
• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus
##### Semesters
 Semester 1 Credit Value: 10 Semester 2 Credit Value: 10 ECTS Credits: 10

#### 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

Please note that module leaders are reviewing the module teaching and assessment methods for Semester 2 modules, in light of the Covid-19 restrictions. There may also be a few further changes to Semester 1 modules. Final information will be available by the end of August 2020 in for Semester 1 modules and the end of October 2020 for Semester 2 modules.

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Guided Independent StudyAssessment preparation and completion301:0030:00Completion of in course assessment
Structured Guided LearningLecture materials361:0036:00Non-Synchronous Activities
Scheduled Learning And Teaching ActivitiesLecture91:009:00Synchronous On Line Material
Scheduled Learning And Teaching ActivitiesLecture91:009:00Present in Person
Structured Guided LearningStructured non-synchronous discussion181:0018:00Non Synchronous Discussion of Lecture Material
Scheduled Learning And Teaching ActivitiesDrop-in/surgery41:004:00Office Hour or Discussion Board Activity
Guided Independent StudyIndependent study941:0094:00Lecture preparation, background reading, coursework review
Total200:00
##### 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

Please note that module leaders are reviewing the module teaching and assessment methods for Semester 2 modules, in light of the Covid-19 restrictions. There may also be a few further changes to Semester 1 modules. Final information will be available by the end of August 2020 in for Semester 1 modules and the end of October 2020 for Semester 2 modules.

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

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination1202A80Alternative assessment - class test
##### Other Assessment
Description Semester When Set Percentage Comment
Written exercise1M10written exercises
Written exercise2M10written 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.