MA307-6-AU-CO:
Advanced Ordinary Differential Equations
2020/21
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Undergraduate: Level 6
Current
Thursday 08 October 2020
Friday 18 December 2020
15
16 July 2020
Requisites for this module
MA202
(none)
(none)
(none)
(none)
The subject of Ordinary Differential Equations (ODEs) is a very important and fascinating branch in Mathematics. An abundance of phenomena in Physics, Biology, Engineering, Chemistry, Finance and Neuroscience to name a few, may be described and studied using such equations. The module will introduce students to advanced topics and theories in ODEs and dynamical systems.
The aim of the module is to familiarise students with advanced concepts and theories in Ordinary Differential Equations and serve as an introduction to dynamical systems. It will also equip students with the knowledge and skills to solve such equations by using advanced analytical approaches.
1. Use methods to solve linear ODEs with variable coefficients
2. Systematic understanding of phase planes, population dynamics and prey-predator systems
3. Systematic understanding of Liapunov’s method and function
4. Systematic understanding of periodic solutions and limit cycles
5. Critical awareness on plane autonomous systems and linearisation
6. Systematic understanding of autonomous systems, almost linear systems, nonlinear systems, and stability/instability
7. Use of power-series to construct solutions of ODEs
The module will be based on the following syllabus:
1. Linear ODEs with variable coefficients (Power-series solutions. Regular singular points)
2. The phase plane, population dynamics: competing species, prey-predator systems
3. Liapunov's method, Liapunov function
4. Periodic solutions and limit cycles. The Poincaré-Bendixson theorem
5. Plane autonomous systems and linearisation
6. Autonomous systems, almost linear systems, nonlinear systems, stability/instability
Teaching will be delivered in a way that blends face-to-face classes, for those students that can be present on campus, with a range of online lectures, teaching, learning and collaborative support.
- Boyce, William E. (2013) '9.7', in Student Solutions Manual to Accompany Elementary Differential Equations, Hoboken, NJ: Wiley., pp.253-258
- Meade, Douglas B. (2017) '9.7 Periodic Solutions and Limit Cycles', in Boyce's elementary differential equations and boundary value problems, Hoboken, NJ: Wiley., pp.565-577
- Hirsch, Morris W.; Smale, Stephen; Devaney, Robert L. (2013) Differential equations, dynamical systems, and an introduction to chaos, Amsterdam: Academic Press.
- Meade, Douglas B. (2017) '9.6 Liapunov’s Second Method', in Boyce's elementary differential equations and boundary value problems, Hoboken, NJ: Wiley., pp.554-565
- Boyce, William E. (2013) '9.5', in Student Solutions Manual to Accompany Elementary Differential Equations, Hoboken, NJ: Wiley., pp.247-250
- Boyce, William E. (2013) '9.2', in Student solutions manual to accompany Elementary differential equations, Hoboken, NJ: Wiley., pp.224-228
- Meade, Douglas B. (2017) '9.3 Locally Linear Systems', in Boyce's elementary differential equations and boundary value problems, Hoboken, NJ: Wiley., pp.519-531
- Boyce, William E. (2013) '9.8', in Student Solutions Manual to Accompany Elementary Differential Equations, Hoboken, NJ: Wiley., pp.258-263
- Meade, Douglas B. (2017) '9.5 Predator–Prey Equations', in Boyce's elementary differential equations and boundary value problems, Hoboken, NJ: Wiley., pp.544-554
- Meade, Douglas B. (2017) '9.2 Autonomous Systems and Stability', in Boyce's elementary differential equations and boundary value problems, Hoboken, NJ: Wiley., pp.508-518
- Boyce, William E. (2013) '9.6', in Student Solutions Manual to Accompany Elementary Differential Equations, Hoboken, NJ: Wiley., pp.250-253
- Boyce, William E. (2013) '9.3', in Student solutions manual to accompany Elementary differential equations, Hoboken, NJ: Wiley., pp.228-237
- Meade, Douglas B. (2017) '9.4 Competing Species', in Boyce's elementary differential equations and boundary value problems, Hoboken, NJ: Wiley., pp.531-544
- Meade, Douglas B. (2017) '9.8 Chaos and Strange Attractors: The Lorenz Equations', in Boyce's elementary differential equations and boundary value problems, Hoboken, NJ: Wiley., pp.577-586
- Boyce, William E.; DiPrima, Richard C.; Meade, Douglas B. (2017) Boyce's elementary differential equations and boundary value problems, Hoboken, NJ: Wiley.
- Boyce, William E. (2013) '9.4', in Student Solutions Manual to Accompany Elementary Differential Equations, Hoboken, NJ: Wiley., pp.237-246
- Meade, Douglas B. (2017) '9.1 The Phase Plane: Linear Systems', in Boyce's elementary differential equations and boundary value problems, Hoboken, NJ: Wiley., pp.495-508
- Boyce, William E. (2013) '9.1', in Student Solutions Manual to Accompany Elementary Differential Equations, Hoboken, NJ: Wiley., pp.219-224
- D. Jordan; P. Smith. (2007) Nonlinear Ordinary Differential Equations: Problems and Solutions: Oxford University Press, USA.
- Jordan, D. W.; Smith, P. (2007) Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Oxford: Oxford University Press.
The above list is indicative of the essential reading for the course. The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students. Further reading can be obtained from this module's reading list.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Test |
|
|
| Exam |
Main exam: 180 minutes during Summer (Main Period)
|
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Chris Antonopoulos, email: canton@essex.ac.uk.
Dr Chris Antonopoulos & Dr Georgi Grahovski
Dr Chris Antonopoulos (canton@essex.ac.uk), Dr Georgi Grahovski (grah@essex.ac.uk)
Yes
No
No
Prof Stephen Langdon
Brunel University London
Professor
Available via Moodle
Of 2050 hours, 2 (0.1%) hours available to students:
2048 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).
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