MAS2802 : Differential Equations, Transforms and Waves
- Offered for Year: 2019/20
- Module Leader(s): Professor Anvar Shukurov
- Owning School: Mathematics, Statistics and Physics
- Teaching Location: Newcastle City Campus
Semester 2 Credit Value:
To introduce a range of advanced concepts and methods for solving ordinary and partial differential equations.
Many applications of mathematics lead to problems where the unknown is a function, a law expressing the dependence of a certain variable on others. Fundamental laws of nature often involve not only functions that characterise a certain phenomenon but also their rates of change in space and/or time. In such problems, the governing equation contains not only unknown functions but also their derivatives: these are differential equations.
This course continues the exploration of differential equation that started in Stage 1, with emphasis on methods to solve them, both exact and approximate. The essential elements in the theory of ordinary and partial differential equations, and their methods of solution, introduced in this course, provide the basis for specific studies in other modules. The methods that will be introduced, justified and practiced apply to a wide range of ordinary and partial differential equations. The course will focus on understanding and solving differential equations rather than on extensive physical interpretation (although this aspect certainly will be mentioned).
Outline Of Syllabus
• A review of the fundamental concepts of ordinary and partial differential equations.
• Series solutions.
• Elements of the Sturm-Liouville theory.
• Fourier series and Fourier transforms.
• Wave equation: simple derivation in one spatial dimension, generalisation to three dimensions, fundamental properties (D’Alembert’s solution, phase speed, plane waves, superposition, standing and travelling waves), wave packets.
• Second-order partial differential equations. Separation of variables in Cartesian coordinates: application to the wave, heat, Laplace’s and Poisson’s equations.
|Scheduled Learning And Teaching Activities||Lecture||1||1:00||1:00||Class test|
|Scheduled Learning And Teaching Activities||Lecture||3||1:00||3:00||Problem classes|
|Scheduled Learning And Teaching Activities||Lecture||2||1:00||2:00||Revision lectures|
|Scheduled Learning And Teaching Activities||Lecture||25||1:00||25:00||Formal lectures|
|Guided Independent Study||Assessment preparation and completion||1||6:00||6:00||Revision for class test|
|Guided Independent Study||Assessment preparation and completion||1||13:00||13:00||Revision for unseen exam|
|Guided Independent Study||Assessment preparation and completion||1||2:00||2:00||Unseen exam|
|Scheduled Learning And Teaching Activities||Drop-in/surgery||4||1:00||4:00||Tutorials in the lecture room|
|Scheduled Learning And Teaching Activities||Drop-in/surgery||12||0:10||2:00||Office hours|
|Guided Independent Study||Independent study||1||21:00||21:00||Studying, practising and gaining understanding of course material|
|Guided Independent Study||Independent study||3||3:00||9:00||Review of coursework assignments and class test|
|Guided Independent Study||Independent study||2||6:00||12:00||Preparation for coursework assignments|
Jointly Taught With
|PHY2031||Differential Equations, Transforms and Waves|
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. Tutorials are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. In addition, office hours (two per week) 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.
The format of resits will be determined by the Board of Examiners
|Written Examination||40||2||M||10||Class test|
|PHY2031||Differential Equations, Transforms and Waves||2||N/A|
|Prob solv exercises||2||M||5||Coursework assignments|
Assessment Rationale And Relationship
A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments will consist of two written assignment of approximately equal weight. The coursework assignments and the (in class, therefore 40 minute) coursework test 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.