MA323-6-SP-CO:
Partial Differential Equations
2020/21
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Spring
Undergraduate: Level 6
Current
Sunday 17 January 2021
Friday 26 March 2021
15
16 July 2020
Requisites for this module
MA201 and MA202 and MA210
(none)
(none)
(none)
(none)
BSC G1F3 Mathematics with Physics,
BSC G1F4 Mathematics with Physics (Including Placement Year),
BSC GCF3 Mathematics with Physics (Including Year Abroad)
The main majority of physical processes and phenomena can be described by partial differential equations, i.e. equations involving partial derivatives (e.g. the Navier-Stokes equations of Fluid Dynamics, Maxwell's equations of Electromagnetism, Schrodinger equation in Quantum Mechanics, Einstein equations in General relativity). This module considers the properties of the most common first and second order PDEs, the mathematical concepts behind them and analytical methods of solution for such equations.
The main difference from the case of ordinary differential equations is that here there is no analogue of existence and uniqueness theorem for a generic PDE. Instead of this, there is a variety of initial and boundary value problems one can impose for finding solutions of wide classes of equations. In addition, some important classes of nonlinear PDEs are also considered.
In this module we shall concentrate on second order linear PDEs (known as Equations of Mathematical Physics): elliptic equations (Laplace's equation), parabolic equations (heat equations) and hyperbolic equations (wave equations). We will study also various topics from real and complex analysis used for solving such equations: the Sturm-Liouville problem, maximum principle for harmonic functions, Fourier series, generalised functions (distributions).
The module aims to provide a general understanding of the theory of linear PDEs and methods of solution the most important types of such equations arising in applications to Physics and Geometry.
On completion of the module students should be able to:
1. Use the method of characteristics to solve first-order partial differential equations;
2. Classify a second order PDE as elliptic, parabolic or hyperbolic;
3. Use separation of variables for suitable boundary value problems for second order linear PDE;
4. Have a basic knowledge and understanding of the theory of Fourier transforms and distributions and applying them to obtain fundamental solutions of Sturm-Liouville problems;
5. Use Green's functions to solve elliptic equations.
Syllabus
1. Linear Differential Operators
2. Method of Characteristics
3. The one-dimensional wave equation
4. The Sturm-Liouville problem
5. Fourier transforms and distributions (generalised functions)
6. Parabolic equations
7. Hyperbolic equations
8. Elliptic equations
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.
- Olver, Peter John. (2014) Introduction to partial differntial equations, Cham: Springer.
- Walter A. Strauss. (2008) Partial Differential Equations: An Introduction, New York, NY: Wiley.
- Pinchover, Yehuda; Rubinstein, Jacob. (2005) Introduction to partial differential equations, New York: Cambridge University Press.
- Hillen, Thomas; Leonard, I. Ed; Van Roessel, Henry. (2012) Partial differential equations: theory and completely solved problems, Hoboken, N.J.: Wiley.
- Levandosky, Julie L.; Strauss, Walter A.; Levandosky, Steven. (2008) Solutions manual for partial differential equations: an introduction, Danvers, MA: John Wiley & Sons.
- Levandosky, Julie L.; Strauss, Walter A.; Levandosky, Steven. (2008) Solutions manual for partial differential equations: an introduction, Danvers, MA: John Wiley & Sons.
- Strauss, Walter A. (2008) Partial differential equations: an introduction, New York: Wiley.
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 Georgi Grahovski, email: gggrah@essex.ac.uk.
Dr Georgi Grahovski & Dr Georgios Papamikos
Dr Georgi Grahovski (gggrah@essex.ac.uk), Dr Georgios Papamikos (g.papamikos@essex.ac.uk)
Yes
No
No
Dr Tania Clare Dunning
The University of Kent
Reader in Applied Mathematics
Prof Stephen Langdon
Brunel University London
Professor
Available via Moodle
Of 1055 hours, 0 (0%) hours available to students:
1055 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).
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