# PHY2031 : 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
##### Semesters
 Semester 2 Credit Value: 10 ECTS Credits: 5

#### Aims

To introduce a range of advanced concepts and methods for solving ordinary and partial differential
equations.

Module summary

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.

#### Teaching Methods

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture11:001:00Class test
Scheduled Learning And Teaching ActivitiesLecture31:003:00Problem classes
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Scheduled Learning And Teaching ActivitiesLecture251:0025:00Formal lectures
Guided Independent StudyAssessment preparation and completion16:006:00Revision for class test
Guided Independent StudyAssessment preparation and completion113:0013:00Revision for unseen exam
Guided Independent StudyAssessment preparation and completion12:002:00Unseen exam
Scheduled Learning And Teaching ActivitiesDrop-in/surgery41:004:00Tutorials in the lecture room
Scheduled Learning And Teaching ActivitiesDrop-in/surgery120:102:00Office hours
Guided Independent StudyIndependent study121:0021:00Studying, practising and gaining understanding of course material
Guided Independent StudyIndependent study33:009:00Review of coursework assignments and class test
Guided Independent StudyIndependent study26:0012:00Preparation for coursework assignments
Total100:00
##### Jointly Taught With
Code Title
MAS2802Differential 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.

#### Assessment Methods

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

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination1202A85unseen
Written Examination402M10Class test
##### Exam Pairings
Module Code Module Title Semester Comment
MAS2802Differential Equations, Transforms and Waves1N/A
##### Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M5Coursework 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.